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sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[1, 9, 5, 8, 2, 6, 7, 4, 3], [2, 7, 3, 4, 9, 5, 1, 8, 6], [4, 8, 6, 7, 1, 3, 5, 9, 2], [7, 3, 0, 0, 0, 0, 0, 0, 9], [6, 0, 0, 0, 0, 7, 8, 5, 4], [0, 0, 2, 1, 5, 9, 0, 6, 0], [0, 2, 0, 5, 0, 0, 0, 3, 0], [0, 6, 8, 0, 0, 0, 4, 2, 0], [0, 0, 0, 3, 8, 0, 0, 7, 1]]
Explored next state that leads to an unsolvable path:
[[1, 9, 5, 8, 2, 6, 7, 4, 3], [2, 7, 3, 4, 9, 5, 1, 8, 6], [4, 8, 6, 7, 1, 3, 5, 9, 2], [7, 3, 1, 0, 0, 0, 0, 0, 9], [6, 0, 0, 0, 0, 7, 8, 5, 4], [0, 0, 2, 1, 5, 9, 0, 6, 0], [0, 2, 0, 5, 0, 0, 0, 3, 0], [0, 6, 8, 0, 0, 0, 4, 2, 0], [0, 0, 0, 3, 8, 0, 0, 7, 1]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
game24_lookahead | You are given four numbers and the current calculation state for the Game of 24.
Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if the remaining numbers can be combined using basic arithmetic operations (+ - * /) to reach exactly 24. You must use each number exactly once.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Numbers:**
['2', '6', '9', '11']
**Current calculation state:**
['(6 / 2)', '9', '11']
**Explored next state that leads to an unsolvable path:**
['((6 / 2) + 9)', '11']
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (A) | null |
game24_lookahead | You are given four numbers and the current calculation state for the Game of 24.
Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if the remaining numbers can be combined using basic arithmetic operations (+ - * /) to reach exactly 24. You must use each number exactly once.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Numbers:**
['4', '7', '12', '12']
**Current calculation state:**
['(7 - 4)', '12', '12']
**Explored next state that leads to an unsolvable path:**
['((7 - 4) + 12)', '12']
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (A) | null |
sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[2, 4, 9, 7, 6, 3, 8, 5, 1], [3, 8, 1, 9, 2, 5, 7, 4, 6], [5, 6, 7, 4, 8, 1, 2, 9, 3], [4, 9, 6, 1, 3, 0, 0, 2, 0], [0, 2, 3, 0, 5, 0, 1, 0, 0], [7, 0, 0, 0, 0, 9, 0, 0, 8], [0, 0, 0, 3, 0, 7, 0, 0, 0], [6, 0, 8, 0, 0, 0, 9, 0, 0], [0, 3, 0, 8, 4, 0, 5, 1, 7]]
Explored next state that leads to an unsolvable path:
[[2, 4, 9, 7, 6, 3, 8, 5, 1], [3, 8, 1, 9, 2, 5, 7, 4, 6], [5, 6, 7, 4, 8, 1, 2, 9, 3], [4, 9, 6, 1, 3, 8, 0, 2, 0], [0, 2, 3, 0, 5, 0, 1, 0, 0], [7, 0, 0, 0, 0, 9, 0, 0, 8], [0, 0, 0, 3, 0, 7, 0, 0, 0], [6, 0, 8, 0, 0, 0, 9, 0, 0], [0, 3, 0, 8, 4, 0, 5, 1, 7]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[2, 5, 9, 4, 1, 3, 8, 6, 7], [4, 6, 8, 2, 7, 9, 1, 5, 3], [3, 7, 1, 5, 8, 6, 4, 2, 9], [1, 9, 7, 6, 4, 2, 3, 8, 5], [8, 0, 0, 0, 0, 0, 0, 9, 0], [0, 3, 0, 9, 0, 0, 0, 0, 4], [0, 0, 3, 0, 6, 7, 9, 0, 2], [0, 0, 0, 8, 0, 5, 0, 3, 1], [0, 1, 6, 0, 0, 0, 5, 0, 0]]
Explored next state that leads to an unsolvable path:
[[2, 5, 9, 4, 1, 3, 8, 6, 7], [4, 6, 8, 2, 7, 9, 1, 5, 3], [3, 7, 1, 5, 8, 6, 4, 2, 9], [1, 9, 7, 6, 4, 2, 3, 8, 5], [8, 2, 0, 0, 0, 0, 0, 9, 0], [0, 3, 0, 9, 0, 0, 0, 0, 4], [0, 0, 3, 0, 6, 7, 9, 0, 2], [0, 0, 0, 8, 0, 5, 0, 3, 1], [0, 1, 6, 0, 0, 0, 5, 0, 0]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (A) | null |
game24_statetransition | You are given an initial Game of 24 configuration S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid solution requires using each of the four initial numbers exactly once, using only basic arithmetic operations (+ - * /), and ultimately evaluating to 24.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Applying an operation: Combining two expressions using a basic arithmetic operation (+ - * /), reducing the number of expressions by 1.
2. Reverting an operation: Removing the last operation applied to the expressions, increasing the number of expressions by 1.
**Initial configuration:**
S(0) = ['1', '1', '5', '8']
L(0) = Solvable
**Current state:**
S(i) = ['(1 + 1)', '5', '8']
L(i) = Solvable
**Explored next state:**
S(i+1) = ['((1 + 1) + 5)', '8']
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{expressions\}", where \{expressions\} is in the same python list format as the previous states. | null | {'parent_state': ['1', '1', '5', '8'], 'current_state': ['(1 + 1)', '5', '8'], 'current_status': 'Solvable', 'unsolvable_child': ['((1 + 1) + 5)', '8'], 'numbers': [1, 1, 5, 8]} |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[7, 0, 9, 0, 0, 0, 0, 0, 4], [0, 6, 0, 0, 7, 0, 0, 0, 3], [0, 0, 0, 9, 0, 5, 8, 0, 1], [1, 0, 0, 0, 4, 8, 0, 0, 0], [9, 0, 0, 0, 2, 0, 5, 0, 6], [2, 0, 5, 3, 0, 0, 0, 4, 0], [0, 0, 8, 7, 0, 0, 0, 6, 0], [0, 0, 4, 2, 8, 3, 0, 0, 5], [0, 2, 7, 6, 0, 0, 3, 9, 0]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[7, 5, 9, 8, 3, 1, 6, 2, 4], [8, 6, 1, 4, 7, 2, 9, 5, 3], [4, 3, 2, 9, 6, 5, 8, 7, 1], [1, 7, 6, 5, 4, 8, 2, 3, 9], [9, 4, 3, 1, 2, 7, 5, 8, 6], [2, 8, 5, 3, 9, 6, 1, 4, 7], [0, 0, 8, 7, 0, 0, 0, 6, 0], [0, 0, 4, 2, 8, 3, 0, 0, 5], [0, 2, 7, 6, 0, 0, 3, 9, 0]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[7, 5, 9, 8, 3, 1, 6, 2, 4], [8, 6, 1, 4, 7, 2, 9, 5, 3], [4, 3, 2, 9, 6, 5, 8, 7, 1], [1, 7, 6, 5, 4, 8, 2, 3, 9], [9, 4, 3, 1, 2, 7, 5, 8, 6], [2, 8, 5, 3, 9, 6, 1, 4, 7], [3, 0, 8, 7, 0, 0, 0, 6, 0], [0, 0, 4, 2, 8, 3, 0, 0, 5], [0, 2, 7, 6, 0, 0, 3, 9, 0]]
L(i-1) = Solvable
**Current state:**
S(i) = [[7, 5, 9, 8, 3, 1, 6, 2, 4], [8, 6, 1, 4, 7, 2, 9, 5, 3], [4, 3, 2, 9, 6, 5, 8, 7, 1], [1, 7, 6, 5, 4, 8, 2, 3, 9], [9, 4, 3, 1, 2, 7, 5, 8, 6], [2, 8, 5, 3, 9, 6, 1, 4, 7], [3, 9, 8, 7, 0, 0, 0, 6, 0], [0, 0, 4, 2, 8, 3, 0, 0, 5], [0, 2, 7, 6, 0, 0, 3, 9, 0]]
L(i) = Unsolvable
**Explored next state:**
S(i+1) = [[7, 5, 9, 8, 3, 1, 6, 2, 4], [8, 6, 1, 4, 7, 2, 9, 5, 3], [4, 3, 2, 9, 6, 5, 8, 7, 1], [1, 7, 6, 5, 4, 8, 2, 3, 9], [9, 4, 3, 1, 2, 7, 5, 8, 6], [2, 8, 5, 3, 9, 6, 1, 4, 7], [3, 9, 8, 7, 1, 0, 0, 6, 0], [0, 0, 4, 2, 8, 3, 0, 0, 5], [0, 2, 7, 6, 0, 0, 3, 9, 0]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[7, 5, 9, 8, 3, 1, 6, 2, 4], [8, 6, 1, 4, 7, 2, 9, 5, 3], [4, 3, 2, 9, 6, 5, 8, 7, 1], [1, 7, 6, 5, 4, 8, 2, 3, 9], [9, 4, 3, 1, 2, 7, 5, 8, 6], [2, 8, 5, 3, 9, 6, 1, 4, 7], [3, 0, 8, 7, 0, 0, 0, 6, 0], [0, 0, 4, 2, 8, 3, 0, 0, 5], [0, 2, 7, 6, 0, 0, 3, 9, 0]], 'current_state': [[7, 5, 9, 8, 3, 1, 6, 2, 4], [8, 6, 1, 4, 7, 2, 9, 5, 3], [4, 3, 2, 9, 6, 5, 8, 7, 1], [1, 7, 6, 5, 4, 8, 2, 3, 9], [9, 4, 3, 1, 2, 7, 5, 8, 6], [2, 8, 5, 3, 9, 6, 1, 4, 7], [3, 9, 8, 7, 0, 0, 0, 6, 0], [0, 0, 4, 2, 8, 3, 0, 0, 5], [0, 2, 7, 6, 0, 0, 3, 9, 0]], 'current_status': 'Unsolvable', 'unsolvable_child': [[7, 5, 9, 8, 3, 1, 6, 2, 4], [8, 6, 1, 4, 7, 2, 9, 5, 3], [4, 3, 2, 9, 6, 5, 8, 7, 1], [1, 7, 6, 5, 4, 8, 2, 3, 9], [9, 4, 3, 1, 2, 7, 5, 8, 6], [2, 8, 5, 3, 9, 6, 1, 4, 7], [3, 9, 8, 7, 1, 0, 0, 6, 0], [0, 0, 4, 2, 8, 3, 0, 0, 5], [0, 2, 7, 6, 0, 0, 3, 9, 0]]} |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[0, 7, 0, 0, 0, 0, 2, 0, 6], [8, 0, 3, 0, 0, 5, 7, 0, 4], [0, 0, 9, 2, 4, 0, 5, 0, 0], [4, 0, 0, 0, 5, 8, 0, 2, 0], [0, 0, 1, 4, 0, 6, 0, 0, 9], [3, 0, 0, 0, 1, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 9, 5, 1], [0, 6, 0, 0, 9, 3, 0, 0, 8], [0, 0, 0, 7, 8, 1, 0, 0, 0]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[5, 7, 4, 8, 3, 9, 2, 1, 6], [8, 2, 3, 1, 6, 5, 7, 9, 4], [6, 1, 9, 2, 4, 7, 5, 8, 3], [4, 9, 6, 3, 5, 8, 1, 2, 7], [0, 0, 1, 4, 0, 6, 0, 0, 9], [3, 0, 0, 0, 1, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 9, 5, 1], [0, 6, 0, 0, 9, 3, 0, 0, 8], [0, 0, 0, 7, 8, 1, 0, 0, 0]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[5, 7, 4, 8, 3, 9, 2, 1, 6], [8, 2, 3, 1, 6, 5, 7, 9, 4], [6, 1, 9, 2, 4, 7, 5, 8, 3], [4, 9, 6, 3, 5, 8, 1, 2, 7], [2, 0, 1, 4, 0, 6, 0, 0, 9], [3, 0, 0, 0, 1, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 9, 5, 1], [0, 6, 0, 0, 9, 3, 0, 0, 8], [0, 0, 0, 7, 8, 1, 0, 0, 0]]
L(i-1) = Solvable
**Current state:**
S(i) = [[5, 7, 4, 8, 3, 9, 2, 1, 6], [8, 2, 3, 1, 6, 5, 7, 9, 4], [6, 1, 9, 2, 4, 7, 5, 8, 3], [4, 9, 6, 3, 5, 8, 1, 2, 7], [2, 8, 1, 4, 0, 6, 0, 0, 9], [3, 0, 0, 0, 1, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 9, 5, 1], [0, 6, 0, 0, 9, 3, 0, 0, 8], [0, 0, 0, 7, 8, 1, 0, 0, 0]]
L(i) = Unsolvable
**Explored next state:**
S(i+1) = [[5, 7, 4, 8, 3, 9, 2, 1, 6], [8, 2, 3, 1, 6, 5, 7, 9, 4], [6, 1, 9, 2, 4, 7, 5, 8, 3], [4, 9, 6, 3, 5, 8, 1, 2, 7], [2, 8, 1, 4, 7, 6, 0, 0, 9], [3, 0, 0, 0, 1, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 9, 5, 1], [0, 6, 0, 0, 9, 3, 0, 0, 8], [0, 0, 0, 7, 8, 1, 0, 0, 0]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[5, 7, 4, 8, 3, 9, 2, 1, 6], [8, 2, 3, 1, 6, 5, 7, 9, 4], [6, 1, 9, 2, 4, 7, 5, 8, 3], [4, 9, 6, 3, 5, 8, 1, 2, 7], [2, 0, 1, 4, 0, 6, 0, 0, 9], [3, 0, 0, 0, 1, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 9, 5, 1], [0, 6, 0, 0, 9, 3, 0, 0, 8], [0, 0, 0, 7, 8, 1, 0, 0, 0]], 'current_state': [[5, 7, 4, 8, 3, 9, 2, 1, 6], [8, 2, 3, 1, 6, 5, 7, 9, 4], [6, 1, 9, 2, 4, 7, 5, 8, 3], [4, 9, 6, 3, 5, 8, 1, 2, 7], [2, 8, 1, 4, 0, 6, 0, 0, 9], [3, 0, 0, 0, 1, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 9, 5, 1], [0, 6, 0, 0, 9, 3, 0, 0, 8], [0, 0, 0, 7, 8, 1, 0, 0, 0]], 'current_status': 'Unsolvable', 'unsolvable_child': [[5, 7, 4, 8, 3, 9, 2, 1, 6], [8, 2, 3, 1, 6, 5, 7, 9, 4], [6, 1, 9, 2, 4, 7, 5, 8, 3], [4, 9, 6, 3, 5, 8, 1, 2, 7], [2, 8, 1, 4, 7, 6, 0, 0, 9], [3, 0, 0, 0, 1, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 9, 5, 1], [0, 6, 0, 0, 9, 3, 0, 0, 8], [0, 0, 0, 7, 8, 1, 0, 0, 0]]} |
graphcoloring_lookahead | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given the current coloring state of the graph in a list, where each index represents the corresonding vertex, and the number at that index represents its color (0 indicates an uncolored vertex).
Your task is to determine if this current state can lead to a valid coloring. Specifically, use lookahead techniques to determine if it's possible to color the remaining vertices such that no two adjacent vertices share the same color, using no more than 7 colors in total.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Graph adjacency list:**
[[1, 3, 4, 5, 6, 9, 11, 12, 13, 15, 16], [0, 2, 3, 4, 5, 6, 7, 10, 12, 13, 14, 16], [1, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 4, 5, 6, 7, 8, 10, 12, 13, 14, 16], [0, 1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 2, 3, 6, 9, 11, 13, 15, 16], [0, 1, 2, 3, 4, 5, 7, 10, 12, 13, 14, 15, 16], [1, 2, 3, 4, 6, 8, 10, 11, 14, 15], [3, 4, 7, 9, 11, 12, 13, 15, 16], [0, 2, 4, 5, 8, 10, 12, 13, 14, 15, 16], [1, 2, 3, 4, 6, 7, 9, 12, 15, 16], [0, 2, 4, 5, 7, 8, 13, 15, 16], [0, 1, 2, 3, 4, 6, 8, 9, 10, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 16], [1, 2, 3, 4, 6, 7, 9, 12, 13, 15, 16], [0, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16], [0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15]]
**Current coloring state:**
[1, 2, 1, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
**Explored next state that leads to an uncolorable path:**
[1, 2, 1, 3, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Colorable (B) Uncolorable. | (B) | null |
game24_lookahead | You are given four numbers and the current calculation state for the Game of 24.
Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if the remaining numbers can be combined using basic arithmetic operations (+ - * /) to reach exactly 24. You must use each number exactly once.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Numbers:**
['1', '3', '3', '7']
**Current calculation state:**
['(7 - 1)', '3', '3']
**Explored next state that leads to an unsolvable path:**
['((7 - 1) + 3)', '3']
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[9, 3, 6, 8, 4, 5, 2, 7, 1], [2, 7, 8, 1, 9, 6, 3, 5, 4], [1, 5, 4, 7, 3, 2, 6, 9, 8], [8, 1, 9, 5, 7, 3, 4, 2, 6], [4, 2, 3, 6, 1, 9, 5, 8, 7], [5, 6, 7, 2, 8, 4, 1, 3, 9], [3, 8, 1, 9, 5, 0, 0, 6, 0], [0, 0, 0, 3, 0, 0, 0, 4, 0], [6, 4, 5, 0, 0, 8, 7, 0, 0]]
Explored next state that leads to an unsolvable path:
[[9, 3, 6, 8, 4, 5, 2, 7, 1], [2, 7, 8, 1, 9, 6, 3, 5, 4], [1, 5, 4, 7, 3, 2, 6, 9, 8], [8, 1, 9, 5, 7, 3, 4, 2, 6], [4, 2, 3, 6, 1, 9, 5, 8, 7], [5, 6, 7, 2, 8, 4, 1, 3, 9], [3, 8, 1, 9, 5, 7, 0, 6, 0], [0, 0, 0, 3, 0, 0, 0, 4, 0], [6, 4, 5, 0, 0, 8, 7, 0, 0]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[9, 0, 0, 7, 0, 0, 8, 0, 2], [0, 8, 0, 0, 3, 0, 0, 0, 6], [2, 7, 0, 0, 0, 0, 0, 1, 0], [0, 0, 4, 9, 0, 3, 5, 0, 0], [0, 0, 0, 2, 0, 1, 6, 8, 4], [5, 0, 1, 0, 8, 0, 0, 7, 0], [0, 0, 2, 4, 6, 0, 0, 0, 8], [1, 0, 0, 0, 0, 5, 9, 0, 0], [0, 3, 9, 8, 0, 0, 0, 4, 0]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[9, 1, 0, 7, 0, 0, 8, 0, 2], [0, 8, 0, 0, 3, 0, 0, 0, 6], [2, 7, 0, 0, 0, 0, 0, 1, 0], [0, 0, 4, 9, 0, 3, 5, 0, 0], [0, 0, 0, 2, 0, 1, 6, 8, 4], [5, 0, 1, 0, 8, 0, 0, 7, 0], [0, 0, 2, 4, 6, 0, 0, 0, 8], [1, 0, 0, 0, 0, 5, 9, 0, 0], [0, 3, 9, 8, 0, 0, 0, 4, 0]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[9, 1, 3, 7, 0, 0, 8, 0, 2], [0, 8, 0, 0, 3, 0, 0, 0, 6], [2, 7, 0, 0, 0, 0, 0, 1, 0], [0, 0, 4, 9, 0, 3, 5, 0, 0], [0, 0, 0, 2, 0, 1, 6, 8, 4], [5, 0, 1, 0, 8, 0, 0, 7, 0], [0, 0, 2, 4, 6, 0, 0, 0, 8], [1, 0, 0, 0, 0, 5, 9, 0, 0], [0, 3, 9, 8, 0, 0, 0, 4, 0]]
L(i-1) = Solvable
**Current state:**
S(i) = [[9, 1, 3, 7, 5, 0, 8, 0, 2], [0, 8, 0, 0, 3, 0, 0, 0, 6], [2, 7, 0, 0, 0, 0, 0, 1, 0], [0, 0, 4, 9, 0, 3, 5, 0, 0], [0, 0, 0, 2, 0, 1, 6, 8, 4], [5, 0, 1, 0, 8, 0, 0, 7, 0], [0, 0, 2, 4, 6, 0, 0, 0, 8], [1, 0, 0, 0, 0, 5, 9, 0, 0], [0, 3, 9, 8, 0, 0, 0, 4, 0]]
L(i) = Unsolvable
**Explored next state:**
S(i+1) = [[9, 1, 3, 7, 5, 4, 8, 0, 2], [0, 8, 0, 0, 3, 0, 0, 0, 6], [2, 7, 0, 0, 0, 0, 0, 1, 0], [0, 0, 4, 9, 0, 3, 5, 0, 0], [0, 0, 0, 2, 0, 1, 6, 8, 4], [5, 0, 1, 0, 8, 0, 0, 7, 0], [0, 0, 2, 4, 6, 0, 0, 0, 8], [1, 0, 0, 0, 0, 5, 9, 0, 0], [0, 3, 9, 8, 0, 0, 0, 4, 0]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[9, 1, 3, 7, 0, 0, 8, 0, 2], [0, 8, 0, 0, 3, 0, 0, 0, 6], [2, 7, 0, 0, 0, 0, 0, 1, 0], [0, 0, 4, 9, 0, 3, 5, 0, 0], [0, 0, 0, 2, 0, 1, 6, 8, 4], [5, 0, 1, 0, 8, 0, 0, 7, 0], [0, 0, 2, 4, 6, 0, 0, 0, 8], [1, 0, 0, 0, 0, 5, 9, 0, 0], [0, 3, 9, 8, 0, 0, 0, 4, 0]], 'current_state': [[9, 1, 3, 7, 5, 0, 8, 0, 2], [0, 8, 0, 0, 3, 0, 0, 0, 6], [2, 7, 0, 0, 0, 0, 0, 1, 0], [0, 0, 4, 9, 0, 3, 5, 0, 0], [0, 0, 0, 2, 0, 1, 6, 8, 4], [5, 0, 1, 0, 8, 0, 0, 7, 0], [0, 0, 2, 4, 6, 0, 0, 0, 8], [1, 0, 0, 0, 0, 5, 9, 0, 0], [0, 3, 9, 8, 0, 0, 0, 4, 0]], 'current_status': 'Unsolvable', 'unsolvable_child': [[9, 1, 3, 7, 5, 4, 8, 0, 2], [0, 8, 0, 0, 3, 0, 0, 0, 6], [2, 7, 0, 0, 0, 0, 0, 1, 0], [0, 0, 4, 9, 0, 3, 5, 0, 0], [0, 0, 0, 2, 0, 1, 6, 8, 4], [5, 0, 1, 0, 8, 0, 0, 7, 0], [0, 0, 2, 4, 6, 0, 0, 0, 8], [1, 0, 0, 0, 0, 5, 9, 0, 0], [0, 3, 9, 8, 0, 0, 0, 4, 0]]} |
sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[4, 7, 3, 9, 8, 5, 2, 6, 1], [1, 9, 5, 7, 2, 6, 4, 8, 3], [6, 2, 8, 4, 1, 3, 5, 7, 9], [2, 3, 0, 0, 0, 1, 7, 0, 0], [0, 0, 9, 5, 0, 7, 0, 0, 4], [0, 0, 0, 0, 9, 0, 0, 5, 6], [0, 0, 1, 8, 7, 0, 0, 3, 0], [0, 0, 4, 0, 0, 0, 0, 1, 0], [0, 6, 2, 0, 3, 9, 0, 0, 5]]
Explored next state that leads to an unsolvable path:
[[4, 7, 3, 9, 8, 5, 2, 6, 1], [1, 9, 5, 7, 2, 6, 4, 8, 3], [6, 2, 8, 4, 1, 3, 5, 7, 9], [2, 3, 6, 0, 0, 1, 7, 0, 0], [0, 0, 9, 5, 0, 7, 0, 0, 4], [0, 0, 0, 0, 9, 0, 0, 5, 6], [0, 0, 1, 8, 7, 0, 0, 3, 0], [0, 0, 4, 0, 0, 0, 0, 1, 0], [0, 6, 2, 0, 3, 9, 0, 0, 5]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
game24_lookahead | You are given four numbers and the current calculation state for the Game of 24.
Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if the remaining numbers can be combined using basic arithmetic operations (+ - * /) to reach exactly 24. You must use each number exactly once.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Numbers:**
['1', '3', '5', '8']
**Current calculation state:**
['(5 - 1)', '3', '8']
**Explored next state that leads to an unsolvable path:**
['((5 - 1) + 3)', '8']
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[8, 5, 3, 4, 7, 1, 2, 6, 9], [2, 4, 6, 9, 8, 5, 3, 7, 1], [7, 1, 9, 3, 2, 6, 4, 5, 8], [4, 8, 7, 6, 5, 9, 1, 2, 3], [0, 0, 1, 0, 3, 8, 6, 0, 0], [6, 0, 0, 1, 0, 2, 0, 8, 7], [0, 0, 0, 0, 0, 0, 0, 0, 4], [3, 7, 2, 0, 9, 0, 8, 0, 0], [0, 9, 4, 0, 6, 0, 5, 0, 0]]
Explored next state that leads to an unsolvable path:
[[8, 5, 3, 4, 7, 1, 2, 6, 9], [2, 4, 6, 9, 8, 5, 3, 7, 1], [7, 1, 9, 3, 2, 6, 4, 5, 8], [4, 8, 7, 6, 5, 9, 1, 2, 3], [5, 0, 1, 0, 3, 8, 6, 0, 0], [6, 0, 0, 1, 0, 2, 0, 8, 7], [0, 0, 0, 0, 0, 0, 0, 0, 4], [3, 7, 2, 0, 9, 0, 8, 0, 0], [0, 9, 4, 0, 6, 0, 5, 0, 0]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (A) | null |
game24_lookahead | You are given four numbers and the current calculation state for the Game of 24.
Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if the remaining numbers can be combined using basic arithmetic operations (+ - * /) to reach exactly 24. You must use each number exactly once.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Numbers:**
['10', '10', '12', '12']
**Current calculation state:**
['(10 * 12)', '10', '12']
**Explored next state that leads to an unsolvable path:**
['((10 * 12) + 10)', '12']
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (A) | null |
game24_lookahead | You are given four numbers and the current calculation state for the Game of 24.
Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if the remaining numbers can be combined using basic arithmetic operations (+ - * /) to reach exactly 24. You must use each number exactly once.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Numbers:**
['4', '4', '4', '7']
**Current calculation state:**
['(4 * 4)', '4', '7']
**Explored next state that leads to an unsolvable path:**
['((4 * 4) + 4)', '7']
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
graphcoloring_lookahead | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given the current coloring state of the graph in a list, where each index represents the corresonding vertex, and the number at that index represents its color (0 indicates an uncolored vertex).
Your task is to determine if this current state can lead to a valid coloring. Specifically, use lookahead techniques to determine if it's possible to color the remaining vertices such that no two adjacent vertices share the same color, using no more than 7 colors in total.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Graph adjacency list:**
[[1, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 22, 24], [0, 3, 4, 5, 7, 12, 14, 15, 16, 17, 19, 21, 23], [4, 5, 6, 11, 12, 13, 14, 15, 16, 19, 22], [0, 1, 5, 7, 9, 10, 11, 13, 15, 16, 20, 23], [1, 2, 5, 9, 10, 11, 14, 16, 21, 23, 24], [1, 2, 3, 4, 7, 8, 10, 14, 17], [0, 2, 7, 11, 12, 13, 14, 15, 19, 20, 21, 22], [0, 1, 3, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23], [0, 5, 7, 9, 10, 12, 13, 14, 15, 16, 17, 19, 23], [0, 3, 4, 7, 8, 11, 14, 16, 18, 19, 20, 22], [0, 3, 4, 5, 7, 8, 11, 13, 15, 18, 19, 20, 21, 22, 23], [0, 2, 3, 4, 6, 7, 9, 10, 14, 15, 16, 18, 19, 20, 21, 22, 23], [0, 1, 2, 6, 7, 8, 15, 17, 18, 19, 20, 22, 23], [0, 2, 3, 6, 8, 10, 14, 17, 18, 19, 20, 22, 23], [0, 1, 2, 4, 5, 6, 7, 8, 9, 11, 13, 15, 21, 22], [1, 2, 3, 6, 7, 8, 10, 11, 12, 14, 17, 18, 21, 22, 23, 24], [0, 1, 2, 3, 4, 7, 8, 9, 11, 17, 18, 19, 22, 23, 24], [0, 1, 5, 7, 8, 12, 13, 15, 16, 18, 19, 22, 23, 24], [0, 7, 9, 10, 11, 12, 13, 15, 16, 17, 20, 21, 22, 23, 24], [1, 2, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 20, 21, 22, 23, 24], [3, 6, 9, 10, 11, 12, 13, 18, 19, 21], [1, 4, 6, 7, 10, 11, 14, 15, 18, 19, 20, 22, 24], [0, 2, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23], [1, 3, 4, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 22, 24], [0, 4, 15, 16, 17, 18, 19, 21, 23]]
**Current coloring state:**
[1, 2, 1, 3, 4, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
**Explored next state that leads to an uncolorable path:**
[1, 2, 1, 3, 4, 5, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Colorable (B) Uncolorable. | (A) | null |
game24_statetransition | You are given an initial Game of 24 configuration S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid solution requires using each of the four initial numbers exactly once, using only basic arithmetic operations (+ - * /), and ultimately evaluating to 24.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Applying an operation: Combining two expressions using a basic arithmetic operation (+ - * /), reducing the number of expressions by 1.
2. Reverting an operation: Removing the last operation applied to the expressions, increasing the number of expressions by 1.
**Initial configuration:**
S(0) = ['1', '5', '6', '12']
L(0) = Solvable
**Current state:**
S(i) = ['(1 * 6)', '5', '12']
L(i) = Unsolvable
**Explored next state:**
S(i+1) = ['((1 * 6) + 5)', '12']
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{expressions\}", where \{expressions\} is in the same python list format as the previous states. | null | {'parent_state': ['1', '5', '6', '12'], 'current_state': ['(1 * 6)', '5', '12'], 'current_status': 'Unsolvable', 'unsolvable_child': ['((1 * 6) + 5)', '12'], 'numbers': [1, 5, 6, 12]} |
game24_statetransition | You are given an initial Game of 24 configuration S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid solution requires using each of the four initial numbers exactly once, using only basic arithmetic operations (+ - * /), and ultimately evaluating to 24.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Applying an operation: Combining two expressions using a basic arithmetic operation (+ - * /), reducing the number of expressions by 1.
2. Reverting an operation: Removing the last operation applied to the expressions, increasing the number of expressions by 1.
**Initial configuration:**
S(0) = ['1', '1', '4', '9']
L(0) = Solvable
**Current state:**
S(i) = ['(1 + 4)', '1', '9']
L(i) = Unsolvable
**Explored next state:**
S(i+1) = ['((1 + 4) + 1)', '9']
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{expressions\}", where \{expressions\} is in the same python list format as the previous states. | null | {'parent_state': ['1', '1', '4', '9'], 'current_state': ['(1 + 4)', '1', '9'], 'current_status': 'Unsolvable', 'unsolvable_child': ['((1 + 4) + 1)', '9'], 'numbers': [1, 1, 4, 9]} |
graphcoloring_statetransition | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given a sequence of partial coloring states leading to the current coloring state S(i). The coloring state is a list, where each index represents the corresonding vertex in the graph, and the number at that index represents its color (0 indicates an uncolored vertex). Alongside each state, its colorability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid coloring with no more than 8 colors. A valid coloring requires that no two adjacent vertices share the same color.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Coloring: Replacing a 0 in exactly one uncolored vertex with a value from 1 to 8.
2. Removing a color: Replacing a value in exactly one colored vertex with 0.
**Graph adjacency list:**
[[1, 5, 7, 8, 9, 11, 13, 14, 15, 19, 20, 21, 22], [0, 2, 4, 6, 9, 10, 12, 16, 17, 20, 21, 22], [1, 5, 8, 9, 10, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22], [4, 7, 8, 9, 10, 12, 14, 15, 16, 17, 23], [1, 3, 5, 6, 9, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 23], [0, 2, 4, 7, 9, 11, 17, 20, 22, 23], [1, 4, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 23], [0, 3, 5, 8, 9, 10, 14, 16, 18, 19, 20, 21, 22, 23], [0, 2, 3, 6, 7, 9, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 17, 20, 23], [1, 2, 3, 6, 7, 9, 12, 14, 15, 16, 18, 23], [0, 2, 4, 5, 6, 8, 12, 15, 16, 17, 19, 22], [1, 3, 4, 6, 8, 9, 10, 11, 13, 14, 18, 19, 22, 23], [0, 2, 4, 6, 8, 12, 14, 15, 20, 21, 22], [0, 3, 4, 6, 7, 8, 9, 10, 12, 13, 15, 17, 18, 19, 23], [0, 2, 3, 6, 8, 10, 11, 13, 14, 16, 22, 23], [1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 15, 17, 19], [1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 21, 22], [2, 4, 7, 10, 12, 14, 21, 23], [0, 2, 6, 7, 8, 11, 12, 14, 16, 21, 23], [0, 1, 2, 4, 5, 7, 8, 9, 13, 23], [0, 1, 2, 4, 6, 7, 8, 13, 17, 18, 19, 23], [0, 1, 2, 4, 5, 6, 7, 11, 12, 13, 15, 17, 23], [3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 19, 20, 21, 22]]
**Two moves ago:**
S(i-2) = [1, 2, 1, 1, 3, 2, 1, 3, 2, 4, 5, 4, 6, 4, 7, 3, 6, 5, 2, 0, 0, 0, 0, 0]
L(i-2) = Colorable
**One move ago:**
S(i-1) = [1, 2, 1, 1, 3, 2, 1, 3, 2, 4, 5, 4, 6, 4, 7, 3, 6, 5, 2, 5, 0, 0, 0, 0]
L(i-1) = Colorable
**Current coloring state:**
S(i) = [1, 2, 1, 1, 3, 2, 1, 3, 2, 4, 5, 4, 6, 4, 7, 3, 6, 5, 2, 5, 5, 0, 0, 0]
L(i) = Colorable
**Explored next state:**
S(i+1) = [1, 2, 1, 1, 3, 2, 1, 3, 2, 4, 5, 4, 6, 4, 7, 3, 6, 5, 2, 5, 5, 8, 0, 0]
L(i+1) = Uncolorable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{coloring\}", where \{coloring\} is in the same python list format as the previous states. | null | {'parent_state': [1, 2, 1, 1, 3, 2, 1, 3, 2, 4, 5, 4, 6, 4, 7, 3, 6, 5, 2, 5, 0, 0, 0, 0], 'current_state': [1, 2, 1, 1, 3, 2, 1, 3, 2, 4, 5, 4, 6, 4, 7, 3, 6, 5, 2, 5, 5, 0, 0, 0], 'current_status': 'Solvable', 'unsolvable_child': [1, 2, 1, 1, 3, 2, 1, 3, 2, 4, 5, 4, 6, 4, 7, 3, 6, 5, 2, 5, 5, 8, 0, 0], 'graph': [[1, 5, 7, 8, 9, 11, 13, 14, 15, 19, 20, 21, 22], [0, 2, 4, 6, 9, 10, 12, 16, 17, 20, 21, 22], [1, 5, 8, 9, 10, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22], [4, 7, 8, 9, 10, 12, 14, 15, 16, 17, 23], [1, 3, 5, 6, 9, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 23], [0, 2, 4, 7, 9, 11, 17, 20, 22, 23], [1, 4, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 22, 23], [0, 3, 5, 8, 9, 10, 14, 16, 18, 19, 20, 21, 22, 23], [0, 2, 3, 6, 7, 9, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21], [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 17, 20, 23], [1, 2, 3, 6, 7, 9, 12, 14, 15, 16, 18, 23], [0, 2, 4, 5, 6, 8, 12, 15, 16, 17, 19, 22], [1, 3, 4, 6, 8, 9, 10, 11, 13, 14, 18, 19, 22, 23], [0, 2, 4, 6, 8, 12, 14, 15, 20, 21, 22], [0, 3, 4, 6, 7, 8, 9, 10, 12, 13, 15, 17, 18, 19, 23], [0, 2, 3, 6, 8, 10, 11, 13, 14, 16, 22, 23], [1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 15, 17, 19], [1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 21, 22], [2, 4, 7, 10, 12, 14, 21, 23], [0, 2, 6, 7, 8, 11, 12, 14, 16, 21, 23], [0, 1, 2, 4, 5, 7, 8, 9, 13, 23], [0, 1, 2, 4, 6, 7, 8, 13, 17, 18, 19, 23], [0, 1, 2, 4, 5, 6, 7, 11, 12, 13, 15, 17, 23], [3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 19, 20, 21, 22]], 'chromatic_number': 8} |
graphcoloring_statetransition | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given a sequence of partial coloring states leading to the current coloring state S(i). The coloring state is a list, where each index represents the corresonding vertex in the graph, and the number at that index represents its color (0 indicates an uncolored vertex). Alongside each state, its colorability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid coloring with no more than 6 colors. A valid coloring requires that no two adjacent vertices share the same color.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Coloring: Replacing a 0 in exactly one uncolored vertex with a value from 1 to 6.
2. Removing a color: Replacing a value in exactly one colored vertex with 0.
**Graph adjacency list:**
[[1, 8, 11, 13, 16, 17], [0, 2, 3, 4, 6, 9, 11, 12, 15, 16, 17], [1, 3, 4, 7, 10, 11, 12, 14, 15, 17], [1, 2, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15], [1, 2, 3, 6, 7, 8, 10, 11, 12, 17], [3, 6, 9, 10, 12, 14, 16], [1, 3, 4, 5, 10, 11, 12, 13, 15, 16, 17], [2, 3, 4, 8, 10, 11, 12, 16], [0, 3, 4, 7, 9, 10, 11, 12, 15], [1, 5, 8, 14, 15, 16, 17], [2, 4, 5, 6, 7, 8, 12, 13, 17], [0, 1, 2, 3, 4, 6, 7, 8, 14, 15, 17], [1, 2, 3, 4, 5, 6, 7, 8, 10, 14, 15, 17], [0, 3, 6, 10, 14, 15, 16], [2, 3, 5, 9, 11, 12, 13, 17], [1, 2, 3, 6, 8, 9, 11, 12, 13], [0, 1, 5, 6, 7, 9, 13, 17], [0, 1, 2, 4, 6, 9, 10, 11, 12, 14, 16]]
**Two moves ago:**
S(i-2) = [1, 2, 1, 3, 4, 2, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i-2) = Colorable
**One move ago:**
S(i-1) = [1, 2, 1, 3, 4, 2, 1, 2, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i-1) = Colorable
**Current coloring state:**
S(i) = [1, 2, 1, 3, 4, 2, 1, 2, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0]
L(i) = Colorable
**Explored next state:**
S(i+1) = [1, 2, 1, 3, 4, 2, 1, 2, 5, 1, 6, 0, 0, 0, 0, 0, 0, 0]
L(i+1) = Uncolorable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{coloring\}", where \{coloring\} is in the same python list format as the previous states. | null | {'parent_state': [1, 2, 1, 3, 4, 2, 1, 2, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'current_state': [1, 2, 1, 3, 4, 2, 1, 2, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0], 'current_status': 'Solvable', 'unsolvable_child': [1, 2, 1, 3, 4, 2, 1, 2, 5, 1, 6, 0, 0, 0, 0, 0, 0, 0], 'graph': [[1, 8, 11, 13, 16, 17], [0, 2, 3, 4, 6, 9, 11, 12, 15, 16, 17], [1, 3, 4, 7, 10, 11, 12, 14, 15, 17], [1, 2, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15], [1, 2, 3, 6, 7, 8, 10, 11, 12, 17], [3, 6, 9, 10, 12, 14, 16], [1, 3, 4, 5, 10, 11, 12, 13, 15, 16, 17], [2, 3, 4, 8, 10, 11, 12, 16], [0, 3, 4, 7, 9, 10, 11, 12, 15], [1, 5, 8, 14, 15, 16, 17], [2, 4, 5, 6, 7, 8, 12, 13, 17], [0, 1, 2, 3, 4, 6, 7, 8, 14, 15, 17], [1, 2, 3, 4, 5, 6, 7, 8, 10, 14, 15, 17], [0, 3, 6, 10, 14, 15, 16], [2, 3, 5, 9, 11, 12, 13, 17], [1, 2, 3, 6, 8, 9, 11, 12, 13], [0, 1, 5, 6, 7, 9, 13, 17], [0, 1, 2, 4, 6, 9, 10, 11, 12, 14, 16]], 'chromatic_number': 6} |
game24_lookahead | You are given four numbers and the current calculation state for the Game of 24.
Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if the remaining numbers can be combined using basic arithmetic operations (+ - * /) to reach exactly 24. You must use each number exactly once.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Numbers:**
['2', '2', '2', '8']
**Current calculation state:**
['(2 * 8)', '2', '2']
**Explored next state that leads to an unsolvable path:**
['((2 * 8) + 2)', '2']
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
game24_lookahead | You are given four numbers and the current calculation state for the Game of 24.
Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if the remaining numbers can be combined using basic arithmetic operations (+ - * /) to reach exactly 24. You must use each number exactly once.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Numbers:**
['4', '5', '12', '12']
**Current calculation state:**
['(5 * 12)', '4', '12']
**Explored next state that leads to an unsolvable path:**
['((5 * 12) + 4)', '12']
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
graphcoloring_lookahead | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given the current coloring state of the graph in a list, where each index represents the corresonding vertex, and the number at that index represents its color (0 indicates an uncolored vertex).
Your task is to determine if this current state can lead to a valid coloring. Specifically, use lookahead techniques to determine if it's possible to color the remaining vertices such that no two adjacent vertices share the same color, using no more than 7 colors in total.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Graph adjacency list:**
[[3, 4, 7, 9, 10, 14, 15, 17, 18, 19, 20, 21, 23], [3, 5, 6, 7, 9, 10, 11, 12, 13, 17, 19, 20, 22, 24], [3, 4, 8, 11, 13, 14, 15, 22], [0, 1, 2, 4, 7, 8, 9, 11, 14, 15, 16, 19, 21, 23], [0, 2, 3, 6, 7, 12, 13, 14, 15, 19, 20, 23, 24], [1, 6, 7, 9, 11, 15, 16, 17, 21], [1, 4, 5, 7, 11, 15, 17, 18, 20, 21, 22, 23], [0, 1, 3, 4, 5, 6, 13, 16, 20, 23], [2, 3, 9, 10, 11, 12, 17, 18, 19, 22, 23], [0, 1, 3, 5, 8, 10, 11, 12, 13, 14, 15, 16, 18, 22], [0, 1, 8, 9, 12, 13, 15, 16, 18, 19, 22], [1, 2, 3, 5, 6, 8, 9, 13, 14, 17, 19, 22, 23], [1, 4, 8, 9, 10, 14, 17, 21, 22], [1, 2, 4, 7, 9, 10, 11, 14, 19, 20, 21, 24], [0, 2, 3, 4, 9, 11, 12, 13, 15, 16, 17, 18, 20, 21, 24], [0, 2, 3, 4, 5, 6, 9, 10, 14, 16, 17, 18, 19, 20], [3, 5, 7, 9, 10, 14, 15, 17, 18, 23, 24], [0, 1, 5, 6, 8, 11, 12, 14, 15, 16, 19, 20, 22, 24], [0, 6, 8, 9, 10, 14, 15, 16, 19, 21, 23, 24], [0, 1, 3, 4, 8, 10, 11, 13, 15, 17, 18, 22, 23], [0, 1, 4, 6, 7, 13, 14, 15, 17, 23], [0, 3, 5, 6, 12, 13, 14, 18, 22], [1, 2, 6, 8, 9, 10, 11, 12, 17, 19, 21, 24], [0, 3, 4, 6, 7, 8, 11, 16, 18, 19, 20, 24], [1, 4, 13, 14, 16, 17, 18, 22, 23]]
**Current coloring state:**
[1, 1, 1, 2, 3, 2, 4, 5, 3, 4, 2, 5, 5, 6, 7, 5, 1, 6, 6, 7, 0, 0, 0, 0, 0]
**Explored next state that leads to an uncolorable path:**
[1, 1, 1, 2, 3, 2, 4, 5, 3, 4, 2, 5, 5, 6, 7, 5, 1, 6, 6, 7, 2, 0, 0, 0, 0]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Colorable (B) Uncolorable. | (B) | null |
sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[1, 3, 9, 5, 2, 6, 8, 4, 7], [4, 7, 8, 3, 1, 9, 5, 2, 6], [2, 5, 6, 7, 8, 4, 3, 1, 9], [7, 6, 2, 4, 9, 8, 1, 5, 3], [5, 9, 1, 6, 3, 2, 4, 7, 8], [8, 4, 3, 1, 7, 5, 9, 6, 2], [9, 8, 7, 2, 5, 1, 6, 3, 4], [0, 0, 0, 0, 0, 7, 0, 8, 0], [0, 2, 5, 0, 0, 3, 0, 0, 1]]
Explored next state that leads to an unsolvable path:
[[1, 3, 9, 5, 2, 6, 8, 4, 7], [4, 7, 8, 3, 1, 9, 5, 2, 6], [2, 5, 6, 7, 8, 4, 3, 1, 9], [7, 6, 2, 4, 9, 8, 1, 5, 3], [5, 9, 1, 6, 3, 2, 4, 7, 8], [8, 4, 3, 1, 7, 5, 9, 6, 2], [9, 8, 7, 2, 5, 1, 6, 3, 4], [6, 0, 0, 0, 0, 7, 0, 8, 0], [0, 2, 5, 0, 0, 3, 0, 0, 1]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (A) | null |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[0, 2, 0, 0, 0, 0, 0, 0, 8], [0, 8, 0, 9, 4, 6, 5, 0, 0], [0, 3, 0, 0, 2, 1, 0, 6, 7], [0, 0, 5, 3, 6, 0, 7, 0, 0], [0, 0, 0, 0, 7, 5, 9, 0, 2], [0, 1, 3, 0, 0, 0, 0, 0, 0], [9, 7, 6, 0, 0, 2, 0, 8, 5], [0, 0, 1, 0, 0, 0, 0, 4, 0], [8, 4, 0, 0, 3, 0, 0, 0, 1]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[6, 2, 4, 7, 5, 3, 1, 9, 8], [1, 8, 7, 9, 4, 6, 5, 2, 3], [5, 3, 9, 8, 2, 1, 4, 6, 7], [0, 0, 5, 3, 6, 0, 7, 0, 0], [0, 0, 0, 0, 7, 5, 9, 0, 2], [0, 1, 3, 0, 0, 0, 0, 0, 0], [9, 7, 6, 0, 0, 2, 0, 8, 5], [0, 0, 1, 0, 0, 0, 0, 4, 0], [8, 4, 0, 0, 3, 0, 0, 0, 1]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[6, 2, 4, 7, 5, 3, 1, 9, 8], [1, 8, 7, 9, 4, 6, 5, 2, 3], [5, 3, 9, 8, 2, 1, 4, 6, 7], [2, 0, 5, 3, 6, 0, 7, 0, 0], [0, 0, 0, 0, 7, 5, 9, 0, 2], [0, 1, 3, 0, 0, 0, 0, 0, 0], [9, 7, 6, 0, 0, 2, 0, 8, 5], [0, 0, 1, 0, 0, 0, 0, 4, 0], [8, 4, 0, 0, 3, 0, 0, 0, 1]]
L(i-1) = Solvable
**Current state:**
S(i) = [[6, 2, 4, 7, 5, 3, 1, 9, 8], [1, 8, 7, 9, 4, 6, 5, 2, 3], [5, 3, 9, 8, 2, 1, 4, 6, 7], [2, 9, 5, 3, 6, 0, 7, 0, 0], [0, 0, 0, 0, 7, 5, 9, 0, 2], [0, 1, 3, 0, 0, 0, 0, 0, 0], [9, 7, 6, 0, 0, 2, 0, 8, 5], [0, 0, 1, 0, 0, 0, 0, 4, 0], [8, 4, 0, 0, 3, 0, 0, 0, 1]]
L(i) = Solvable
**Explored next state:**
S(i+1) = [[6, 2, 4, 7, 5, 3, 1, 9, 8], [1, 8, 7, 9, 4, 6, 5, 2, 3], [5, 3, 9, 8, 2, 1, 4, 6, 7], [2, 9, 5, 3, 6, 4, 7, 0, 0], [0, 0, 0, 0, 7, 5, 9, 0, 2], [0, 1, 3, 0, 0, 0, 0, 0, 0], [9, 7, 6, 0, 0, 2, 0, 8, 5], [0, 0, 1, 0, 0, 0, 0, 4, 0], [8, 4, 0, 0, 3, 0, 0, 0, 1]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[6, 2, 4, 7, 5, 3, 1, 9, 8], [1, 8, 7, 9, 4, 6, 5, 2, 3], [5, 3, 9, 8, 2, 1, 4, 6, 7], [2, 0, 5, 3, 6, 0, 7, 0, 0], [0, 0, 0, 0, 7, 5, 9, 0, 2], [0, 1, 3, 0, 0, 0, 0, 0, 0], [9, 7, 6, 0, 0, 2, 0, 8, 5], [0, 0, 1, 0, 0, 0, 0, 4, 0], [8, 4, 0, 0, 3, 0, 0, 0, 1]], 'current_state': [[6, 2, 4, 7, 5, 3, 1, 9, 8], [1, 8, 7, 9, 4, 6, 5, 2, 3], [5, 3, 9, 8, 2, 1, 4, 6, 7], [2, 9, 5, 3, 6, 0, 7, 0, 0], [0, 0, 0, 0, 7, 5, 9, 0, 2], [0, 1, 3, 0, 0, 0, 0, 0, 0], [9, 7, 6, 0, 0, 2, 0, 8, 5], [0, 0, 1, 0, 0, 0, 0, 4, 0], [8, 4, 0, 0, 3, 0, 0, 0, 1]], 'current_status': 'Solvable', 'unsolvable_child': [[6, 2, 4, 7, 5, 3, 1, 9, 8], [1, 8, 7, 9, 4, 6, 5, 2, 3], [5, 3, 9, 8, 2, 1, 4, 6, 7], [2, 9, 5, 3, 6, 4, 7, 0, 0], [0, 0, 0, 0, 7, 5, 9, 0, 2], [0, 1, 3, 0, 0, 0, 0, 0, 0], [9, 7, 6, 0, 0, 2, 0, 8, 5], [0, 0, 1, 0, 0, 0, 0, 4, 0], [8, 4, 0, 0, 3, 0, 0, 0, 1]]} |
graphcoloring_lookahead | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given the current coloring state of the graph in a list, where each index represents the corresonding vertex, and the number at that index represents its color (0 indicates an uncolored vertex).
Your task is to determine if this current state can lead to a valid coloring. Specifically, use lookahead techniques to determine if it's possible to color the remaining vertices such that no two adjacent vertices share the same color, using no more than 6 colors in total.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Graph adjacency list:**
[[1, 8, 11, 13, 16, 17], [0, 2, 3, 4, 6, 9, 11, 12, 15, 16, 17], [1, 3, 4, 7, 10, 11, 12, 14, 15, 17], [1, 2, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15], [1, 2, 3, 6, 7, 8, 10, 11, 12, 17], [3, 6, 9, 10, 12, 14, 16], [1, 3, 4, 5, 10, 11, 12, 13, 15, 16, 17], [2, 3, 4, 8, 10, 11, 12, 16], [0, 3, 4, 7, 9, 10, 11, 12, 15], [1, 5, 8, 14, 15, 16, 17], [2, 4, 5, 6, 7, 8, 12, 13, 17], [0, 1, 2, 3, 4, 6, 7, 8, 14, 15, 17], [1, 2, 3, 4, 5, 6, 7, 8, 10, 14, 15, 17], [0, 3, 6, 10, 14, 15, 16], [2, 3, 5, 9, 11, 12, 13, 17], [1, 2, 3, 6, 8, 9, 11, 12, 13], [0, 1, 5, 6, 7, 9, 13, 17], [0, 1, 2, 4, 6, 9, 10, 11, 12, 14, 16]]
**Current coloring state:**
[1, 2, 1, 3, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
**Explored next state that leads to an uncolorable path:**
[1, 2, 1, 3, 4, 2, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Colorable (B) Uncolorable. | (A) | null |
game24_lookahead | You are given four numbers and the current calculation state for the Game of 24.
Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if the remaining numbers can be combined using basic arithmetic operations (+ - * /) to reach exactly 24. You must use each number exactly once.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Numbers:**
['1', '5', '5', '6']
**Current calculation state:**
['(5 / 1)', '5', '6']
**Explored next state that leads to an unsolvable path:**
['((5 / 1) + 5)', '6']
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[0, 6, 3, 5, 0, 0, 9, 0, 4], [0, 8, 4, 7, 3, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 0, 0, 0, 6, 9, 0, 3, 1], [0, 0, 0, 8, 0, 7, 5, 0, 0], [2, 9, 0, 0, 0, 0, 0, 0, 8], [5, 0, 2, 4, 0, 0, 0, 0, 0], [0, 7, 0, 3, 1, 0, 0, 6, 5], [0, 0, 6, 0, 8, 0, 1, 7, 0]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[7, 6, 3, 5, 2, 8, 9, 1, 4], [9, 8, 4, 7, 3, 1, 2, 5, 6], [1, 2, 5, 6, 9, 4, 3, 8, 7], [4, 5, 8, 2, 6, 9, 7, 3, 1], [6, 3, 0, 8, 0, 7, 5, 0, 0], [2, 9, 0, 0, 0, 0, 0, 0, 8], [5, 0, 2, 4, 0, 0, 0, 0, 0], [0, 7, 0, 3, 1, 0, 0, 6, 5], [0, 0, 6, 0, 8, 0, 1, 7, 0]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[7, 6, 3, 5, 2, 8, 9, 1, 4], [9, 8, 4, 7, 3, 1, 2, 5, 6], [1, 2, 5, 6, 9, 4, 3, 8, 7], [4, 5, 8, 2, 6, 9, 7, 3, 1], [6, 3, 1, 8, 0, 7, 5, 0, 0], [2, 9, 0, 0, 0, 0, 0, 0, 8], [5, 0, 2, 4, 0, 0, 0, 0, 0], [0, 7, 0, 3, 1, 0, 0, 6, 5], [0, 0, 6, 0, 8, 0, 1, 7, 0]]
L(i-1) = Solvable
**Current state:**
S(i) = [[7, 6, 3, 5, 2, 8, 9, 1, 4], [9, 8, 4, 7, 3, 1, 2, 5, 6], [1, 2, 5, 6, 9, 4, 3, 8, 7], [4, 5, 8, 2, 6, 9, 7, 3, 1], [6, 3, 1, 8, 4, 7, 5, 0, 0], [2, 9, 0, 0, 0, 0, 0, 0, 8], [5, 0, 2, 4, 0, 0, 0, 0, 0], [0, 7, 0, 3, 1, 0, 0, 6, 5], [0, 0, 6, 0, 8, 0, 1, 7, 0]]
L(i) = Solvable
**Explored next state:**
S(i+1) = [[7, 6, 3, 5, 2, 8, 9, 1, 4], [9, 8, 4, 7, 3, 1, 2, 5, 6], [1, 2, 5, 6, 9, 4, 3, 8, 7], [4, 5, 8, 2, 6, 9, 7, 3, 1], [6, 3, 1, 8, 4, 7, 5, 9, 0], [2, 9, 0, 0, 0, 0, 0, 0, 8], [5, 0, 2, 4, 0, 0, 0, 0, 0], [0, 7, 0, 3, 1, 0, 0, 6, 5], [0, 0, 6, 0, 8, 0, 1, 7, 0]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[7, 6, 3, 5, 2, 8, 9, 1, 4], [9, 8, 4, 7, 3, 1, 2, 5, 6], [1, 2, 5, 6, 9, 4, 3, 8, 7], [4, 5, 8, 2, 6, 9, 7, 3, 1], [6, 3, 1, 8, 0, 7, 5, 0, 0], [2, 9, 0, 0, 0, 0, 0, 0, 8], [5, 0, 2, 4, 0, 0, 0, 0, 0], [0, 7, 0, 3, 1, 0, 0, 6, 5], [0, 0, 6, 0, 8, 0, 1, 7, 0]], 'current_state': [[7, 6, 3, 5, 2, 8, 9, 1, 4], [9, 8, 4, 7, 3, 1, 2, 5, 6], [1, 2, 5, 6, 9, 4, 3, 8, 7], [4, 5, 8, 2, 6, 9, 7, 3, 1], [6, 3, 1, 8, 4, 7, 5, 0, 0], [2, 9, 0, 0, 0, 0, 0, 0, 8], [5, 0, 2, 4, 0, 0, 0, 0, 0], [0, 7, 0, 3, 1, 0, 0, 6, 5], [0, 0, 6, 0, 8, 0, 1, 7, 0]], 'current_status': 'Solvable', 'unsolvable_child': [[7, 6, 3, 5, 2, 8, 9, 1, 4], [9, 8, 4, 7, 3, 1, 2, 5, 6], [1, 2, 5, 6, 9, 4, 3, 8, 7], [4, 5, 8, 2, 6, 9, 7, 3, 1], [6, 3, 1, 8, 4, 7, 5, 9, 0], [2, 9, 0, 0, 0, 0, 0, 0, 8], [5, 0, 2, 4, 0, 0, 0, 0, 0], [0, 7, 0, 3, 1, 0, 0, 6, 5], [0, 0, 6, 0, 8, 0, 1, 7, 0]]} |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[0, 0, 6, 0, 9, 2, 8, 4, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0], [8, 0, 9, 0, 0, 5, 0, 3, 1], [0, 0, 2, 0, 0, 0, 0, 8, 3], [0, 0, 0, 6, 4, 0, 0, 7, 5], [3, 7, 0, 0, 1, 8, 9, 0, 0], [0, 5, 0, 7, 6, 9, 0, 0, 0], [7, 0, 4, 0, 8, 0, 0, 2, 0], [1, 0, 0, 3, 0, 0, 0, 5, 9]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[5, 3, 6, 1, 9, 2, 8, 4, 7], [4, 1, 7, 8, 0, 0, 0, 0, 0], [8, 0, 9, 0, 0, 5, 0, 3, 1], [0, 0, 2, 0, 0, 0, 0, 8, 3], [0, 0, 0, 6, 4, 0, 0, 7, 5], [3, 7, 0, 0, 1, 8, 9, 0, 0], [0, 5, 0, 7, 6, 9, 0, 0, 0], [7, 0, 4, 0, 8, 0, 0, 2, 0], [1, 0, 0, 3, 0, 0, 0, 5, 9]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[5, 3, 6, 1, 9, 2, 8, 4, 7], [4, 1, 7, 8, 3, 0, 0, 0, 0], [8, 0, 9, 0, 0, 5, 0, 3, 1], [0, 0, 2, 0, 0, 0, 0, 8, 3], [0, 0, 0, 6, 4, 0, 0, 7, 5], [3, 7, 0, 0, 1, 8, 9, 0, 0], [0, 5, 0, 7, 6, 9, 0, 0, 0], [7, 0, 4, 0, 8, 0, 0, 2, 0], [1, 0, 0, 3, 0, 0, 0, 5, 9]]
L(i-1) = Solvable
**Current state:**
S(i) = [[5, 3, 6, 1, 9, 2, 8, 4, 7], [4, 1, 7, 8, 3, 6, 0, 0, 0], [8, 0, 9, 0, 0, 5, 0, 3, 1], [0, 0, 2, 0, 0, 0, 0, 8, 3], [0, 0, 0, 6, 4, 0, 0, 7, 5], [3, 7, 0, 0, 1, 8, 9, 0, 0], [0, 5, 0, 7, 6, 9, 0, 0, 0], [7, 0, 4, 0, 8, 0, 0, 2, 0], [1, 0, 0, 3, 0, 0, 0, 5, 9]]
L(i) = Solvable
**Explored next state:**
S(i+1) = [[5, 3, 6, 1, 9, 2, 8, 4, 7], [4, 1, 7, 8, 3, 6, 2, 0, 0], [8, 0, 9, 0, 0, 5, 0, 3, 1], [0, 0, 2, 0, 0, 0, 0, 8, 3], [0, 0, 0, 6, 4, 0, 0, 7, 5], [3, 7, 0, 0, 1, 8, 9, 0, 0], [0, 5, 0, 7, 6, 9, 0, 0, 0], [7, 0, 4, 0, 8, 0, 0, 2, 0], [1, 0, 0, 3, 0, 0, 0, 5, 9]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[5, 3, 6, 1, 9, 2, 8, 4, 7], [4, 1, 7, 8, 3, 0, 0, 0, 0], [8, 0, 9, 0, 0, 5, 0, 3, 1], [0, 0, 2, 0, 0, 0, 0, 8, 3], [0, 0, 0, 6, 4, 0, 0, 7, 5], [3, 7, 0, 0, 1, 8, 9, 0, 0], [0, 5, 0, 7, 6, 9, 0, 0, 0], [7, 0, 4, 0, 8, 0, 0, 2, 0], [1, 0, 0, 3, 0, 0, 0, 5, 9]], 'current_state': [[5, 3, 6, 1, 9, 2, 8, 4, 7], [4, 1, 7, 8, 3, 6, 0, 0, 0], [8, 0, 9, 0, 0, 5, 0, 3, 1], [0, 0, 2, 0, 0, 0, 0, 8, 3], [0, 0, 0, 6, 4, 0, 0, 7, 5], [3, 7, 0, 0, 1, 8, 9, 0, 0], [0, 5, 0, 7, 6, 9, 0, 0, 0], [7, 0, 4, 0, 8, 0, 0, 2, 0], [1, 0, 0, 3, 0, 0, 0, 5, 9]], 'current_status': 'Solvable', 'unsolvable_child': [[5, 3, 6, 1, 9, 2, 8, 4, 7], [4, 1, 7, 8, 3, 6, 2, 0, 0], [8, 0, 9, 0, 0, 5, 0, 3, 1], [0, 0, 2, 0, 0, 0, 0, 8, 3], [0, 0, 0, 6, 4, 0, 0, 7, 5], [3, 7, 0, 0, 1, 8, 9, 0, 0], [0, 5, 0, 7, 6, 9, 0, 0, 0], [7, 0, 4, 0, 8, 0, 0, 2, 0], [1, 0, 0, 3, 0, 0, 0, 5, 9]]} |
graphcoloring_statetransition | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given a sequence of partial coloring states leading to the current coloring state S(i). The coloring state is a list, where each index represents the corresonding vertex in the graph, and the number at that index represents its color (0 indicates an uncolored vertex). Alongside each state, its colorability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid coloring with no more than 6 colors. A valid coloring requires that no two adjacent vertices share the same color.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Coloring: Replacing a 0 in exactly one uncolored vertex with a value from 1 to 6.
2. Removing a color: Replacing a value in exactly one colored vertex with 0.
**Graph adjacency list:**
[[2, 3, 4, 9, 14, 19, 21, 23, 24], [3, 4, 5, 6, 7, 8, 10, 11, 16, 17, 19, 21, 22, 23, 24], [0, 4, 5, 6, 8, 9, 10, 16, 18, 19, 22, 23], [0, 1, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22], [0, 1, 2, 6, 8, 9, 10, 14, 16, 17, 18, 19, 20, 22, 23], [1, 2, 6, 8, 10, 11, 12, 14, 15, 16, 17, 18], [1, 2, 4, 5, 7, 12, 14, 17, 18, 19, 21, 23, 24], [1, 3, 6, 8, 9, 12, 13, 15, 18, 21, 23], [1, 2, 3, 4, 5, 7, 9, 13, 14, 15, 17, 21, 23], [0, 2, 3, 4, 7, 8, 12, 13, 14, 15, 19, 20, 23, 24], [1, 2, 3, 4, 5, 12, 15, 16, 18, 19, 20, 23], [1, 3, 5, 14, 17, 19, 21, 23, 24], [3, 5, 6, 7, 9, 10, 14, 15, 16, 20], [3, 7, 8, 9, 16, 18, 19, 22], [0, 3, 4, 5, 6, 8, 9, 11, 12, 16, 17, 18, 19, 20, 21], [3, 5, 7, 8, 9, 10, 12, 16, 17, 20, 21, 22, 23, 24], [1, 2, 3, 4, 5, 10, 12, 13, 14, 15, 20, 22, 23], [1, 3, 4, 5, 6, 8, 11, 14, 15, 18, 21, 22], [2, 3, 4, 5, 6, 7, 10, 13, 14, 17, 23], [0, 1, 2, 4, 6, 9, 10, 11, 13, 14, 20, 22, 23], [3, 4, 9, 10, 12, 14, 15, 16, 19, 21, 23], [0, 1, 3, 6, 7, 8, 11, 14, 15, 17, 20, 22, 23, 24], [1, 2, 3, 4, 13, 15, 16, 17, 19, 21, 23, 24], [0, 1, 2, 4, 6, 7, 8, 9, 10, 11, 15, 16, 18, 19, 20, 21, 22, 24], [0, 1, 6, 9, 11, 15, 21, 22, 23]]
**Two moves ago:**
S(i-2) = [1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i-2) = Colorable
**One move ago:**
S(i-1) = [1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i-1) = Colorable
**Current coloring state:**
S(i) = [1, 2, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i) = Colorable
**Explored next state:**
S(i+1) = [1, 2, 2, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i+1) = Uncolorable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{coloring\}", where \{coloring\} is in the same python list format as the previous states. | null | {'parent_state': [1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'current_state': [1, 2, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'current_status': 'Solvable', 'unsolvable_child': [1, 2, 2, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'graph': [[2, 3, 4, 9, 14, 19, 21, 23, 24], [3, 4, 5, 6, 7, 8, 10, 11, 16, 17, 19, 21, 22, 23, 24], [0, 4, 5, 6, 8, 9, 10, 16, 18, 19, 22, 23], [0, 1, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22], [0, 1, 2, 6, 8, 9, 10, 14, 16, 17, 18, 19, 20, 22, 23], [1, 2, 6, 8, 10, 11, 12, 14, 15, 16, 17, 18], [1, 2, 4, 5, 7, 12, 14, 17, 18, 19, 21, 23, 24], [1, 3, 6, 8, 9, 12, 13, 15, 18, 21, 23], [1, 2, 3, 4, 5, 7, 9, 13, 14, 15, 17, 21, 23], [0, 2, 3, 4, 7, 8, 12, 13, 14, 15, 19, 20, 23, 24], [1, 2, 3, 4, 5, 12, 15, 16, 18, 19, 20, 23], [1, 3, 5, 14, 17, 19, 21, 23, 24], [3, 5, 6, 7, 9, 10, 14, 15, 16, 20], [3, 7, 8, 9, 16, 18, 19, 22], [0, 3, 4, 5, 6, 8, 9, 11, 12, 16, 17, 18, 19, 20, 21], [3, 5, 7, 8, 9, 10, 12, 16, 17, 20, 21, 22, 23, 24], [1, 2, 3, 4, 5, 10, 12, 13, 14, 15, 20, 22, 23], [1, 3, 4, 5, 6, 8, 11, 14, 15, 18, 21, 22], [2, 3, 4, 5, 6, 7, 10, 13, 14, 17, 23], [0, 1, 2, 4, 6, 9, 10, 11, 13, 14, 20, 22, 23], [3, 4, 9, 10, 12, 14, 15, 16, 19, 21, 23], [0, 1, 3, 6, 7, 8, 11, 14, 15, 17, 20, 22, 23, 24], [1, 2, 3, 4, 13, 15, 16, 17, 19, 21, 23, 24], [0, 1, 2, 4, 6, 7, 8, 9, 10, 11, 15, 16, 18, 19, 20, 21, 22, 24], [0, 1, 6, 9, 11, 15, 21, 22, 23]], 'chromatic_number': 6} |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[0, 9, 0, 2, 8, 6, 0, 0, 4], [5, 4, 8, 0, 0, 3, 7, 0, 0], [6, 0, 0, 0, 0, 0, 9, 0, 0], [0, 8, 0, 0, 1, 7, 0, 5, 0], [9, 6, 0, 0, 2, 8, 0, 0, 3], [7, 0, 3, 0, 0, 0, 0, 1, 0], [3, 0, 0, 6, 5, 0, 0, 0, 0], [0, 0, 0, 9, 0, 0, 1, 2, 6], [0, 2, 0, 0, 7, 0, 0, 9, 0]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[1, 9, 7, 2, 8, 6, 5, 3, 4], [5, 4, 8, 1, 9, 3, 7, 6, 2], [6, 3, 2, 7, 4, 5, 9, 8, 1], [2, 8, 4, 3, 1, 7, 6, 5, 9], [9, 6, 1, 5, 2, 8, 4, 7, 3], [7, 5, 3, 4, 6, 9, 2, 1, 8], [3, 0, 0, 6, 5, 0, 0, 0, 0], [0, 0, 0, 9, 0, 0, 1, 2, 6], [0, 2, 0, 0, 7, 0, 0, 9, 0]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[1, 9, 7, 2, 8, 6, 5, 3, 4], [5, 4, 8, 1, 9, 3, 7, 6, 2], [6, 3, 2, 7, 4, 5, 9, 8, 1], [2, 8, 4, 3, 1, 7, 6, 5, 9], [9, 6, 1, 5, 2, 8, 4, 7, 3], [7, 5, 3, 4, 6, 9, 2, 1, 8], [3, 1, 0, 6, 5, 0, 0, 0, 0], [0, 0, 0, 9, 0, 0, 1, 2, 6], [0, 2, 0, 0, 7, 0, 0, 9, 0]]
L(i-1) = Solvable
**Current state:**
S(i) = [[1, 9, 7, 2, 8, 6, 5, 3, 4], [5, 4, 8, 1, 9, 3, 7, 6, 2], [6, 3, 2, 7, 4, 5, 9, 8, 1], [2, 8, 4, 3, 1, 7, 6, 5, 9], [9, 6, 1, 5, 2, 8, 4, 7, 3], [7, 5, 3, 4, 6, 9, 2, 1, 8], [3, 1, 9, 6, 5, 0, 0, 0, 0], [0, 0, 0, 9, 0, 0, 1, 2, 6], [0, 2, 0, 0, 7, 0, 0, 9, 0]]
L(i) = Solvable
**Explored next state:**
S(i+1) = [[1, 9, 7, 2, 8, 6, 5, 3, 4], [5, 4, 8, 1, 9, 3, 7, 6, 2], [6, 3, 2, 7, 4, 5, 9, 8, 1], [2, 8, 4, 3, 1, 7, 6, 5, 9], [9, 6, 1, 5, 2, 8, 4, 7, 3], [7, 5, 3, 4, 6, 9, 2, 1, 8], [3, 1, 9, 6, 5, 4, 0, 0, 0], [0, 0, 0, 9, 0, 0, 1, 2, 6], [0, 2, 0, 0, 7, 0, 0, 9, 0]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[1, 9, 7, 2, 8, 6, 5, 3, 4], [5, 4, 8, 1, 9, 3, 7, 6, 2], [6, 3, 2, 7, 4, 5, 9, 8, 1], [2, 8, 4, 3, 1, 7, 6, 5, 9], [9, 6, 1, 5, 2, 8, 4, 7, 3], [7, 5, 3, 4, 6, 9, 2, 1, 8], [3, 1, 0, 6, 5, 0, 0, 0, 0], [0, 0, 0, 9, 0, 0, 1, 2, 6], [0, 2, 0, 0, 7, 0, 0, 9, 0]], 'current_state': [[1, 9, 7, 2, 8, 6, 5, 3, 4], [5, 4, 8, 1, 9, 3, 7, 6, 2], [6, 3, 2, 7, 4, 5, 9, 8, 1], [2, 8, 4, 3, 1, 7, 6, 5, 9], [9, 6, 1, 5, 2, 8, 4, 7, 3], [7, 5, 3, 4, 6, 9, 2, 1, 8], [3, 1, 9, 6, 5, 0, 0, 0, 0], [0, 0, 0, 9, 0, 0, 1, 2, 6], [0, 2, 0, 0, 7, 0, 0, 9, 0]], 'current_status': 'Solvable', 'unsolvable_child': [[1, 9, 7, 2, 8, 6, 5, 3, 4], [5, 4, 8, 1, 9, 3, 7, 6, 2], [6, 3, 2, 7, 4, 5, 9, 8, 1], [2, 8, 4, 3, 1, 7, 6, 5, 9], [9, 6, 1, 5, 2, 8, 4, 7, 3], [7, 5, 3, 4, 6, 9, 2, 1, 8], [3, 1, 9, 6, 5, 4, 0, 0, 0], [0, 0, 0, 9, 0, 0, 1, 2, 6], [0, 2, 0, 0, 7, 0, 0, 9, 0]]} |
sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[9, 1, 2, 7, 6, 4, 8, 5, 3], [6, 4, 8, 5, 3, 1, 7, 2, 9], [3, 5, 7, 8, 9, 2, 4, 1, 6], [5, 6, 1, 9, 8, 7, 3, 4, 2], [2, 7, 4, 6, 0, 3, 9, 0, 0], [8, 0, 0, 2, 4, 0, 0, 0, 1], [0, 0, 0, 0, 1, 8, 0, 0, 7], [0, 0, 3, 0, 0, 0, 0, 8, 0], [1, 0, 6, 4, 7, 0, 2, 0, 5]]
Explored next state that leads to an unsolvable path:
[[9, 1, 2, 7, 6, 4, 8, 5, 3], [6, 4, 8, 5, 3, 1, 7, 2, 9], [3, 5, 7, 8, 9, 2, 4, 1, 6], [5, 6, 1, 9, 8, 7, 3, 4, 2], [2, 7, 4, 6, 5, 3, 9, 0, 0], [8, 0, 0, 2, 4, 0, 0, 0, 1], [0, 0, 0, 0, 1, 8, 0, 0, 7], [0, 0, 3, 0, 0, 0, 0, 8, 0], [1, 0, 6, 4, 7, 0, 2, 0, 5]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
game24_statetransition | You are given an initial Game of 24 configuration S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid solution requires using each of the four initial numbers exactly once, using only basic arithmetic operations (+ - * /), and ultimately evaluating to 24.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Applying an operation: Combining two expressions using a basic arithmetic operation (+ - * /), reducing the number of expressions by 1.
2. Reverting an operation: Removing the last operation applied to the expressions, increasing the number of expressions by 1.
**Initial configuration:**
S(0) = ['1', '5', '5', '13']
L(0) = Solvable
**Current state:**
S(i) = ['(5 * 13)', '1', '5']
L(i) = Unsolvable
**Explored next state:**
S(i+1) = ['((5 * 13) + 1)', '5']
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{expressions\}", where \{expressions\} is in the same python list format as the previous states. | null | {'parent_state': ['1', '5', '5', '13'], 'current_state': ['(5 * 13)', '1', '5'], 'current_status': 'Unsolvable', 'unsolvable_child': ['((5 * 13) + 1)', '5'], 'numbers': [1, 5, 5, 13]} |
graphcoloring_statetransition | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given a sequence of partial coloring states leading to the current coloring state S(i). The coloring state is a list, where each index represents the corresonding vertex in the graph, and the number at that index represents its color (0 indicates an uncolored vertex). Alongside each state, its colorability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid coloring with no more than 5 colors. A valid coloring requires that no two adjacent vertices share the same color.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Coloring: Replacing a 0 in exactly one uncolored vertex with a value from 1 to 5.
2. Removing a color: Replacing a value in exactly one colored vertex with 0.
**Graph adjacency list:**
[[2, 5, 8, 10, 11, 12, 13, 16, 19, 21, 23], [3, 5, 9, 12, 13, 15, 16, 20, 22], [0, 3, 9, 12, 18], [1, 2, 6, 11, 12, 16], [5, 8, 12, 13, 17, 19, 20], [0, 1, 4, 7, 12, 13, 17, 18, 19, 20, 21, 23], [3, 7, 8, 9, 11, 12, 17, 20], [5, 6, 8, 9, 11, 13, 14, 19, 20, 21, 23], [0, 4, 6, 7, 9, 10, 11, 14, 17, 21, 22, 23], [1, 2, 6, 7, 8, 11, 12, 13, 14, 16, 17, 18, 23], [0, 8, 11, 12, 14, 17, 21, 22], [0, 3, 6, 7, 8, 9, 10, 15, 16, 17, 19, 21], [0, 1, 2, 3, 4, 5, 6, 9, 10, 14, 18, 19], [0, 1, 4, 5, 7, 9, 15, 16, 17, 18, 21, 23], [7, 8, 9, 10, 12, 18, 20], [1, 11, 13, 18, 21], [0, 1, 3, 9, 11, 13, 20], [4, 5, 6, 8, 9, 10, 11, 13, 19], [2, 5, 9, 12, 13, 14, 15, 19, 20, 21], [0, 4, 5, 7, 11, 12, 17, 18, 23], [1, 4, 5, 6, 7, 14, 16, 18, 21, 23], [0, 5, 7, 8, 10, 11, 13, 15, 18, 20], [1, 8, 10], [0, 5, 7, 8, 9, 13, 19, 20]]
**Two moves ago:**
S(i-2) = [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i-2) = Colorable
**One move ago:**
S(i-1) = [1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i-1) = Colorable
**Current coloring state:**
S(i) = [1, 1, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i) = Colorable
**Explored next state:**
S(i+1) = [1, 1, 2, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i+1) = Uncolorable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{coloring\}", where \{coloring\} is in the same python list format as the previous states. | null | {'parent_state': [1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'current_state': [1, 1, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'current_status': 'Solvable', 'unsolvable_child': [1, 1, 2, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'graph': [[2, 5, 8, 10, 11, 12, 13, 16, 19, 21, 23], [3, 5, 9, 12, 13, 15, 16, 20, 22], [0, 3, 9, 12, 18], [1, 2, 6, 11, 12, 16], [5, 8, 12, 13, 17, 19, 20], [0, 1, 4, 7, 12, 13, 17, 18, 19, 20, 21, 23], [3, 7, 8, 9, 11, 12, 17, 20], [5, 6, 8, 9, 11, 13, 14, 19, 20, 21, 23], [0, 4, 6, 7, 9, 10, 11, 14, 17, 21, 22, 23], [1, 2, 6, 7, 8, 11, 12, 13, 14, 16, 17, 18, 23], [0, 8, 11, 12, 14, 17, 21, 22], [0, 3, 6, 7, 8, 9, 10, 15, 16, 17, 19, 21], [0, 1, 2, 3, 4, 5, 6, 9, 10, 14, 18, 19], [0, 1, 4, 5, 7, 9, 15, 16, 17, 18, 21, 23], [7, 8, 9, 10, 12, 18, 20], [1, 11, 13, 18, 21], [0, 1, 3, 9, 11, 13, 20], [4, 5, 6, 8, 9, 10, 11, 13, 19], [2, 5, 9, 12, 13, 14, 15, 19, 20, 21], [0, 4, 5, 7, 11, 12, 17, 18, 23], [1, 4, 5, 6, 7, 14, 16, 18, 21, 23], [0, 5, 7, 8, 10, 11, 13, 15, 18, 20], [1, 8, 10], [0, 5, 7, 8, 9, 13, 19, 20]], 'chromatic_number': 5} |
sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[4, 1, 3, 5, 8, 2, 7, 9, 6], [8, 2, 5, 7, 9, 6, 4, 1, 3], [9, 6, 7, 3, 1, 4, 8, 2, 5], [6, 5, 4, 1, 3, 7, 2, 8, 9], [3, 8, 2, 4, 6, 9, 5, 7, 1], [1, 7, 9, 2, 5, 8, 3, 6, 4], [7, 4, 1, 8, 0, 0, 6, 0, 0], [0, 9, 0, 6, 4, 5, 1, 0, 7], [0, 3, 0, 0, 7, 0, 0, 0, 2]]
Explored next state that leads to an unsolvable path:
[[4, 1, 3, 5, 8, 2, 7, 9, 6], [8, 2, 5, 7, 9, 6, 4, 1, 3], [9, 6, 7, 3, 1, 4, 8, 2, 5], [6, 5, 4, 1, 3, 7, 2, 8, 9], [3, 8, 2, 4, 6, 9, 5, 7, 1], [1, 7, 9, 2, 5, 8, 3, 6, 4], [7, 4, 1, 8, 2, 0, 6, 0, 0], [0, 9, 0, 6, 4, 5, 1, 0, 7], [0, 3, 0, 0, 7, 0, 0, 0, 2]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
game24_statetransition | You are given an initial Game of 24 configuration S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid solution requires using each of the four initial numbers exactly once, using only basic arithmetic operations (+ - * /), and ultimately evaluating to 24.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Applying an operation: Combining two expressions using a basic arithmetic operation (+ - * /), reducing the number of expressions by 1.
2. Reverting an operation: Removing the last operation applied to the expressions, increasing the number of expressions by 1.
**Initial configuration:**
S(0) = ['3', '3', '3', '9']
L(0) = Solvable
**Current state:**
S(i) = ['(3 / 3)', '3', '9']
L(i) = Solvable
**Explored next state:**
S(i+1) = ['((3 / 3) + 3)', '9']
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{expressions\}", where \{expressions\} is in the same python list format as the previous states. | null | {'parent_state': ['3', '3', '3', '9'], 'current_state': ['(3 / 3)', '3', '9'], 'current_status': 'Solvable', 'unsolvable_child': ['((3 / 3) + 3)', '9'], 'numbers': [3, 3, 3, 9]} |
game24_statetransition | You are given an initial Game of 24 configuration S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid solution requires using each of the four initial numbers exactly once, using only basic arithmetic operations (+ - * /), and ultimately evaluating to 24.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Applying an operation: Combining two expressions using a basic arithmetic operation (+ - * /), reducing the number of expressions by 1.
2. Reverting an operation: Removing the last operation applied to the expressions, increasing the number of expressions by 1.
**Initial configuration:**
S(0) = ['4', '4', '4', '12']
L(0) = Solvable
**Current state:**
S(i) = ['(12 - 4)', '4', '4']
L(i) = Solvable
**Explored next state:**
S(i+1) = ['((12 - 4) + 4)', '4']
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{expressions\}", where \{expressions\} is in the same python list format as the previous states. | null | {'parent_state': ['4', '4', '4', '12'], 'current_state': ['(12 - 4)', '4', '4'], 'current_status': 'Solvable', 'unsolvable_child': ['((12 - 4) + 4)', '4'], 'numbers': [4, 4, 4, 12]} |
graphcoloring_statetransition | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given a sequence of partial coloring states leading to the current coloring state S(i). The coloring state is a list, where each index represents the corresonding vertex in the graph, and the number at that index represents its color (0 indicates an uncolored vertex). Alongside each state, its colorability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid coloring with no more than 7 colors. A valid coloring requires that no two adjacent vertices share the same color.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Coloring: Replacing a 0 in exactly one uncolored vertex with a value from 1 to 7.
2. Removing a color: Replacing a value in exactly one colored vertex with 0.
**Graph adjacency list:**
[[1, 3, 4, 5, 6, 9, 10, 11, 15, 16, 17, 18, 20], [0, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 16, 17], [1, 3, 4, 6, 8, 11, 12, 16, 21], [0, 1, 2, 4, 9, 10, 12, 13, 15, 17, 19, 20], [0, 1, 2, 3, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21], [0, 1, 7, 8, 11, 14, 16, 17, 18, 20, 21], [0, 1, 2, 4, 7, 8, 11, 12, 15, 16, 18, 21], [1, 4, 5, 6, 8, 9, 11, 12, 14, 16, 17, 19, 20, 21], [1, 2, 5, 6, 7, 10, 11, 12, 14, 15, 17, 20], [0, 1, 3, 4, 7, 12, 13, 16, 17, 18, 20], [0, 3, 4, 8, 11, 12, 15, 18], [0, 2, 4, 5, 6, 7, 8, 10, 13, 14, 18, 19, 20], [2, 3, 4, 6, 7, 8, 9, 10, 15, 16], [1, 3, 9, 11, 14, 16, 17, 18, 21], [1, 4, 5, 7, 8, 11, 13, 16, 21], [0, 3, 4, 6, 8, 10, 12, 16, 17, 19], [0, 1, 2, 4, 5, 6, 7, 9, 12, 13, 14, 15, 18, 21], [0, 1, 3, 4, 5, 7, 8, 9, 13, 15, 19, 20, 21], [0, 4, 5, 6, 9, 10, 11, 13, 16, 20, 21], [3, 4, 7, 11, 15, 17, 20], [0, 3, 5, 7, 8, 9, 11, 17, 18, 19], [2, 4, 5, 6, 7, 13, 14, 16, 17, 18]]
**Two moves ago:**
S(i-2) = [1, 2, 1, 3, 4, 3, 3, 1, 4, 5, 2, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i-2) = Colorable
**One move ago:**
S(i-1) = [1, 2, 1, 3, 4, 3, 3, 1, 4, 5, 2, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i-1) = Colorable
**Current coloring state:**
S(i) = [1, 2, 1, 3, 4, 3, 3, 1, 4, 5, 2, 5, 6, 7, 0, 0, 0, 0, 0, 0, 0, 0]
L(i) = Uncolorable
**Explored next state:**
S(i+1) = [1, 2, 1, 3, 4, 3, 3, 1, 4, 5, 2, 5, 6, 7, 6, 0, 0, 0, 0, 0, 0, 0]
L(i+1) = Uncolorable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{coloring\}", where \{coloring\} is in the same python list format as the previous states. | null | {'parent_state': [1, 2, 1, 3, 4, 3, 3, 1, 4, 5, 2, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'current_state': [1, 2, 1, 3, 4, 3, 3, 1, 4, 5, 2, 5, 6, 7, 0, 0, 0, 0, 0, 0, 0, 0], 'current_status': 'Unsolvable', 'unsolvable_child': [1, 2, 1, 3, 4, 3, 3, 1, 4, 5, 2, 5, 6, 7, 6, 0, 0, 0, 0, 0, 0, 0], 'graph': [[1, 3, 4, 5, 6, 9, 10, 11, 15, 16, 17, 18, 20], [0, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 16, 17], [1, 3, 4, 6, 8, 11, 12, 16, 21], [0, 1, 2, 4, 9, 10, 12, 13, 15, 17, 19, 20], [0, 1, 2, 3, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21], [0, 1, 7, 8, 11, 14, 16, 17, 18, 20, 21], [0, 1, 2, 4, 7, 8, 11, 12, 15, 16, 18, 21], [1, 4, 5, 6, 8, 9, 11, 12, 14, 16, 17, 19, 20, 21], [1, 2, 5, 6, 7, 10, 11, 12, 14, 15, 17, 20], [0, 1, 3, 4, 7, 12, 13, 16, 17, 18, 20], [0, 3, 4, 8, 11, 12, 15, 18], [0, 2, 4, 5, 6, 7, 8, 10, 13, 14, 18, 19, 20], [2, 3, 4, 6, 7, 8, 9, 10, 15, 16], [1, 3, 9, 11, 14, 16, 17, 18, 21], [1, 4, 5, 7, 8, 11, 13, 16, 21], [0, 3, 4, 6, 8, 10, 12, 16, 17, 19], [0, 1, 2, 4, 5, 6, 7, 9, 12, 13, 14, 15, 18, 21], [0, 1, 3, 4, 5, 7, 8, 9, 13, 15, 19, 20, 21], [0, 4, 5, 6, 9, 10, 11, 13, 16, 20, 21], [3, 4, 7, 11, 15, 17, 20], [0, 3, 5, 7, 8, 9, 11, 17, 18, 19], [2, 4, 5, 6, 7, 13, 14, 16, 17, 18]], 'chromatic_number': 7} |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[0, 4, 5, 0, 0, 0, 0, 0, 8], [9, 0, 2, 8, 0, 0, 0, 5, 0], [0, 7, 0, 0, 2, 0, 0, 6, 0], [0, 0, 0, 1, 0, 0, 5, 0, 7], [0, 1, 7, 0, 0, 2, 0, 0, 6], [4, 5, 0, 0, 6, 3, 0, 0, 9], [0, 0, 0, 0, 9, 0, 3, 1, 0], [0, 6, 0, 0, 7, 0, 9, 0, 2], [1, 8, 0, 0, 3, 5, 0, 0, 4]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[6, 4, 5, 3, 1, 7, 2, 9, 8], [9, 3, 2, 8, 4, 6, 7, 5, 1], [8, 7, 1, 5, 2, 9, 4, 6, 3], [2, 9, 6, 1, 8, 4, 5, 3, 7], [3, 1, 7, 9, 5, 2, 8, 4, 6], [4, 5, 8, 7, 6, 3, 0, 0, 9], [0, 0, 0, 0, 9, 0, 3, 1, 0], [0, 6, 0, 0, 7, 0, 9, 0, 2], [1, 8, 0, 0, 3, 5, 0, 0, 4]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[6, 4, 5, 3, 1, 7, 2, 9, 8], [9, 3, 2, 8, 4, 6, 7, 5, 1], [8, 7, 1, 5, 2, 9, 4, 6, 3], [2, 9, 6, 1, 8, 4, 5, 3, 7], [3, 1, 7, 9, 5, 2, 8, 4, 6], [4, 5, 8, 7, 6, 3, 1, 0, 9], [0, 0, 0, 0, 9, 0, 3, 1, 0], [0, 6, 0, 0, 7, 0, 9, 0, 2], [1, 8, 0, 0, 3, 5, 0, 0, 4]]
L(i-1) = Solvable
**Current state:**
S(i) = [[6, 4, 5, 3, 1, 7, 2, 9, 8], [9, 3, 2, 8, 4, 6, 7, 5, 1], [8, 7, 1, 5, 2, 9, 4, 6, 3], [2, 9, 6, 1, 8, 4, 5, 3, 7], [3, 1, 7, 9, 5, 2, 8, 4, 6], [4, 5, 8, 7, 6, 3, 1, 2, 9], [0, 0, 0, 0, 9, 0, 3, 1, 0], [0, 6, 0, 0, 7, 0, 9, 0, 2], [1, 8, 0, 0, 3, 5, 0, 0, 4]]
L(i) = Solvable
**Explored next state:**
S(i+1) = [[6, 4, 5, 3, 1, 7, 2, 9, 8], [9, 3, 2, 8, 4, 6, 7, 5, 1], [8, 7, 1, 5, 2, 9, 4, 6, 3], [2, 9, 6, 1, 8, 4, 5, 3, 7], [3, 1, 7, 9, 5, 2, 8, 4, 6], [4, 5, 8, 7, 6, 3, 1, 2, 9], [5, 0, 0, 0, 9, 0, 3, 1, 0], [0, 6, 0, 0, 7, 0, 9, 0, 2], [1, 8, 0, 0, 3, 5, 0, 0, 4]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[6, 4, 5, 3, 1, 7, 2, 9, 8], [9, 3, 2, 8, 4, 6, 7, 5, 1], [8, 7, 1, 5, 2, 9, 4, 6, 3], [2, 9, 6, 1, 8, 4, 5, 3, 7], [3, 1, 7, 9, 5, 2, 8, 4, 6], [4, 5, 8, 7, 6, 3, 1, 0, 9], [0, 0, 0, 0, 9, 0, 3, 1, 0], [0, 6, 0, 0, 7, 0, 9, 0, 2], [1, 8, 0, 0, 3, 5, 0, 0, 4]], 'current_state': [[6, 4, 5, 3, 1, 7, 2, 9, 8], [9, 3, 2, 8, 4, 6, 7, 5, 1], [8, 7, 1, 5, 2, 9, 4, 6, 3], [2, 9, 6, 1, 8, 4, 5, 3, 7], [3, 1, 7, 9, 5, 2, 8, 4, 6], [4, 5, 8, 7, 6, 3, 1, 2, 9], [0, 0, 0, 0, 9, 0, 3, 1, 0], [0, 6, 0, 0, 7, 0, 9, 0, 2], [1, 8, 0, 0, 3, 5, 0, 0, 4]], 'current_status': 'Solvable', 'unsolvable_child': [[6, 4, 5, 3, 1, 7, 2, 9, 8], [9, 3, 2, 8, 4, 6, 7, 5, 1], [8, 7, 1, 5, 2, 9, 4, 6, 3], [2, 9, 6, 1, 8, 4, 5, 3, 7], [3, 1, 7, 9, 5, 2, 8, 4, 6], [4, 5, 8, 7, 6, 3, 1, 2, 9], [5, 0, 0, 0, 9, 0, 3, 1, 0], [0, 6, 0, 0, 7, 0, 9, 0, 2], [1, 8, 0, 0, 3, 5, 0, 0, 4]]} |
game24_lookahead | You are given four numbers and the current calculation state for the Game of 24.
Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if the remaining numbers can be combined using basic arithmetic operations (+ - * /) to reach exactly 24. You must use each number exactly once.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Numbers:**
['5', '9', '9', '11']
**Current calculation state:**
['(9 + 11)', '5', '9']
**Explored next state that leads to an unsolvable path:**
['((9 + 11) + 5)', '9']
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (A) | null |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[9, 0, 0, 0, 0, 1, 0, 0, 0], [0, 1, 0, 5, 4, 0, 0, 0, 2], [0, 4, 0, 9, 0, 0, 0, 8, 0], [0, 0, 0, 4, 3, 6, 2, 1, 0], [0, 0, 0, 0, 9, 0, 0, 7, 3], [6, 8, 0, 1, 0, 2, 5, 0, 0], [0, 0, 7, 0, 0, 0, 3, 4, 0], [3, 0, 0, 6, 0, 7, 0, 0, 5], [0, 0, 2, 0, 8, 0, 0, 0, 9]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[9, 3, 8, 7, 2, 1, 4, 5, 6], [7, 1, 6, 5, 4, 8, 9, 3, 2], [2, 4, 5, 9, 6, 3, 1, 8, 7], [5, 7, 9, 4, 3, 6, 2, 1, 8], [4, 2, 1, 8, 9, 5, 6, 7, 3], [6, 8, 3, 1, 7, 2, 5, 9, 0], [0, 0, 7, 0, 0, 0, 3, 4, 0], [3, 0, 0, 6, 0, 7, 0, 0, 5], [0, 0, 2, 0, 8, 0, 0, 0, 9]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[9, 3, 8, 7, 2, 1, 4, 5, 6], [7, 1, 6, 5, 4, 8, 9, 3, 2], [2, 4, 5, 9, 6, 3, 1, 8, 7], [5, 7, 9, 4, 3, 6, 2, 1, 8], [4, 2, 1, 8, 9, 5, 6, 7, 3], [6, 8, 3, 1, 7, 2, 5, 9, 4], [0, 0, 7, 0, 0, 0, 3, 4, 0], [3, 0, 0, 6, 0, 7, 0, 0, 5], [0, 0, 2, 0, 8, 0, 0, 0, 9]]
L(i-1) = Solvable
**Current state:**
S(i) = [[9, 3, 8, 7, 2, 1, 4, 5, 6], [7, 1, 6, 5, 4, 8, 9, 3, 2], [2, 4, 5, 9, 6, 3, 1, 8, 7], [5, 7, 9, 4, 3, 6, 2, 1, 8], [4, 2, 1, 8, 9, 5, 6, 7, 3], [6, 8, 3, 1, 7, 2, 5, 9, 4], [1, 0, 7, 0, 0, 0, 3, 4, 0], [3, 0, 0, 6, 0, 7, 0, 0, 5], [0, 0, 2, 0, 8, 0, 0, 0, 9]]
L(i) = Unsolvable
**Explored next state:**
S(i+1) = [[9, 3, 8, 7, 2, 1, 4, 5, 6], [7, 1, 6, 5, 4, 8, 9, 3, 2], [2, 4, 5, 9, 6, 3, 1, 8, 7], [5, 7, 9, 4, 3, 6, 2, 1, 8], [4, 2, 1, 8, 9, 5, 6, 7, 3], [6, 8, 3, 1, 7, 2, 5, 9, 4], [1, 5, 7, 0, 0, 0, 3, 4, 0], [3, 0, 0, 6, 0, 7, 0, 0, 5], [0, 0, 2, 0, 8, 0, 0, 0, 9]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[9, 3, 8, 7, 2, 1, 4, 5, 6], [7, 1, 6, 5, 4, 8, 9, 3, 2], [2, 4, 5, 9, 6, 3, 1, 8, 7], [5, 7, 9, 4, 3, 6, 2, 1, 8], [4, 2, 1, 8, 9, 5, 6, 7, 3], [6, 8, 3, 1, 7, 2, 5, 9, 4], [0, 0, 7, 0, 0, 0, 3, 4, 0], [3, 0, 0, 6, 0, 7, 0, 0, 5], [0, 0, 2, 0, 8, 0, 0, 0, 9]], 'current_state': [[9, 3, 8, 7, 2, 1, 4, 5, 6], [7, 1, 6, 5, 4, 8, 9, 3, 2], [2, 4, 5, 9, 6, 3, 1, 8, 7], [5, 7, 9, 4, 3, 6, 2, 1, 8], [4, 2, 1, 8, 9, 5, 6, 7, 3], [6, 8, 3, 1, 7, 2, 5, 9, 4], [1, 0, 7, 0, 0, 0, 3, 4, 0], [3, 0, 0, 6, 0, 7, 0, 0, 5], [0, 0, 2, 0, 8, 0, 0, 0, 9]], 'current_status': 'Unsolvable', 'unsolvable_child': [[9, 3, 8, 7, 2, 1, 4, 5, 6], [7, 1, 6, 5, 4, 8, 9, 3, 2], [2, 4, 5, 9, 6, 3, 1, 8, 7], [5, 7, 9, 4, 3, 6, 2, 1, 8], [4, 2, 1, 8, 9, 5, 6, 7, 3], [6, 8, 3, 1, 7, 2, 5, 9, 4], [1, 5, 7, 0, 0, 0, 3, 4, 0], [3, 0, 0, 6, 0, 7, 0, 0, 5], [0, 0, 2, 0, 8, 0, 0, 0, 9]]} |
game24_lookahead | You are given four numbers and the current calculation state for the Game of 24.
Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if the remaining numbers can be combined using basic arithmetic operations (+ - * /) to reach exactly 24. You must use each number exactly once.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Numbers:**
['5', '5', '11', '11']
**Current calculation state:**
['(5 + 11)', '5', '11']
**Explored next state that leads to an unsolvable path:**
['((5 + 11) + 5)', '11']
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[5, 9, 1, 8, 6, 2, 7, 3, 4], [7, 6, 2, 1, 4, 3, 8, 5, 9], [3, 8, 4, 9, 7, 5, 6, 2, 1], [4, 5, 9, 7, 2, 8, 1, 6, 3], [1, 7, 6, 3, 5, 9, 4, 8, 2], [2, 3, 8, 6, 1, 4, 5, 9, 7], [8, 2, 5, 4, 9, 1, 3, 7, 6], [9, 1, 7, 2, 0, 0, 0, 4, 8], [6, 0, 3, 0, 0, 0, 0, 1, 0]]
Explored next state that leads to an unsolvable path:
[[5, 9, 1, 8, 6, 2, 7, 3, 4], [7, 6, 2, 1, 4, 3, 8, 5, 9], [3, 8, 4, 9, 7, 5, 6, 2, 1], [4, 5, 9, 7, 2, 8, 1, 6, 3], [1, 7, 6, 3, 5, 9, 4, 8, 2], [2, 3, 8, 6, 1, 4, 5, 9, 7], [8, 2, 5, 4, 9, 1, 3, 7, 6], [9, 1, 7, 2, 3, 0, 0, 4, 8], [6, 0, 3, 0, 0, 0, 0, 1, 0]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
game24_statetransition | You are given an initial Game of 24 configuration S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid solution requires using each of the four initial numbers exactly once, using only basic arithmetic operations (+ - * /), and ultimately evaluating to 24.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Applying an operation: Combining two expressions using a basic arithmetic operation (+ - * /), reducing the number of expressions by 1.
2. Reverting an operation: Removing the last operation applied to the expressions, increasing the number of expressions by 1.
**Initial configuration:**
S(0) = ['8', '8', '8', '13']
L(0) = Solvable
**Current state:**
S(i) = ['(8 / 8)', '8', '13']
L(i) = Unsolvable
**Explored next state:**
S(i+1) = ['((8 / 8) + 8)', '13']
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{expressions\}", where \{expressions\} is in the same python list format as the previous states. | null | {'parent_state': ['8', '8', '8', '13'], 'current_state': ['(8 / 8)', '8', '13'], 'current_status': 'Unsolvable', 'unsolvable_child': ['((8 / 8) + 8)', '13'], 'numbers': [8, 8, 8, 13]} |
graphcoloring_lookahead | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given the current coloring state of the graph in a list, where each index represents the corresonding vertex, and the number at that index represents its color (0 indicates an uncolored vertex).
Your task is to determine if this current state can lead to a valid coloring. Specifically, use lookahead techniques to determine if it's possible to color the remaining vertices such that no two adjacent vertices share the same color, using no more than 8 colors in total.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Graph adjacency list:**
[[3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16], [2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 15], [1, 3, 6, 8, 9, 10, 11, 12, 14, 15], [0, 1, 2, 4, 5, 6, 7, 9, 10, 12, 13, 14, 15, 16], [0, 3, 6, 7, 9, 10, 11, 13, 14, 15, 16], [0, 1, 3, 7, 8, 9, 10, 11, 12, 14, 15], [0, 1, 2, 3, 4, 7, 9, 10, 13, 14, 15, 16], [0, 1, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16], [0, 1, 2, 5, 7, 9, 11, 12, 13, 15, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 15], [0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 13, 14, 15], [2, 4, 5, 7, 8, 9, 10, 13, 16], [0, 1, 2, 3, 5, 7, 8, 9, 10, 16], [0, 1, 3, 4, 6, 7, 8, 10, 11, 14], [0, 2, 3, 4, 5, 6, 7, 10, 13, 15, 16], [1, 2, 3, 4, 5, 6, 8, 9, 10, 14, 16], [0, 3, 4, 6, 7, 8, 11, 12, 14, 15]]
**Current coloring state:**
[1, 1, 2, 3, 2, 2, 4, 5, 6, 7, 6, 1, 8, 8, 7, 0, 0]
**Explored next state that leads to an uncolorable path:**
[1, 1, 2, 3, 2, 2, 4, 5, 6, 7, 6, 1, 8, 8, 7, 5, 0]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Colorable (B) Uncolorable. | (B) | null |
game24_statetransition | You are given an initial Game of 24 configuration S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid solution requires using each of the four initial numbers exactly once, using only basic arithmetic operations (+ - * /), and ultimately evaluating to 24.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Applying an operation: Combining two expressions using a basic arithmetic operation (+ - * /), reducing the number of expressions by 1.
2. Reverting an operation: Removing the last operation applied to the expressions, increasing the number of expressions by 1.
**Initial configuration:**
S(0) = ['9', '10', '11', '12']
L(0) = Solvable
**Current state:**
S(i) = ['(11 - 10)', '9', '12']
L(i) = Unsolvable
**Explored next state:**
S(i+1) = ['((11 - 10) + 9)', '12']
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{expressions\}", where \{expressions\} is in the same python list format as the previous states. | null | {'parent_state': ['9', '10', '11', '12'], 'current_state': ['(11 - 10)', '9', '12'], 'current_status': 'Unsolvable', 'unsolvable_child': ['((11 - 10) + 9)', '12'], 'numbers': [9, 10, 11, 12]} |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[0, 6, 5, 0, 1, 0, 0, 0, 2], [0, 0, 0, 7, 8, 0, 0, 4, 0], [3, 0, 4, 0, 0, 2, 8, 1, 0], [0, 3, 9, 0, 0, 8, 7, 0, 0], [6, 4, 0, 5, 0, 0, 0, 2, 8], [7, 0, 0, 6, 0, 0, 0, 9, 1], [2, 0, 0, 0, 0, 1, 0, 3, 6], [0, 5, 0, 0, 7, 0, 0, 0, 9], [1, 0, 0, 4, 0, 9, 0, 0, 0]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[8, 6, 5, 3, 1, 4, 9, 7, 2], [9, 1, 2, 7, 8, 5, 6, 4, 3], [3, 7, 4, 9, 6, 2, 8, 1, 5], [5, 3, 9, 1, 2, 8, 7, 6, 4], [6, 4, 1, 5, 9, 7, 3, 2, 8], [7, 2, 8, 6, 4, 3, 5, 9, 1], [2, 9, 7, 8, 5, 1, 4, 3, 6], [0, 5, 0, 0, 7, 0, 0, 0, 9], [1, 0, 0, 4, 0, 9, 0, 0, 0]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[8, 6, 5, 3, 1, 4, 9, 7, 2], [9, 1, 2, 7, 8, 5, 6, 4, 3], [3, 7, 4, 9, 6, 2, 8, 1, 5], [5, 3, 9, 1, 2, 8, 7, 6, 4], [6, 4, 1, 5, 9, 7, 3, 2, 8], [7, 2, 8, 6, 4, 3, 5, 9, 1], [2, 9, 7, 8, 5, 1, 4, 3, 6], [4, 5, 0, 0, 7, 0, 0, 0, 9], [1, 0, 0, 4, 0, 9, 0, 0, 0]]
L(i-1) = Solvable
**Current state:**
S(i) = [[8, 6, 5, 3, 1, 4, 9, 7, 2], [9, 1, 2, 7, 8, 5, 6, 4, 3], [3, 7, 4, 9, 6, 2, 8, 1, 5], [5, 3, 9, 1, 2, 8, 7, 6, 4], [6, 4, 1, 5, 9, 7, 3, 2, 8], [7, 2, 8, 6, 4, 3, 5, 9, 1], [2, 9, 7, 8, 5, 1, 4, 3, 6], [4, 5, 6, 0, 7, 0, 0, 0, 9], [1, 0, 0, 4, 0, 9, 0, 0, 0]]
L(i) = Unsolvable
**Explored next state:**
S(i+1) = [[8, 6, 5, 3, 1, 4, 9, 7, 2], [9, 1, 2, 7, 8, 5, 6, 4, 3], [3, 7, 4, 9, 6, 2, 8, 1, 5], [5, 3, 9, 1, 2, 8, 7, 6, 4], [6, 4, 1, 5, 9, 7, 3, 2, 8], [7, 2, 8, 6, 4, 3, 5, 9, 1], [2, 9, 7, 8, 5, 1, 4, 3, 6], [4, 5, 6, 2, 7, 0, 0, 0, 9], [1, 0, 0, 4, 0, 9, 0, 0, 0]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[8, 6, 5, 3, 1, 4, 9, 7, 2], [9, 1, 2, 7, 8, 5, 6, 4, 3], [3, 7, 4, 9, 6, 2, 8, 1, 5], [5, 3, 9, 1, 2, 8, 7, 6, 4], [6, 4, 1, 5, 9, 7, 3, 2, 8], [7, 2, 8, 6, 4, 3, 5, 9, 1], [2, 9, 7, 8, 5, 1, 4, 3, 6], [4, 5, 0, 0, 7, 0, 0, 0, 9], [1, 0, 0, 4, 0, 9, 0, 0, 0]], 'current_state': [[8, 6, 5, 3, 1, 4, 9, 7, 2], [9, 1, 2, 7, 8, 5, 6, 4, 3], [3, 7, 4, 9, 6, 2, 8, 1, 5], [5, 3, 9, 1, 2, 8, 7, 6, 4], [6, 4, 1, 5, 9, 7, 3, 2, 8], [7, 2, 8, 6, 4, 3, 5, 9, 1], [2, 9, 7, 8, 5, 1, 4, 3, 6], [4, 5, 6, 0, 7, 0, 0, 0, 9], [1, 0, 0, 4, 0, 9, 0, 0, 0]], 'current_status': 'Unsolvable', 'unsolvable_child': [[8, 6, 5, 3, 1, 4, 9, 7, 2], [9, 1, 2, 7, 8, 5, 6, 4, 3], [3, 7, 4, 9, 6, 2, 8, 1, 5], [5, 3, 9, 1, 2, 8, 7, 6, 4], [6, 4, 1, 5, 9, 7, 3, 2, 8], [7, 2, 8, 6, 4, 3, 5, 9, 1], [2, 9, 7, 8, 5, 1, 4, 3, 6], [4, 5, 6, 2, 7, 0, 0, 0, 9], [1, 0, 0, 4, 0, 9, 0, 0, 0]]} |
game24_statetransition | You are given an initial Game of 24 configuration S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid solution requires using each of the four initial numbers exactly once, using only basic arithmetic operations (+ - * /), and ultimately evaluating to 24.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Applying an operation: Combining two expressions using a basic arithmetic operation (+ - * /), reducing the number of expressions by 1.
2. Reverting an operation: Removing the last operation applied to the expressions, increasing the number of expressions by 1.
**Initial configuration:**
S(0) = ['2', '2', '5', '11']
L(0) = Solvable
**Current state:**
S(i) = ['(5 - 2)', '2', '11']
L(i) = Unsolvable
**Explored next state:**
S(i+1) = ['((5 - 2) + 2)', '11']
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{expressions\}", where \{expressions\} is in the same python list format as the previous states. | null | {'parent_state': ['2', '2', '5', '11'], 'current_state': ['(5 - 2)', '2', '11'], 'current_status': 'Unsolvable', 'unsolvable_child': ['((5 - 2) + 2)', '11'], 'numbers': [2, 2, 5, 11]} |
game24_statetransition | You are given an initial Game of 24 configuration S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid solution requires using each of the four initial numbers exactly once, using only basic arithmetic operations (+ - * /), and ultimately evaluating to 24.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Applying an operation: Combining two expressions using a basic arithmetic operation (+ - * /), reducing the number of expressions by 1.
2. Reverting an operation: Removing the last operation applied to the expressions, increasing the number of expressions by 1.
**Initial configuration:**
S(0) = ['3', '4', '10', '13']
L(0) = Solvable
**Current state:**
S(i) = ['(10 + 13)', '3', '4']
L(i) = Solvable
**Explored next state:**
S(i+1) = ['((10 + 13) + 3)', '4']
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{expressions\}", where \{expressions\} is in the same python list format as the previous states. | null | {'parent_state': ['3', '4', '10', '13'], 'current_state': ['(10 + 13)', '3', '4'], 'current_status': 'Solvable', 'unsolvable_child': ['((10 + 13) + 3)', '4'], 'numbers': [3, 4, 10, 13]} |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[0, 0, 0, 0, 9, 6, 3, 0, 0], [7, 2, 3, 8, 0, 0, 6, 0, 0], [9, 6, 0, 2, 0, 0, 0, 4, 0], [4, 0, 5, 0, 7, 3, 0, 8, 9], [0, 0, 0, 0, 0, 0, 1, 3, 7], [0, 0, 0, 0, 1, 0, 5, 0, 0], [0, 0, 7, 3, 5, 0, 9, 0, 0], [6, 0, 0, 0, 0, 1, 0, 0, 8], [8, 0, 2, 4, 0, 9, 0, 1, 0]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[5, 8, 4, 1, 9, 6, 3, 7, 2], [7, 2, 3, 8, 4, 5, 6, 9, 1], [9, 6, 1, 2, 3, 7, 8, 4, 5], [4, 1, 5, 6, 7, 3, 2, 8, 9], [2, 9, 6, 5, 8, 4, 1, 3, 7], [3, 7, 8, 9, 1, 2, 5, 6, 4], [1, 4, 7, 3, 5, 8, 9, 0, 0], [6, 0, 0, 0, 0, 1, 0, 0, 8], [8, 0, 2, 4, 0, 9, 0, 1, 0]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[5, 8, 4, 1, 9, 6, 3, 7, 2], [7, 2, 3, 8, 4, 5, 6, 9, 1], [9, 6, 1, 2, 3, 7, 8, 4, 5], [4, 1, 5, 6, 7, 3, 2, 8, 9], [2, 9, 6, 5, 8, 4, 1, 3, 7], [3, 7, 8, 9, 1, 2, 5, 6, 4], [1, 4, 7, 3, 5, 8, 9, 2, 0], [6, 0, 0, 0, 0, 1, 0, 0, 8], [8, 0, 2, 4, 0, 9, 0, 1, 0]]
L(i-1) = Solvable
**Current state:**
S(i) = [[5, 8, 4, 1, 9, 6, 3, 7, 2], [7, 2, 3, 8, 4, 5, 6, 9, 1], [9, 6, 1, 2, 3, 7, 8, 4, 5], [4, 1, 5, 6, 7, 3, 2, 8, 9], [2, 9, 6, 5, 8, 4, 1, 3, 7], [3, 7, 8, 9, 1, 2, 5, 6, 4], [1, 4, 7, 3, 5, 8, 9, 2, 6], [6, 0, 0, 0, 0, 1, 0, 0, 8], [8, 0, 2, 4, 0, 9, 0, 1, 0]]
L(i) = Solvable
**Explored next state:**
S(i+1) = [[5, 8, 4, 1, 9, 6, 3, 7, 2], [7, 2, 3, 8, 4, 5, 6, 9, 1], [9, 6, 1, 2, 3, 7, 8, 4, 5], [4, 1, 5, 6, 7, 3, 2, 8, 9], [2, 9, 6, 5, 8, 4, 1, 3, 7], [3, 7, 8, 9, 1, 2, 5, 6, 4], [1, 4, 7, 3, 5, 8, 9, 2, 6], [6, 5, 0, 0, 0, 1, 0, 0, 8], [8, 0, 2, 4, 0, 9, 0, 1, 0]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[5, 8, 4, 1, 9, 6, 3, 7, 2], [7, 2, 3, 8, 4, 5, 6, 9, 1], [9, 6, 1, 2, 3, 7, 8, 4, 5], [4, 1, 5, 6, 7, 3, 2, 8, 9], [2, 9, 6, 5, 8, 4, 1, 3, 7], [3, 7, 8, 9, 1, 2, 5, 6, 4], [1, 4, 7, 3, 5, 8, 9, 2, 0], [6, 0, 0, 0, 0, 1, 0, 0, 8], [8, 0, 2, 4, 0, 9, 0, 1, 0]], 'current_state': [[5, 8, 4, 1, 9, 6, 3, 7, 2], [7, 2, 3, 8, 4, 5, 6, 9, 1], [9, 6, 1, 2, 3, 7, 8, 4, 5], [4, 1, 5, 6, 7, 3, 2, 8, 9], [2, 9, 6, 5, 8, 4, 1, 3, 7], [3, 7, 8, 9, 1, 2, 5, 6, 4], [1, 4, 7, 3, 5, 8, 9, 2, 6], [6, 0, 0, 0, 0, 1, 0, 0, 8], [8, 0, 2, 4, 0, 9, 0, 1, 0]], 'current_status': 'Solvable', 'unsolvable_child': [[5, 8, 4, 1, 9, 6, 3, 7, 2], [7, 2, 3, 8, 4, 5, 6, 9, 1], [9, 6, 1, 2, 3, 7, 8, 4, 5], [4, 1, 5, 6, 7, 3, 2, 8, 9], [2, 9, 6, 5, 8, 4, 1, 3, 7], [3, 7, 8, 9, 1, 2, 5, 6, 4], [1, 4, 7, 3, 5, 8, 9, 2, 6], [6, 5, 0, 0, 0, 1, 0, 0, 8], [8, 0, 2, 4, 0, 9, 0, 1, 0]]} |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[0, 0, 3, 0, 0, 1, 0, 0, 9], [0, 1, 4, 6, 0, 0, 0, 7, 0], [0, 2, 5, 0, 0, 9, 6, 0, 3], [7, 6, 0, 0, 0, 0, 0, 2, 0], [0, 0, 8, 2, 9, 0, 1, 5, 0], [0, 0, 0, 4, 0, 6, 8, 0, 0], [0, 4, 9, 0, 5, 3, 0, 0, 0], [0, 0, 0, 1, 0, 0, 3, 0, 8], [0, 0, 0, 0, 7, 2, 0, 4, 1]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[6, 7, 3, 5, 2, 1, 4, 8, 9], [9, 1, 4, 6, 3, 8, 2, 7, 5], [8, 2, 5, 7, 4, 9, 6, 1, 3], [7, 6, 1, 3, 8, 5, 9, 2, 4], [4, 3, 8, 2, 9, 7, 1, 5, 0], [0, 0, 0, 4, 0, 6, 8, 0, 0], [0, 4, 9, 0, 5, 3, 0, 0, 0], [0, 0, 0, 1, 0, 0, 3, 0, 8], [0, 0, 0, 0, 7, 2, 0, 4, 1]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[6, 7, 3, 5, 2, 1, 4, 8, 9], [9, 1, 4, 6, 3, 8, 2, 7, 5], [8, 2, 5, 7, 4, 9, 6, 1, 3], [7, 6, 1, 3, 8, 5, 9, 2, 4], [4, 3, 8, 2, 9, 7, 1, 5, 6], [0, 0, 0, 4, 0, 6, 8, 0, 0], [0, 4, 9, 0, 5, 3, 0, 0, 0], [0, 0, 0, 1, 0, 0, 3, 0, 8], [0, 0, 0, 0, 7, 2, 0, 4, 1]]
L(i-1) = Solvable
**Current state:**
S(i) = [[6, 7, 3, 5, 2, 1, 4, 8, 9], [9, 1, 4, 6, 3, 8, 2, 7, 5], [8, 2, 5, 7, 4, 9, 6, 1, 3], [7, 6, 1, 3, 8, 5, 9, 2, 4], [4, 3, 8, 2, 9, 7, 1, 5, 6], [2, 0, 0, 4, 0, 6, 8, 0, 0], [0, 4, 9, 0, 5, 3, 0, 0, 0], [0, 0, 0, 1, 0, 0, 3, 0, 8], [0, 0, 0, 0, 7, 2, 0, 4, 1]]
L(i) = Unsolvable
**Explored next state:**
S(i+1) = [[6, 7, 3, 5, 2, 1, 4, 8, 9], [9, 1, 4, 6, 3, 8, 2, 7, 5], [8, 2, 5, 7, 4, 9, 6, 1, 3], [7, 6, 1, 3, 8, 5, 9, 2, 4], [4, 3, 8, 2, 9, 7, 1, 5, 6], [2, 5, 0, 4, 0, 6, 8, 0, 0], [0, 4, 9, 0, 5, 3, 0, 0, 0], [0, 0, 0, 1, 0, 0, 3, 0, 8], [0, 0, 0, 0, 7, 2, 0, 4, 1]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[6, 7, 3, 5, 2, 1, 4, 8, 9], [9, 1, 4, 6, 3, 8, 2, 7, 5], [8, 2, 5, 7, 4, 9, 6, 1, 3], [7, 6, 1, 3, 8, 5, 9, 2, 4], [4, 3, 8, 2, 9, 7, 1, 5, 6], [0, 0, 0, 4, 0, 6, 8, 0, 0], [0, 4, 9, 0, 5, 3, 0, 0, 0], [0, 0, 0, 1, 0, 0, 3, 0, 8], [0, 0, 0, 0, 7, 2, 0, 4, 1]], 'current_state': [[6, 7, 3, 5, 2, 1, 4, 8, 9], [9, 1, 4, 6, 3, 8, 2, 7, 5], [8, 2, 5, 7, 4, 9, 6, 1, 3], [7, 6, 1, 3, 8, 5, 9, 2, 4], [4, 3, 8, 2, 9, 7, 1, 5, 6], [2, 0, 0, 4, 0, 6, 8, 0, 0], [0, 4, 9, 0, 5, 3, 0, 0, 0], [0, 0, 0, 1, 0, 0, 3, 0, 8], [0, 0, 0, 0, 7, 2, 0, 4, 1]], 'current_status': 'Unsolvable', 'unsolvable_child': [[6, 7, 3, 5, 2, 1, 4, 8, 9], [9, 1, 4, 6, 3, 8, 2, 7, 5], [8, 2, 5, 7, 4, 9, 6, 1, 3], [7, 6, 1, 3, 8, 5, 9, 2, 4], [4, 3, 8, 2, 9, 7, 1, 5, 6], [2, 5, 0, 4, 0, 6, 8, 0, 0], [0, 4, 9, 0, 5, 3, 0, 0, 0], [0, 0, 0, 1, 0, 0, 3, 0, 8], [0, 0, 0, 0, 7, 2, 0, 4, 1]]} |
graphcoloring_lookahead | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given the current coloring state of the graph in a list, where each index represents the corresonding vertex, and the number at that index represents its color (0 indicates an uncolored vertex).
Your task is to determine if this current state can lead to a valid coloring. Specifically, use lookahead techniques to determine if it's possible to color the remaining vertices such that no two adjacent vertices share the same color, using no more than 6 colors in total.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Graph adjacency list:**
[[1, 2, 3, 6, 7, 11, 12], [0, 2, 3, 5, 6, 8, 10, 12, 13, 14, 16], [0, 1, 3, 5, 9, 10, 12], [0, 1, 2, 4, 6, 9, 10, 12, 14], [3, 5, 7, 10, 11, 15, 16], [1, 2, 4, 8, 9, 10, 14, 15], [0, 1, 3, 7, 8, 9, 10], [0, 4, 6, 8, 9, 12, 14, 15], [1, 5, 6, 7, 9, 10, 12, 13, 14, 15, 16], [2, 3, 5, 6, 7, 8, 10, 11, 12, 16], [1, 2, 3, 4, 5, 6, 8, 9, 15, 16], [0, 4, 9, 13, 15, 16], [0, 1, 2, 3, 7, 8, 9, 14, 16], [1, 8, 11, 15, 16], [1, 3, 5, 7, 8, 12], [4, 5, 7, 8, 10, 11, 13], [1, 4, 8, 9, 10, 11, 12, 13]]
**Current coloring state:**
[1, 2, 3, 4, 1, 4, 3, 2, 1, 5, 6, 3, 6, 0, 0, 0, 0]
**Explored next state that leads to an uncolorable path:**
[1, 2, 3, 4, 1, 4, 3, 2, 1, 5, 6, 3, 6, 4, 0, 0, 0]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Colorable (B) Uncolorable. | (A) | null |
graphcoloring_statetransition | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given a sequence of partial coloring states leading to the current coloring state S(i). The coloring state is a list, where each index represents the corresonding vertex in the graph, and the number at that index represents its color (0 indicates an uncolored vertex). Alongside each state, its colorability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid coloring with no more than 12 colors. A valid coloring requires that no two adjacent vertices share the same color.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Coloring: Replacing a 0 in exactly one uncolored vertex with a value from 1 to 12.
2. Removing a color: Replacing a value in exactly one colored vertex with 0.
**Graph adjacency list:**
[[1, 2, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 20, 21, 22], [0, 2, 3, 4, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22], [0, 1, 3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22], [1, 2, 5, 6, 7, 9, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22], [0, 1, 2, 5, 6, 7, 8, 9, 10, 11, 13, 15, 16, 18, 20, 22], [0, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 19, 20, 21, 22], [3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22], [0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22], [0, 1, 2, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [0, 1, 3, 4, 5, 6, 7, 8, 11, 12, 14, 15, 17, 19, 21, 22], [0, 1, 2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 19, 20, 21], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 20, 21, 22], [1, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22], [1, 2, 3, 5, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, 22], [0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 22], [0, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 22], [0, 1, 2, 3, 6, 8, 9, 10, 12, 13, 14, 15, 16, 19, 20, 21, 22], [0, 1, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22], [1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 21, 22], [0, 1, 2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 17, 18, 21, 22], [0, 1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 15, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20]]
**Two moves ago:**
S(i-2) = [1, 2, 3, 1, 4, 2, 3, 5, 5, 6, 6, 7, 4, 1, 7, 8, 9, 10, 0, 0, 0, 0, 0]
L(i-2) = Colorable
**One move ago:**
S(i-1) = [1, 2, 3, 1, 4, 2, 3, 5, 5, 6, 6, 7, 4, 1, 7, 8, 9, 10, 10, 0, 0, 0, 0]
L(i-1) = Colorable
**Current coloring state:**
S(i) = [1, 2, 3, 1, 4, 2, 3, 5, 5, 6, 6, 7, 4, 1, 7, 8, 9, 10, 10, 11, 0, 0, 0]
L(i) = Colorable
**Explored next state:**
S(i+1) = [1, 2, 3, 1, 4, 2, 3, 5, 5, 6, 6, 7, 4, 1, 7, 8, 9, 10, 10, 11, 12, 0, 0]
L(i+1) = Uncolorable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{coloring\}", where \{coloring\} is in the same python list format as the previous states. | null | {'parent_state': [1, 2, 3, 1, 4, 2, 3, 5, 5, 6, 6, 7, 4, 1, 7, 8, 9, 10, 10, 0, 0, 0, 0], 'current_state': [1, 2, 3, 1, 4, 2, 3, 5, 5, 6, 6, 7, 4, 1, 7, 8, 9, 10, 10, 11, 0, 0, 0], 'current_status': 'Solvable', 'unsolvable_child': [1, 2, 3, 1, 4, 2, 3, 5, 5, 6, 6, 7, 4, 1, 7, 8, 9, 10, 10, 11, 12, 0, 0], 'graph': [[1, 2, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 20, 21, 22], [0, 2, 3, 4, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22], [0, 1, 3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22], [1, 2, 5, 6, 7, 9, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22], [0, 1, 2, 5, 6, 7, 8, 9, 10, 11, 13, 15, 16, 18, 20, 22], [0, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 19, 20, 21, 22], [3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22], [0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22], [0, 1, 2, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [0, 1, 3, 4, 5, 6, 7, 8, 11, 12, 14, 15, 17, 19, 21, 22], [0, 1, 2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 19, 20, 21], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 20, 21, 22], [1, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22], [1, 2, 3, 5, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, 22], [0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 22], [0, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 22], [0, 1, 2, 3, 6, 8, 9, 10, 12, 13, 14, 15, 16, 19, 20, 21, 22], [0, 1, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22], [1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 21, 22], [0, 1, 2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 17, 18, 21, 22], [0, 1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 15, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20]], 'chromatic_number': 12} |
graphcoloring_statetransition | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given a sequence of partial coloring states leading to the current coloring state S(i). The coloring state is a list, where each index represents the corresonding vertex in the graph, and the number at that index represents its color (0 indicates an uncolored vertex). Alongside each state, its colorability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid coloring with no more than 5 colors. A valid coloring requires that no two adjacent vertices share the same color.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Coloring: Replacing a 0 in exactly one uncolored vertex with a value from 1 to 5.
2. Removing a color: Replacing a value in exactly one colored vertex with 0.
**Graph adjacency list:**
[[1, 5, 6, 10, 12, 13], [0, 2, 7, 10, 11, 12, 14], [1, 3, 4, 7, 12, 13], [2, 5, 7, 10, 13], [2, 5, 7, 9, 11, 13], [0, 3, 4, 7, 9, 12], [0, 8, 10, 11, 14], [1, 2, 3, 4, 5, 8, 9, 11, 14], [6, 7, 9, 10, 12, 13], [4, 5, 7, 8, 11, 12, 13, 14], [0, 1, 3, 6, 8, 13], [1, 4, 6, 7, 9], [0, 1, 2, 5, 8, 9], [0, 2, 3, 4, 8, 9, 10, 14], [1, 6, 7, 9, 13]]
**Two moves ago:**
S(i-2) = [1, 2, 1, 2, 2, 3, 3, 4, 2, 1, 4, 0, 0, 0, 0]
L(i-2) = Colorable
**One move ago:**
S(i-1) = [1, 2, 1, 2, 2, 3, 3, 4, 2, 1, 4, 5, 0, 0, 0]
L(i-1) = Colorable
**Current coloring state:**
S(i) = [1, 2, 1, 2, 2, 3, 3, 4, 2, 1, 4, 5, 5, 0, 0]
L(i) = Colorable
**Explored next state:**
S(i+1) = [1, 2, 1, 2, 2, 3, 3, 4, 2, 1, 4, 5, 5, 5, 0]
L(i+1) = Uncolorable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{coloring\}", where \{coloring\} is in the same python list format as the previous states. | null | {'parent_state': [1, 2, 1, 2, 2, 3, 3, 4, 2, 1, 4, 5, 0, 0, 0], 'current_state': [1, 2, 1, 2, 2, 3, 3, 4, 2, 1, 4, 5, 5, 0, 0], 'current_status': 'Solvable', 'unsolvable_child': [1, 2, 1, 2, 2, 3, 3, 4, 2, 1, 4, 5, 5, 5, 0], 'graph': [[1, 5, 6, 10, 12, 13], [0, 2, 7, 10, 11, 12, 14], [1, 3, 4, 7, 12, 13], [2, 5, 7, 10, 13], [2, 5, 7, 9, 11, 13], [0, 3, 4, 7, 9, 12], [0, 8, 10, 11, 14], [1, 2, 3, 4, 5, 8, 9, 11, 14], [6, 7, 9, 10, 12, 13], [4, 5, 7, 8, 11, 12, 13, 14], [0, 1, 3, 6, 8, 13], [1, 4, 6, 7, 9], [0, 1, 2, 5, 8, 9], [0, 2, 3, 4, 8, 9, 10, 14], [1, 6, 7, 9, 13]], 'chromatic_number': 5} |
sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[3, 9, 8, 2, 6, 1, 7, 4, 5], [2, 7, 5, 9, 4, 8, 3, 1, 6], [1, 4, 6, 3, 7, 5, 8, 2, 9], [6, 3, 7, 4, 2, 9, 5, 8, 1], [8, 2, 9, 0, 0, 3, 4, 0, 0], [0, 5, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 8, 0, 2, 1, 9, 0], [0, 1, 4, 0, 0, 6, 0, 5, 0], [0, 8, 0, 0, 9, 0, 0, 7, 3]]
Explored next state that leads to an unsolvable path:
[[3, 9, 8, 2, 6, 1, 7, 4, 5], [2, 7, 5, 9, 4, 8, 3, 1, 6], [1, 4, 6, 3, 7, 5, 8, 2, 9], [6, 3, 7, 4, 2, 9, 5, 8, 1], [8, 2, 9, 1, 0, 3, 4, 0, 0], [0, 5, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 8, 0, 2, 1, 9, 0], [0, 1, 4, 0, 0, 6, 0, 5, 0], [0, 8, 0, 0, 9, 0, 0, 7, 3]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (A) | null |
graphcoloring_statetransition | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given a sequence of partial coloring states leading to the current coloring state S(i). The coloring state is a list, where each index represents the corresonding vertex in the graph, and the number at that index represents its color (0 indicates an uncolored vertex). Alongside each state, its colorability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid coloring with no more than 12 colors. A valid coloring requires that no two adjacent vertices share the same color.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Coloring: Replacing a 0 in exactly one uncolored vertex with a value from 1 to 12.
2. Removing a color: Replacing a value in exactly one colored vertex with 0.
**Graph adjacency list:**
[[1, 2, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 20, 21, 22], [0, 2, 3, 4, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22], [0, 1, 3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22], [1, 2, 5, 6, 7, 9, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22], [0, 1, 2, 5, 6, 7, 8, 9, 10, 11, 13, 15, 16, 18, 20, 22], [0, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 19, 20, 21, 22], [3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22], [0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22], [0, 1, 2, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [0, 1, 3, 4, 5, 6, 7, 8, 11, 12, 14, 15, 17, 19, 21, 22], [0, 1, 2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 19, 20, 21], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 20, 21, 22], [1, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22], [1, 2, 3, 5, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, 22], [0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 22], [0, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 22], [0, 1, 2, 3, 6, 8, 9, 10, 12, 13, 14, 15, 16, 19, 20, 21, 22], [0, 1, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22], [1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 21, 22], [0, 1, 2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 17, 18, 21, 22], [0, 1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 15, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20]]
**Two moves ago:**
S(i-2) = [1, 2, 3, 1, 4, 2, 3, 5, 5, 6, 6, 7, 4, 1, 7, 8, 9, 0, 0, 0, 0, 0, 0]
L(i-2) = Colorable
**One move ago:**
S(i-1) = [1, 2, 3, 1, 4, 2, 3, 5, 5, 6, 6, 7, 4, 1, 7, 8, 9, 10, 0, 0, 0, 0, 0]
L(i-1) = Colorable
**Current coloring state:**
S(i) = [1, 2, 3, 1, 4, 2, 3, 5, 5, 6, 6, 7, 4, 1, 7, 8, 9, 10, 11, 0, 0, 0, 0]
L(i) = Uncolorable
**Explored next state:**
S(i+1) = [1, 2, 3, 1, 4, 2, 3, 5, 5, 6, 6, 7, 4, 1, 7, 8, 9, 10, 11, 12, 0, 0, 0]
L(i+1) = Uncolorable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{coloring\}", where \{coloring\} is in the same python list format as the previous states. | null | {'parent_state': [1, 2, 3, 1, 4, 2, 3, 5, 5, 6, 6, 7, 4, 1, 7, 8, 9, 10, 0, 0, 0, 0, 0], 'current_state': [1, 2, 3, 1, 4, 2, 3, 5, 5, 6, 6, 7, 4, 1, 7, 8, 9, 10, 11, 0, 0, 0, 0], 'current_status': 'Unsolvable', 'unsolvable_child': [1, 2, 3, 1, 4, 2, 3, 5, 5, 6, 6, 7, 4, 1, 7, 8, 9, 10, 11, 12, 0, 0, 0], 'graph': [[1, 2, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 20, 21, 22], [0, 2, 3, 4, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22], [0, 1, 3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22], [1, 2, 5, 6, 7, 9, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22], [0, 1, 2, 5, 6, 7, 8, 9, 10, 11, 13, 15, 16, 18, 20, 22], [0, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 19, 20, 21, 22], [3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22], [0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22], [0, 1, 2, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [0, 1, 3, 4, 5, 6, 7, 8, 11, 12, 14, 15, 17, 19, 21, 22], [0, 1, 2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 19, 20, 21], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 20, 21, 22], [1, 2, 3, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21], [1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22], [1, 2, 3, 5, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, 22], [0, 1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19, 21, 22], [0, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 22], [0, 1, 2, 3, 6, 8, 9, 10, 12, 13, 14, 15, 16, 19, 20, 21, 22], [0, 1, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22], [1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 21, 22], [0, 1, 2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 17, 18, 21, 22], [0, 1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 15, 17, 18, 19, 20], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20]], 'chromatic_number': 12} |
game24_statetransition | You are given an initial Game of 24 configuration S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid solution requires using each of the four initial numbers exactly once, using only basic arithmetic operations (+ - * /), and ultimately evaluating to 24.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Applying an operation: Combining two expressions using a basic arithmetic operation (+ - * /), reducing the number of expressions by 1.
2. Reverting an operation: Removing the last operation applied to the expressions, increasing the number of expressions by 1.
**Initial configuration:**
S(0) = ['3', '7', '7', '8']
L(0) = Solvable
**Current state:**
S(i) = ['(3 * 7)', '7', '8']
L(i) = Solvable
**Explored next state:**
S(i+1) = ['((3 * 7) + 7)', '8']
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{expressions\}", where \{expressions\} is in the same python list format as the previous states. | null | {'parent_state': ['3', '7', '7', '8'], 'current_state': ['(3 * 7)', '7', '8'], 'current_status': 'Solvable', 'unsolvable_child': ['((3 * 7) + 7)', '8'], 'numbers': [3, 7, 7, 8]} |
graphcoloring_lookahead | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given the current coloring state of the graph in a list, where each index represents the corresonding vertex, and the number at that index represents its color (0 indicates an uncolored vertex).
Your task is to determine if this current state can lead to a valid coloring. Specifically, use lookahead techniques to determine if it's possible to color the remaining vertices such that no two adjacent vertices share the same color, using no more than 10 colors in total.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Graph adjacency list:**
[[2, 3, 4, 6, 7, 8, 12, 13, 14, 15, 16, 17], [2, 3, 4, 5, 7, 8, 9, 12, 13, 14, 16, 17, 18, 19], [0, 1, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, 19], [0, 1, 2, 4, 6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19], [1, 2, 4, 6, 8, 9, 10, 11, 13, 15, 16, 17, 18], [0, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19], [0, 1, 2, 4, 6, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19], [0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 16, 17, 19], [3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19], [2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 17, 18, 19], [0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 19], [0, 1, 2, 3, 6, 8, 10, 11, 12, 13, 15, 16, 17, 19], [0, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 16, 17, 19], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19], [0, 1, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19], [1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 16, 17, 19], [1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]]
**Current coloring state:**
[1, 1, 2, 3, 4, 3, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
**Explored next state that leads to an uncolorable path:**
[1, 1, 2, 3, 4, 3, 5, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Colorable (B) Uncolorable. | (B) | null |
game24_lookahead | You are given four numbers and the current calculation state for the Game of 24.
Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if the remaining numbers can be combined using basic arithmetic operations (+ - * /) to reach exactly 24. You must use each number exactly once.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Numbers:**
['2', '4', '5', '8']
**Current calculation state:**
['(2 * 5)', '4', '8']
**Explored next state that leads to an unsolvable path:**
['((2 * 5) + 4)', '8']
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (B) | null |
game24_lookahead | You are given four numbers and the current calculation state for the Game of 24.
Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if the remaining numbers can be combined using basic arithmetic operations (+ - * /) to reach exactly 24. You must use each number exactly once.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Numbers:**
['6', '6', '7', '9']
**Current calculation state:**
['(9 - 7)', '6', '6']
**Explored next state that leads to an unsolvable path:**
['((9 - 7) + 6)', '6']
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (A) | null |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[0, 0, 3, 0, 8, 0, 5, 0, 0], [7, 6, 4, 0, 0, 0, 0, 0, 8], [0, 0, 0, 6, 1, 9, 0, 3, 0], [0, 0, 6, 8, 0, 3, 7, 0, 1], [0, 7, 0, 9, 0, 2, 3, 0, 0], [0, 9, 5, 0, 0, 0, 8, 2, 6], [0, 8, 0, 0, 4, 6, 0, 0, 0], [0, 0, 0, 1, 0, 0, 2, 5, 0], [4, 2, 0, 0, 3, 7, 0, 0, 9]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[9, 1, 3, 7, 8, 4, 5, 6, 2], [7, 6, 4, 3, 2, 5, 9, 1, 8], [8, 5, 2, 6, 1, 9, 4, 3, 7], [2, 4, 6, 8, 5, 3, 7, 9, 1], [1, 7, 8, 9, 6, 2, 3, 4, 5], [3, 9, 5, 4, 7, 1, 8, 2, 6], [0, 8, 0, 0, 4, 6, 0, 0, 0], [0, 0, 0, 1, 0, 0, 2, 5, 0], [4, 2, 0, 0, 3, 7, 0, 0, 9]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[9, 1, 3, 7, 8, 4, 5, 6, 2], [7, 6, 4, 3, 2, 5, 9, 1, 8], [8, 5, 2, 6, 1, 9, 4, 3, 7], [2, 4, 6, 8, 5, 3, 7, 9, 1], [1, 7, 8, 9, 6, 2, 3, 4, 5], [3, 9, 5, 4, 7, 1, 8, 2, 6], [5, 8, 0, 0, 4, 6, 0, 0, 0], [0, 0, 0, 1, 0, 0, 2, 5, 0], [4, 2, 0, 0, 3, 7, 0, 0, 9]]
L(i-1) = Solvable
**Current state:**
S(i) = [[9, 1, 3, 7, 8, 4, 5, 6, 2], [7, 6, 4, 3, 2, 5, 9, 1, 8], [8, 5, 2, 6, 1, 9, 4, 3, 7], [2, 4, 6, 8, 5, 3, 7, 9, 1], [1, 7, 8, 9, 6, 2, 3, 4, 5], [3, 9, 5, 4, 7, 1, 8, 2, 6], [5, 8, 1, 0, 4, 6, 0, 0, 0], [0, 0, 0, 1, 0, 0, 2, 5, 0], [4, 2, 0, 0, 3, 7, 0, 0, 9]]
L(i) = Unsolvable
**Explored next state:**
S(i+1) = [[9, 1, 3, 7, 8, 4, 5, 6, 2], [7, 6, 4, 3, 2, 5, 9, 1, 8], [8, 5, 2, 6, 1, 9, 4, 3, 7], [2, 4, 6, 8, 5, 3, 7, 9, 1], [1, 7, 8, 9, 6, 2, 3, 4, 5], [3, 9, 5, 4, 7, 1, 8, 2, 6], [5, 8, 1, 2, 4, 6, 0, 0, 0], [0, 0, 0, 1, 0, 0, 2, 5, 0], [4, 2, 0, 0, 3, 7, 0, 0, 9]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[9, 1, 3, 7, 8, 4, 5, 6, 2], [7, 6, 4, 3, 2, 5, 9, 1, 8], [8, 5, 2, 6, 1, 9, 4, 3, 7], [2, 4, 6, 8, 5, 3, 7, 9, 1], [1, 7, 8, 9, 6, 2, 3, 4, 5], [3, 9, 5, 4, 7, 1, 8, 2, 6], [5, 8, 0, 0, 4, 6, 0, 0, 0], [0, 0, 0, 1, 0, 0, 2, 5, 0], [4, 2, 0, 0, 3, 7, 0, 0, 9]], 'current_state': [[9, 1, 3, 7, 8, 4, 5, 6, 2], [7, 6, 4, 3, 2, 5, 9, 1, 8], [8, 5, 2, 6, 1, 9, 4, 3, 7], [2, 4, 6, 8, 5, 3, 7, 9, 1], [1, 7, 8, 9, 6, 2, 3, 4, 5], [3, 9, 5, 4, 7, 1, 8, 2, 6], [5, 8, 1, 0, 4, 6, 0, 0, 0], [0, 0, 0, 1, 0, 0, 2, 5, 0], [4, 2, 0, 0, 3, 7, 0, 0, 9]], 'current_status': 'Unsolvable', 'unsolvable_child': [[9, 1, 3, 7, 8, 4, 5, 6, 2], [7, 6, 4, 3, 2, 5, 9, 1, 8], [8, 5, 2, 6, 1, 9, 4, 3, 7], [2, 4, 6, 8, 5, 3, 7, 9, 1], [1, 7, 8, 9, 6, 2, 3, 4, 5], [3, 9, 5, 4, 7, 1, 8, 2, 6], [5, 8, 1, 2, 4, 6, 0, 0, 0], [0, 0, 0, 1, 0, 0, 2, 5, 0], [4, 2, 0, 0, 3, 7, 0, 0, 9]]} |
sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[5, 6, 4, 9, 8, 1, 7, 2, 3], [3, 0, 8, 0, 4, 6, 0, 5, 0], [0, 2, 0, 0, 5, 0, 6, 4, 0], [0, 0, 5, 0, 0, 9, 8, 1, 0], [0, 0, 6, 4, 2, 0, 0, 0, 9], [7, 0, 0, 0, 0, 0, 4, 0, 0], [0, 5, 0, 1, 0, 0, 0, 8, 0], [2, 8, 3, 0, 6, 0, 0, 0, 7], [0, 4, 9, 7, 3, 0, 2, 0, 5]]
Explored next state that leads to an unsolvable path:
[[5, 6, 4, 9, 8, 1, 7, 2, 3], [3, 1, 8, 0, 4, 6, 0, 5, 0], [0, 2, 0, 0, 5, 0, 6, 4, 0], [0, 0, 5, 0, 0, 9, 8, 1, 0], [0, 0, 6, 4, 2, 0, 0, 0, 9], [7, 0, 0, 0, 0, 0, 4, 0, 0], [0, 5, 0, 1, 0, 0, 0, 8, 0], [2, 8, 3, 0, 6, 0, 0, 0, 7], [0, 4, 9, 7, 3, 0, 2, 0, 5]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (A) | null |
graphcoloring_lookahead | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given the current coloring state of the graph in a list, where each index represents the corresonding vertex, and the number at that index represents its color (0 indicates an uncolored vertex).
Your task is to determine if this current state can lead to a valid coloring. Specifically, use lookahead techniques to determine if it's possible to color the remaining vertices such that no two adjacent vertices share the same color, using no more than 6 colors in total.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Graph adjacency list:**
[[2, 3, 5, 7, 9, 10, 11, 12, 15, 16], [3, 4, 12, 14, 15], [0, 3, 4, 6, 7, 9, 10, 11, 14, 15, 16], [0, 1, 2, 4, 6, 7, 11, 13, 15], [1, 2, 3, 5, 6, 7, 8, 10, 12, 14, 15], [0, 4, 6, 7, 10, 11, 12, 14, 15, 16], [2, 3, 4, 5, 7, 11, 16], [0, 2, 3, 4, 5, 6, 10, 11, 12, 14, 15], [4, 9, 10, 12, 14, 15], [0, 2, 8, 12, 15], [0, 2, 4, 5, 7, 8, 15], [0, 2, 3, 5, 6, 7, 16], [0, 1, 4, 5, 7, 8, 9, 16], [3, 14], [1, 2, 4, 5, 7, 8, 13, 15], [0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 14, 16], [0, 2, 5, 6, 11, 12, 15]]
**Current coloring state:**
[1, 1, 2, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
**Explored next state that leads to an uncolorable path:**
[1, 1, 2, 3, 4, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Colorable (B) Uncolorable. | (A) | null |
graphcoloring_lookahead | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given the current coloring state of the graph in a list, where each index represents the corresonding vertex, and the number at that index represents its color (0 indicates an uncolored vertex).
Your task is to determine if this current state can lead to a valid coloring. Specifically, use lookahead techniques to determine if it's possible to color the remaining vertices such that no two adjacent vertices share the same color, using no more than 6 colors in total.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Graph adjacency list:**
[[2, 3, 6, 7, 14, 16, 17, 19], [3, 4, 5, 8, 9, 10, 13, 14, 15, 16, 17, 19], [0, 3, 5, 6, 8, 11, 13, 14, 15, 19], [0, 1, 2, 5, 6, 9, 10, 14, 15, 16, 17, 18, 20], [1, 6, 7, 8, 10, 12, 16, 17, 18], [1, 2, 3, 6, 8, 9, 10, 11, 13, 18], [0, 2, 3, 4, 5, 9, 11, 15, 16, 17, 18], [0, 4, 8, 9, 11, 15, 17, 20], [1, 2, 4, 5, 7, 10, 11, 14, 15, 16, 17, 18], [1, 3, 5, 6, 7, 11, 12, 13, 14, 15, 17, 19, 20], [1, 3, 4, 5, 8, 16, 18], [2, 5, 6, 7, 8, 9, 13, 15, 16, 17, 18], [4, 9, 13, 14, 15, 20], [1, 2, 5, 9, 11, 12, 14, 15, 19, 20], [0, 1, 2, 3, 8, 9, 12, 13, 15, 16, 19], [1, 2, 3, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 19], [0, 1, 3, 4, 6, 8, 10, 11, 14, 15], [0, 1, 3, 4, 6, 7, 8, 9, 11, 15, 19, 20], [3, 4, 5, 6, 8, 10, 11, 19], [0, 1, 2, 9, 13, 14, 15, 17, 18, 20], [3, 7, 9, 12, 13, 17, 19]]
**Current coloring state:**
[1, 1, 2, 3, 2, 4, 5, 3, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
**Explored next state that leads to an uncolorable path:**
[1, 1, 2, 3, 2, 4, 5, 3, 5, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Colorable (B) Uncolorable. | (B) | null |
graphcoloring_lookahead | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given the current coloring state of the graph in a list, where each index represents the corresonding vertex, and the number at that index represents its color (0 indicates an uncolored vertex).
Your task is to determine if this current state can lead to a valid coloring. Specifically, use lookahead techniques to determine if it's possible to color the remaining vertices such that no two adjacent vertices share the same color, using no more than 7 colors in total.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Graph adjacency list:**
[[1, 3, 4, 5, 6, 9, 11, 12, 13, 15, 16], [0, 2, 3, 4, 5, 6, 7, 10, 12, 13, 14, 16], [1, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 4, 5, 6, 7, 8, 10, 12, 13, 14, 16], [0, 1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [0, 1, 2, 3, 6, 9, 11, 13, 15, 16], [0, 1, 2, 3, 4, 5, 7, 10, 12, 13, 14, 15, 16], [1, 2, 3, 4, 6, 8, 10, 11, 14, 15], [3, 4, 7, 9, 11, 12, 13, 15, 16], [0, 2, 4, 5, 8, 10, 12, 13, 14, 15, 16], [1, 2, 3, 4, 6, 7, 9, 12, 15, 16], [0, 2, 4, 5, 7, 8, 13, 15, 16], [0, 1, 2, 3, 4, 6, 8, 9, 10, 14, 15, 16], [0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 16], [1, 2, 3, 4, 6, 7, 9, 12, 13, 15, 16], [0, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16], [0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15]]
**Current coloring state:**
[1, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
**Explored next state that leads to an uncolorable path:**
[1, 2, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Colorable (B) Uncolorable. | (A) | null |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[3, 0, 2, 0, 0, 0, 0, 7, 0], [0, 8, 0, 0, 3, 0, 0, 1, 6], [5, 0, 0, 0, 0, 1, 0, 0, 0], [8, 0, 0, 0, 0, 0, 4, 5, 0], [0, 0, 0, 7, 6, 8, 9, 0, 0], [7, 0, 1, 0, 2, 0, 3, 0, 0], [0, 5, 9, 0, 0, 4, 6, 0, 2], [0, 0, 0, 0, 1, 2, 0, 3, 9], [0, 0, 3, 0, 8, 6, 0, 0, 0]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[3, 1, 2, 6, 5, 9, 8, 7, 4], [9, 8, 4, 2, 3, 7, 5, 1, 6], [5, 6, 7, 8, 4, 1, 2, 9, 3], [8, 2, 6, 1, 9, 3, 4, 5, 7], [4, 3, 5, 7, 6, 8, 9, 2, 1], [7, 9, 1, 4, 2, 5, 3, 6, 8], [1, 5, 9, 3, 7, 4, 6, 0, 2], [0, 0, 0, 0, 1, 2, 0, 3, 9], [0, 0, 3, 0, 8, 6, 0, 0, 0]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[3, 1, 2, 6, 5, 9, 8, 7, 4], [9, 8, 4, 2, 3, 7, 5, 1, 6], [5, 6, 7, 8, 4, 1, 2, 9, 3], [8, 2, 6, 1, 9, 3, 4, 5, 7], [4, 3, 5, 7, 6, 8, 9, 2, 1], [7, 9, 1, 4, 2, 5, 3, 6, 8], [1, 5, 9, 3, 7, 4, 6, 8, 2], [0, 0, 0, 0, 1, 2, 0, 3, 9], [0, 0, 3, 0, 8, 6, 0, 0, 0]]
L(i-1) = Solvable
**Current state:**
S(i) = [[3, 1, 2, 6, 5, 9, 8, 7, 4], [9, 8, 4, 2, 3, 7, 5, 1, 6], [5, 6, 7, 8, 4, 1, 2, 9, 3], [8, 2, 6, 1, 9, 3, 4, 5, 7], [4, 3, 5, 7, 6, 8, 9, 2, 1], [7, 9, 1, 4, 2, 5, 3, 6, 8], [1, 5, 9, 3, 7, 4, 6, 8, 2], [6, 0, 0, 0, 1, 2, 0, 3, 9], [0, 0, 3, 0, 8, 6, 0, 0, 0]]
L(i) = Solvable
**Explored next state:**
S(i+1) = [[3, 1, 2, 6, 5, 9, 8, 7, 4], [9, 8, 4, 2, 3, 7, 5, 1, 6], [5, 6, 7, 8, 4, 1, 2, 9, 3], [8, 2, 6, 1, 9, 3, 4, 5, 7], [4, 3, 5, 7, 6, 8, 9, 2, 1], [7, 9, 1, 4, 2, 5, 3, 6, 8], [1, 5, 9, 3, 7, 4, 6, 8, 2], [6, 7, 0, 0, 1, 2, 0, 3, 9], [0, 0, 3, 0, 8, 6, 0, 0, 0]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[3, 1, 2, 6, 5, 9, 8, 7, 4], [9, 8, 4, 2, 3, 7, 5, 1, 6], [5, 6, 7, 8, 4, 1, 2, 9, 3], [8, 2, 6, 1, 9, 3, 4, 5, 7], [4, 3, 5, 7, 6, 8, 9, 2, 1], [7, 9, 1, 4, 2, 5, 3, 6, 8], [1, 5, 9, 3, 7, 4, 6, 8, 2], [0, 0, 0, 0, 1, 2, 0, 3, 9], [0, 0, 3, 0, 8, 6, 0, 0, 0]], 'current_state': [[3, 1, 2, 6, 5, 9, 8, 7, 4], [9, 8, 4, 2, 3, 7, 5, 1, 6], [5, 6, 7, 8, 4, 1, 2, 9, 3], [8, 2, 6, 1, 9, 3, 4, 5, 7], [4, 3, 5, 7, 6, 8, 9, 2, 1], [7, 9, 1, 4, 2, 5, 3, 6, 8], [1, 5, 9, 3, 7, 4, 6, 8, 2], [6, 0, 0, 0, 1, 2, 0, 3, 9], [0, 0, 3, 0, 8, 6, 0, 0, 0]], 'current_status': 'Solvable', 'unsolvable_child': [[3, 1, 2, 6, 5, 9, 8, 7, 4], [9, 8, 4, 2, 3, 7, 5, 1, 6], [5, 6, 7, 8, 4, 1, 2, 9, 3], [8, 2, 6, 1, 9, 3, 4, 5, 7], [4, 3, 5, 7, 6, 8, 9, 2, 1], [7, 9, 1, 4, 2, 5, 3, 6, 8], [1, 5, 9, 3, 7, 4, 6, 8, 2], [6, 7, 0, 0, 1, 2, 0, 3, 9], [0, 0, 3, 0, 8, 6, 0, 0, 0]]} |
sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[2, 4, 6, 9, 7, 8, 3, 1, 5], [9, 3, 8, 2, 5, 1, 4, 7, 6], [5, 1, 7, 6, 3, 4, 9, 2, 8], [7, 6, 1, 8, 4, 3, 5, 9, 2], [3, 8, 2, 1, 9, 5, 6, 4, 7], [4, 5, 9, 7, 2, 6, 8, 3, 1], [1, 2, 0, 0, 6, 9, 0, 8, 0], [6, 0, 4, 0, 8, 0, 1, 0, 0], [8, 9, 0, 0, 0, 7, 2, 0, 3]]
Explored next state that leads to an unsolvable path:
[[2, 4, 6, 9, 7, 8, 3, 1, 5], [9, 3, 8, 2, 5, 1, 4, 7, 6], [5, 1, 7, 6, 3, 4, 9, 2, 8], [7, 6, 1, 8, 4, 3, 5, 9, 2], [3, 8, 2, 1, 9, 5, 6, 4, 7], [4, 5, 9, 7, 2, 6, 8, 3, 1], [1, 2, 5, 0, 6, 9, 0, 8, 0], [6, 0, 4, 0, 8, 0, 1, 0, 0], [8, 9, 0, 0, 0, 7, 2, 0, 3]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (A) | null |
graphcoloring_statetransition | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given a sequence of partial coloring states leading to the current coloring state S(i). The coloring state is a list, where each index represents the corresonding vertex in the graph, and the number at that index represents its color (0 indicates an uncolored vertex). Alongside each state, its colorability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid coloring with no more than 7 colors. A valid coloring requires that no two adjacent vertices share the same color.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Coloring: Replacing a 0 in exactly one uncolored vertex with a value from 1 to 7.
2. Removing a color: Replacing a value in exactly one colored vertex with 0.
**Graph adjacency list:**
[[1, 4, 6, 10, 11, 12, 16, 20, 21], [0, 3, 4, 5, 6, 7, 9, 11, 12, 13, 15, 16, 18, 20, 21], [3, 6, 7, 11, 12, 14, 15, 17, 20, 21], [1, 2, 7, 8, 11, 12, 15, 16, 18, 19], [0, 1, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21], [1, 6, 7, 11, 12, 14, 16, 17], [0, 1, 2, 4, 5, 7, 10, 11, 13, 14, 15], [1, 2, 3, 4, 5, 6, 12, 15, 16, 17, 19], [3, 4, 10, 14, 15, 16, 20, 21], [1, 4, 11, 12, 13, 14, 15, 16, 17, 18, 20], [0, 4, 6, 8, 13, 15, 18, 21], [0, 1, 2, 3, 4, 5, 6, 9, 16, 17, 21], [0, 1, 2, 3, 5, 7, 9, 13, 14, 17, 20, 21], [1, 4, 6, 9, 10, 12, 14, 15, 16, 17, 18, 19], [2, 4, 5, 6, 8, 9, 12, 13, 16, 18, 20, 21], [1, 2, 3, 4, 6, 7, 8, 9, 10, 13, 18, 20], [0, 1, 3, 4, 5, 7, 8, 9, 11, 13, 14, 18, 19], [2, 4, 5, 7, 9, 11, 12, 13, 18, 19, 20], [1, 3, 9, 10, 13, 14, 15, 16, 17, 20, 21], [3, 4, 7, 13, 16, 17, 21], [0, 1, 2, 4, 8, 9, 12, 14, 15, 17, 18, 21], [0, 1, 2, 4, 8, 10, 11, 12, 14, 18, 19, 20]]
**Two moves ago:**
S(i-2) = [1, 2, 1, 3, 3, 1, 4, 5, 1, 1, 2, 5, 4, 5, 6, 6, 0, 0, 0, 0, 0, 0]
L(i-2) = Colorable
**One move ago:**
S(i-1) = [1, 2, 1, 3, 3, 1, 4, 5, 1, 1, 2, 5, 4, 5, 6, 6, 4, 0, 0, 0, 0, 0]
L(i-1) = Colorable
**Current coloring state:**
S(i) = [1, 2, 1, 3, 3, 1, 4, 5, 1, 1, 2, 5, 4, 5, 6, 6, 4, 2, 0, 0, 0, 0]
L(i) = Uncolorable
**Explored next state:**
S(i+1) = [1, 2, 1, 3, 3, 1, 4, 5, 1, 1, 2, 5, 4, 5, 6, 6, 4, 2, 7, 0, 0, 0]
L(i+1) = Uncolorable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{coloring\}", where \{coloring\} is in the same python list format as the previous states. | null | {'parent_state': [1, 2, 1, 3, 3, 1, 4, 5, 1, 1, 2, 5, 4, 5, 6, 6, 4, 0, 0, 0, 0, 0], 'current_state': [1, 2, 1, 3, 3, 1, 4, 5, 1, 1, 2, 5, 4, 5, 6, 6, 4, 2, 0, 0, 0, 0], 'current_status': 'Unsolvable', 'unsolvable_child': [1, 2, 1, 3, 3, 1, 4, 5, 1, 1, 2, 5, 4, 5, 6, 6, 4, 2, 7, 0, 0, 0], 'graph': [[1, 4, 6, 10, 11, 12, 16, 20, 21], [0, 3, 4, 5, 6, 7, 9, 11, 12, 13, 15, 16, 18, 20, 21], [3, 6, 7, 11, 12, 14, 15, 17, 20, 21], [1, 2, 7, 8, 11, 12, 15, 16, 18, 19], [0, 1, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21], [1, 6, 7, 11, 12, 14, 16, 17], [0, 1, 2, 4, 5, 7, 10, 11, 13, 14, 15], [1, 2, 3, 4, 5, 6, 12, 15, 16, 17, 19], [3, 4, 10, 14, 15, 16, 20, 21], [1, 4, 11, 12, 13, 14, 15, 16, 17, 18, 20], [0, 4, 6, 8, 13, 15, 18, 21], [0, 1, 2, 3, 4, 5, 6, 9, 16, 17, 21], [0, 1, 2, 3, 5, 7, 9, 13, 14, 17, 20, 21], [1, 4, 6, 9, 10, 12, 14, 15, 16, 17, 18, 19], [2, 4, 5, 6, 8, 9, 12, 13, 16, 18, 20, 21], [1, 2, 3, 4, 6, 7, 8, 9, 10, 13, 18, 20], [0, 1, 3, 4, 5, 7, 8, 9, 11, 13, 14, 18, 19], [2, 4, 5, 7, 9, 11, 12, 13, 18, 19, 20], [1, 3, 9, 10, 13, 14, 15, 16, 17, 20, 21], [3, 4, 7, 13, 16, 17, 21], [0, 1, 2, 4, 8, 9, 12, 14, 15, 17, 18, 21], [0, 1, 2, 4, 8, 10, 11, 12, 14, 18, 19, 20]], 'chromatic_number': 7} |
game24_statetransition | You are given an initial Game of 24 configuration S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid solution requires using each of the four initial numbers exactly once, using only basic arithmetic operations (+ - * /), and ultimately evaluating to 24.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Applying an operation: Combining two expressions using a basic arithmetic operation (+ - * /), reducing the number of expressions by 1.
2. Reverting an operation: Removing the last operation applied to the expressions, increasing the number of expressions by 1.
**Initial configuration:**
S(0) = ['1', '1', '5', '5']
L(0) = Solvable
**Current state:**
S(i) = ['(1 * 5)', '1', '5']
L(i) = Solvable
**Explored next state:**
S(i+1) = ['((1 * 5) + 1)', '5']
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{expressions\}", where \{expressions\} is in the same python list format as the previous states. | null | {'parent_state': ['1', '1', '5', '5'], 'current_state': ['(1 * 5)', '1', '5'], 'current_status': 'Solvable', 'unsolvable_child': ['((1 * 5) + 1)', '5'], 'numbers': [1, 1, 5, 5]} |
graphcoloring_statetransition | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given a sequence of partial coloring states leading to the current coloring state S(i). The coloring state is a list, where each index represents the corresonding vertex in the graph, and the number at that index represents its color (0 indicates an uncolored vertex). Alongside each state, its colorability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid coloring with no more than 4 colors. A valid coloring requires that no two adjacent vertices share the same color.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Coloring: Replacing a 0 in exactly one uncolored vertex with a value from 1 to 4.
2. Removing a color: Replacing a value in exactly one colored vertex with 0.
**Graph adjacency list:**
[[3, 4, 7, 8, 9, 13, 15], [8, 10, 11, 13, 16], [6, 7, 10, 11], [0, 8, 9, 12, 13, 16], [0, 5, 6, 8, 11, 12, 13, 14, 15], [4, 8, 10, 15, 16], [2, 4, 8, 10, 11, 12], [0, 2, 9, 10, 11, 12, 13, 14, 16], [0, 1, 3, 4, 5, 6, 12, 16], [0, 3, 7, 10, 16], [1, 2, 5, 6, 7, 9, 11, 12, 14, 16], [1, 2, 4, 6, 7, 10, 14], [3, 4, 6, 7, 8, 10], [0, 1, 3, 4, 7, 14], [4, 7, 10, 11, 13], [0, 4, 5, 16], [1, 3, 5, 7, 8, 9, 10, 15]]
**Two moves ago:**
S(i-2) = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i-2) = Colorable
**One move ago:**
S(i-1) = [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i-1) = Colorable
**Current coloring state:**
S(i) = [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i) = Uncolorable
**Explored next state:**
S(i+1) = [1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
L(i+1) = Uncolorable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{coloring\}", where \{coloring\} is in the same python list format as the previous states. | null | {'parent_state': [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'current_state': [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'current_status': 'Unsolvable', 'unsolvable_child': [1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 'graph': [[3, 4, 7, 8, 9, 13, 15], [8, 10, 11, 13, 16], [6, 7, 10, 11], [0, 8, 9, 12, 13, 16], [0, 5, 6, 8, 11, 12, 13, 14, 15], [4, 8, 10, 15, 16], [2, 4, 8, 10, 11, 12], [0, 2, 9, 10, 11, 12, 13, 14, 16], [0, 1, 3, 4, 5, 6, 12, 16], [0, 3, 7, 10, 16], [1, 2, 5, 6, 7, 9, 11, 12, 14, 16], [1, 2, 4, 6, 7, 10, 14], [3, 4, 6, 7, 8, 10], [0, 1, 3, 4, 7, 14], [4, 7, 10, 11, 13], [0, 4, 5, 16], [1, 3, 5, 7, 8, 9, 10, 15]], 'chromatic_number': 4} |
sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[8, 9, 2, 7, 3, 5, 1, 4, 6], [3, 4, 5, 9, 1, 6, 7, 2, 8], [7, 6, 1, 8, 2, 4, 3, 5, 9], [6, 1, 7, 3, 4, 9, 2, 8, 5], [5, 8, 9, 2, 7, 1, 6, 3, 4], [4, 2, 3, 5, 6, 8, 9, 7, 1], [9, 3, 6, 4, 5, 0, 0, 0, 0], [1, 7, 0, 6, 8, 0, 0, 9, 0], [2, 5, 0, 1, 0, 0, 4, 0, 7]]
Explored next state that leads to an unsolvable path:
[[8, 9, 2, 7, 3, 5, 1, 4, 6], [3, 4, 5, 9, 1, 6, 7, 2, 8], [7, 6, 1, 8, 2, 4, 3, 5, 9], [6, 1, 7, 3, 4, 9, 2, 8, 5], [5, 8, 9, 2, 7, 1, 6, 3, 4], [4, 2, 3, 5, 6, 8, 9, 7, 1], [9, 3, 6, 4, 5, 2, 0, 0, 0], [1, 7, 0, 6, 8, 0, 0, 9, 0], [2, 5, 0, 1, 0, 0, 4, 0, 7]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (A) | null |
sudoku_lookahead | You are given a partially filled 9x9 Sudoku grid represented as a list of lists, where empty cells are represented as 0. Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if it's possible to fill the remaining cells according to standard Sudoku rules, ensuring that each row, column, and 3x3 subgrid contains unique numbers from 1 to 9.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
Current state:
[[9, 4, 6, 7, 3, 8, 1, 2, 5], [5, 2, 1, 9, 4, 6, 3, 7, 8], [7, 8, 3, 1, 5, 2, 4, 6, 9], [1, 6, 7, 2, 8, 9, 5, 3, 4], [8, 5, 2, 3, 7, 4, 6, 9, 1], [4, 3, 9, 5, 6, 1, 7, 8, 2], [2, 9, 0, 0, 1, 7, 0, 0, 0], [0, 0, 0, 8, 0, 0, 0, 5, 0], [3, 7, 8, 4, 9, 0, 0, 1, 0]]
Explored next state that leads to an unsolvable path:
[[9, 4, 6, 7, 3, 8, 1, 2, 5], [5, 2, 1, 9, 4, 6, 3, 7, 8], [7, 8, 3, 1, 5, 2, 4, 6, 9], [1, 6, 7, 2, 8, 9, 5, 3, 4], [8, 5, 2, 3, 7, 4, 6, 9, 1], [4, 3, 9, 5, 6, 1, 7, 8, 2], [2, 9, 4, 0, 1, 7, 0, 0, 0], [0, 0, 0, 8, 0, 0, 0, 5, 0], [3, 7, 8, 4, 9, 0, 0, 1, 0]]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (A) | null |
graphcoloring_statetransition | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given a sequence of partial coloring states leading to the current coloring state S(i). The coloring state is a list, where each index represents the corresonding vertex in the graph, and the number at that index represents its color (0 indicates an uncolored vertex). Alongside each state, its colorability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid coloring with no more than 9 colors. A valid coloring requires that no two adjacent vertices share the same color.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Coloring: Replacing a 0 in exactly one uncolored vertex with a value from 1 to 9.
2. Removing a color: Replacing a value in exactly one colored vertex with 0.
**Graph adjacency list:**
[[3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 16, 17, 18, 20, 22], [2, 3, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 17, 18, 20], [1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 21, 22], [0, 1, 2, 4, 6, 8, 9, 10, 11, 12, 13, 14, 16, 18, 19, 21], [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 22], [0, 1, 2, 4, 6, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 22], [0, 3, 4, 5, 9, 10, 11, 12, 14, 16, 17, 19, 21, 22], [0, 1, 2, 4, 9, 10, 12, 13, 14, 15, 16, 17, 20, 21, 22], [0, 1, 2, 3, 4, 9, 10, 16, 18, 19, 20, 22], [0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16, 18, 20, 21, 22], [3, 4, 5, 6, 7, 8, 11, 12, 14, 15, 18, 20, 21, 22], [0, 1, 2, 3, 5, 6, 9, 10, 12, 14, 16, 18, 22], [1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 14, 15, 16, 17, 19, 20, 21], [0, 2, 3, 4, 5, 7, 15, 16, 18, 19, 20, 21, 22], [0, 1, 3, 4, 5, 6, 7, 10, 11, 12, 15, 18, 19, 20, 21, 22], [1, 5, 7, 9, 10, 12, 13, 14, 16, 18, 19, 21, 22], [0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 15, 17, 18, 21], [0, 1, 2, 6, 7, 12, 16, 19, 20, 21, 22], [0, 1, 2, 3, 5, 8, 9, 10, 11, 13, 14, 15, 16, 19, 20, 21, 22], [3, 5, 6, 8, 12, 13, 14, 15, 17, 18, 20, 21, 22], [0, 1, 5, 7, 8, 9, 10, 12, 13, 14, 17, 18, 19], [2, 3, 6, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19], [0, 2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19]]
**Two moves ago:**
S(i-2) = [1, 1, 2, 3, 4, 3, 2, 3, 5, 6, 1, 4, 5, 5, 6, 2, 7, 8, 0, 0, 0, 0, 0]
L(i-2) = Colorable
**One move ago:**
S(i-1) = [1, 1, 2, 3, 4, 3, 2, 3, 5, 6, 1, 4, 5, 5, 6, 2, 7, 8, 9, 0, 0, 0, 0]
L(i-1) = Colorable
**Current coloring state:**
S(i) = [1, 1, 2, 3, 4, 3, 2, 3, 5, 6, 1, 4, 5, 5, 6, 2, 7, 8, 9, 7, 0, 0, 0]
L(i) = Uncolorable
**Explored next state:**
S(i+1) = [1, 1, 2, 3, 4, 3, 2, 3, 5, 6, 1, 4, 5, 5, 6, 2, 7, 8, 9, 7, 2, 0, 0]
L(i+1) = Uncolorable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{coloring\}", where \{coloring\} is in the same python list format as the previous states. | null | {'parent_state': [1, 1, 2, 3, 4, 3, 2, 3, 5, 6, 1, 4, 5, 5, 6, 2, 7, 8, 9, 0, 0, 0, 0], 'current_state': [1, 1, 2, 3, 4, 3, 2, 3, 5, 6, 1, 4, 5, 5, 6, 2, 7, 8, 9, 7, 0, 0, 0], 'current_status': 'Unsolvable', 'unsolvable_child': [1, 1, 2, 3, 4, 3, 2, 3, 5, 6, 1, 4, 5, 5, 6, 2, 7, 8, 9, 7, 2, 0, 0], 'graph': [[3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 16, 17, 18, 20, 22], [2, 3, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 17, 18, 20], [1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 21, 22], [0, 1, 2, 4, 6, 8, 9, 10, 11, 12, 13, 14, 16, 18, 19, 21], [0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 22], [0, 1, 2, 4, 6, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 22], [0, 3, 4, 5, 9, 10, 11, 12, 14, 16, 17, 19, 21, 22], [0, 1, 2, 4, 9, 10, 12, 13, 14, 15, 16, 17, 20, 21, 22], [0, 1, 2, 3, 4, 9, 10, 16, 18, 19, 20, 22], [0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16, 18, 20, 21, 22], [3, 4, 5, 6, 7, 8, 11, 12, 14, 15, 18, 20, 21, 22], [0, 1, 2, 3, 5, 6, 9, 10, 12, 14, 16, 18, 22], [1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 14, 15, 16, 17, 19, 20, 21], [0, 2, 3, 4, 5, 7, 15, 16, 18, 19, 20, 21, 22], [0, 1, 3, 4, 5, 6, 7, 10, 11, 12, 15, 18, 19, 20, 21, 22], [1, 5, 7, 9, 10, 12, 13, 14, 16, 18, 19, 21, 22], [0, 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 15, 17, 18, 21], [0, 1, 2, 6, 7, 12, 16, 19, 20, 21, 22], [0, 1, 2, 3, 5, 8, 9, 10, 11, 13, 14, 15, 16, 19, 20, 21, 22], [3, 5, 6, 8, 12, 13, 14, 15, 17, 18, 20, 21, 22], [0, 1, 5, 7, 8, 9, 10, 12, 13, 14, 17, 18, 19], [2, 3, 6, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19], [0, 2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19]], 'chromatic_number': 9} |
graphcoloring_lookahead | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given the current coloring state of the graph in a list, where each index represents the corresonding vertex, and the number at that index represents its color (0 indicates an uncolored vertex).
Your task is to determine if this current state can lead to a valid coloring. Specifically, use lookahead techniques to determine if it's possible to color the remaining vertices such that no two adjacent vertices share the same color, using no more than 6 colors in total.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Graph adjacency list:**
[[2, 4, 6, 10, 11, 12, 14, 16], [2, 4, 5, 8, 13, 16], [0, 1, 3, 4, 8, 9, 11, 15, 16], [2, 5, 6, 8, 12, 15], [0, 1, 2, 5, 6, 14, 15, 16], [1, 3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15], [0, 3, 4, 5, 10, 11, 13], [8, 11, 13, 16], [1, 2, 3, 5, 7, 9, 10, 11, 13, 14, 16], [2, 5, 8, 10, 13, 14], [0, 5, 6, 8, 9, 11, 12, 13, 14, 15], [0, 2, 5, 6, 7, 8, 10, 12, 14, 16], [0, 3, 5, 10, 11, 16], [1, 5, 6, 7, 8, 9, 10, 15], [0, 4, 5, 8, 9, 10, 11, 15], [2, 3, 4, 5, 10, 13, 14, 16], [0, 1, 2, 4, 7, 8, 11, 12, 15]]
**Current coloring state:**
[1, 1, 2, 1, 3, 2, 4, 1, 3, 1, 5, 6, 0, 0, 0, 0, 0]
**Explored next state that leads to an uncolorable path:**
[1, 1, 2, 1, 3, 2, 4, 1, 3, 1, 5, 6, 3, 0, 0, 0, 0]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Colorable (B) Uncolorable. | (B) | null |
game24_statetransition | You are given an initial Game of 24 configuration S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid solution requires using each of the four initial numbers exactly once, using only basic arithmetic operations (+ - * /), and ultimately evaluating to 24.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Applying an operation: Combining two expressions using a basic arithmetic operation (+ - * /), reducing the number of expressions by 1.
2. Reverting an operation: Removing the last operation applied to the expressions, increasing the number of expressions by 1.
**Initial configuration:**
S(0) = ['1', '8', '9', '11']
L(0) = Solvable
**Current state:**
S(i) = ['(11 / 1)', '8', '9']
L(i) = Unsolvable
**Explored next state:**
S(i+1) = ['((11 / 1) + 8)', '9']
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{expressions\}", where \{expressions\} is in the same python list format as the previous states. | null | {'parent_state': ['1', '8', '9', '11'], 'current_state': ['(11 / 1)', '8', '9'], 'current_status': 'Unsolvable', 'unsolvable_child': ['((11 / 1) + 8)', '9'], 'numbers': [1, 8, 9, 11]} |
graphcoloring_statetransition | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given a sequence of partial coloring states leading to the current coloring state S(i). The coloring state is a list, where each index represents the corresonding vertex in the graph, and the number at that index represents its color (0 indicates an uncolored vertex). Alongside each state, its colorability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid coloring with no more than 6 colors. A valid coloring requires that no two adjacent vertices share the same color.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Coloring: Replacing a 0 in exactly one uncolored vertex with a value from 1 to 6.
2. Removing a color: Replacing a value in exactly one colored vertex with 0.
**Graph adjacency list:**
[[3, 4, 5, 8, 10, 17], [2, 4, 11, 13, 14, 16], [1, 3, 5, 6, 12, 13, 14, 16], [0, 2, 7, 10, 11, 12, 14, 15, 16], [0, 1, 5, 7, 9, 10, 13, 14, 15], [0, 2, 4, 6, 7, 8, 12, 13, 16], [2, 5, 9, 11, 12, 13, 15, 17], [3, 4, 5, 9, 14, 15], [0, 5, 9, 10, 12, 14, 15, 17], [4, 6, 7, 8, 14, 16, 17], [0, 3, 4, 8, 12, 14], [1, 3, 6, 12, 13, 14], [2, 3, 5, 6, 8, 10, 11, 14, 15, 16], [1, 2, 4, 5, 6, 11, 14, 16], [1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 13, 15], [3, 4, 6, 7, 8, 12, 14], [1, 2, 3, 5, 9, 12, 13], [0, 6, 8, 9]]
**Two moves ago:**
S(i-2) = [1, 1, 2, 3, 2, 3, 1, 1, 2, 3, 4, 0, 0, 0, 0, 0, 0, 0]
L(i-2) = Colorable
**One move ago:**
S(i-1) = [1, 1, 2, 3, 2, 3, 1, 1, 2, 3, 4, 5, 0, 0, 0, 0, 0, 0]
L(i-1) = Colorable
**Current coloring state:**
S(i) = [1, 1, 2, 3, 2, 3, 1, 1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 0]
L(i) = Uncolorable
**Explored next state:**
S(i+1) = [1, 1, 2, 3, 2, 3, 1, 1, 2, 3, 4, 5, 6, 4, 0, 0, 0, 0]
L(i+1) = Uncolorable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{coloring\}", where \{coloring\} is in the same python list format as the previous states. | null | {'parent_state': [1, 1, 2, 3, 2, 3, 1, 1, 2, 3, 4, 5, 0, 0, 0, 0, 0, 0], 'current_state': [1, 1, 2, 3, 2, 3, 1, 1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 0], 'current_status': 'Unsolvable', 'unsolvable_child': [1, 1, 2, 3, 2, 3, 1, 1, 2, 3, 4, 5, 6, 4, 0, 0, 0, 0], 'graph': [[3, 4, 5, 8, 10, 17], [2, 4, 11, 13, 14, 16], [1, 3, 5, 6, 12, 13, 14, 16], [0, 2, 7, 10, 11, 12, 14, 15, 16], [0, 1, 5, 7, 9, 10, 13, 14, 15], [0, 2, 4, 6, 7, 8, 12, 13, 16], [2, 5, 9, 11, 12, 13, 15, 17], [3, 4, 5, 9, 14, 15], [0, 5, 9, 10, 12, 14, 15, 17], [4, 6, 7, 8, 14, 16, 17], [0, 3, 4, 8, 12, 14], [1, 3, 6, 12, 13, 14], [2, 3, 5, 6, 8, 10, 11, 14, 15, 16], [1, 2, 4, 5, 6, 11, 14, 16], [1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 13, 15], [3, 4, 6, 7, 8, 12, 14], [1, 2, 3, 5, 9, 12, 13], [0, 6, 8, 9]], 'chromatic_number': 6} |
game24_lookahead | You are given four numbers and the current calculation state for the Game of 24.
Your task is to determine if this current state can lead to a solvable solution. Specifically, use lookahead techniques to determine if the remaining numbers can be combined using basic arithmetic operations (+ - * /) to reach exactly 24. You must use each number exactly once.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Numbers:**
['5', '6', '10', '10']
**Current calculation state:**
['(10 / 10)', '5', '6']
**Explored next state that leads to an unsolvable path:**
['((10 / 10) + 5)', '6']
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Solvable (B) Unsolvable. | (A) | null |
game24_statetransition | You are given an initial Game of 24 configuration S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid solution requires using each of the four initial numbers exactly once, using only basic arithmetic operations (+ - * /), and ultimately evaluating to 24.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Applying an operation: Combining two expressions using a basic arithmetic operation (+ - * /), reducing the number of expressions by 1.
2. Reverting an operation: Removing the last operation applied to the expressions, increasing the number of expressions by 1.
**Initial configuration:**
S(0) = ['4', '4', '11', '13']
L(0) = Solvable
**Current state:**
S(i) = ['(4 + 11)', '4', '13']
L(i) = Solvable
**Explored next state:**
S(i+1) = ['((4 + 11) + 4)', '13']
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{expressions\}", where \{expressions\} is in the same python list format as the previous states. | null | {'parent_state': ['4', '4', '11', '13'], 'current_state': ['(4 + 11)', '4', '13'], 'current_status': 'Solvable', 'unsolvable_child': ['((4 + 11) + 4)', '13'], 'numbers': [4, 4, 11, 13]} |
graphcoloring_lookahead | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given the current coloring state of the graph in a list, where each index represents the corresonding vertex, and the number at that index represents its color (0 indicates an uncolored vertex).
Your task is to determine if this current state can lead to a valid coloring. Specifically, use lookahead techniques to determine if it's possible to color the remaining vertices such that no two adjacent vertices share the same color, using no more than 6 colors in total.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path and leverage it to make a more informed decision about the current state.
**Graph adjacency list:**
[[2, 4, 6, 10, 11, 12, 14, 16], [2, 4, 5, 8, 13, 16], [0, 1, 3, 4, 8, 9, 11, 15, 16], [2, 5, 6, 8, 12, 15], [0, 1, 2, 5, 6, 14, 15, 16], [1, 3, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15], [0, 3, 4, 5, 10, 11, 13], [8, 11, 13, 16], [1, 2, 3, 5, 7, 9, 10, 11, 13, 14, 16], [2, 5, 8, 10, 13, 14], [0, 5, 6, 8, 9, 11, 12, 13, 14, 15], [0, 2, 5, 6, 7, 8, 10, 12, 14, 16], [0, 3, 5, 10, 11, 16], [1, 5, 6, 7, 8, 9, 10, 15], [0, 4, 5, 8, 9, 10, 11, 15], [2, 3, 4, 5, 10, 13, 14, 16], [0, 1, 2, 4, 7, 8, 11, 12, 15]]
**Current coloring state:**
[1, 1, 2, 1, 3, 2, 4, 1, 4, 1, 3, 5, 0, 0, 0, 0, 0]
**Explored next state that leads to an uncolorable path:**
[1, 1, 2, 1, 3, 2, 4, 1, 4, 1, 3, 5, 6, 0, 0, 0, 0]
Let's think step by step, considering the failed state to avoid unnecessary exploration. Do not solve using programming.
Choose from (A) Colorable (B) Uncolorable. | (A) | null |
graphcoloring_statetransition | You are given a graph represented as an adjacency list, where each index corresponds to a vertex, and the list at that index represents its adjacent vertices. You are also given a sequence of partial coloring states leading to the current coloring state S(i). The coloring state is a list, where each index represents the corresonding vertex in the graph, and the number at that index represents its color (0 indicates an uncolored vertex). Alongside each state, its colorability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid coloring with no more than 9 colors. A valid coloring requires that no two adjacent vertices share the same color.
Additionally, you are provided with a previously explored next state that has been proven to be uncolorable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Coloring: Replacing a 0 in exactly one uncolored vertex with a value from 1 to 9.
2. Removing a color: Replacing a value in exactly one colored vertex with 0.
**Graph adjacency list:**
[[1, 2, 3, 4, 5, 7, 9, 10, 11, 12, 15, 17], [0, 2, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17], [0, 2, 4, 5, 6, 7, 8, 9, 12, 13, 15, 16], [0, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 17], [0, 2, 3, 4, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17], [2, 3, 4, 8, 9, 11, 12, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 8, 9, 10, 11, 13, 14, 15, 16, 17], [1, 2, 3, 4, 6, 7, 9, 10, 12, 14, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 17], [0, 1, 2, 4, 5, 7, 8, 9, 11, 12, 15, 16, 17], [0, 1, 2, 5, 6, 7, 9, 10, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 7, 11, 12, 14, 15, 16, 17], [1, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 16, 17], [1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15], [0, 1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15]]
**Two moves ago:**
S(i-2) = [1, 2, 3, 2, 4, 5, 1, 6, 5, 7, 0, 0, 0, 0, 0, 0, 0, 0]
L(i-2) = Colorable
**One move ago:**
S(i-1) = [1, 2, 3, 2, 4, 5, 1, 6, 5, 7, 8, 0, 0, 0, 0, 0, 0, 0]
L(i-1) = Colorable
**Current coloring state:**
S(i) = [1, 2, 3, 2, 4, 5, 1, 6, 5, 7, 8, 4, 0, 0, 0, 0, 0, 0]
L(i) = Colorable
**Explored next state:**
S(i+1) = [1, 2, 3, 2, 4, 5, 1, 6, 5, 7, 8, 4, 9, 0, 0, 0, 0, 0]
L(i+1) = Uncolorable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{coloring\}", where \{coloring\} is in the same python list format as the previous states. | null | {'parent_state': [1, 2, 3, 2, 4, 5, 1, 6, 5, 7, 8, 0, 0, 0, 0, 0, 0, 0], 'current_state': [1, 2, 3, 2, 4, 5, 1, 6, 5, 7, 8, 4, 0, 0, 0, 0, 0, 0], 'current_status': 'Solvable', 'unsolvable_child': [1, 2, 3, 2, 4, 5, 1, 6, 5, 7, 8, 4, 9, 0, 0, 0, 0, 0], 'graph': [[1, 2, 3, 4, 5, 7, 9, 10, 11, 12, 15, 17], [0, 2, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], [0, 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17], [0, 2, 4, 5, 6, 7, 8, 9, 12, 13, 15, 16], [0, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 17], [0, 2, 3, 4, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17], [2, 3, 4, 8, 9, 11, 12, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 8, 9, 10, 11, 13, 14, 15, 16, 17], [1, 2, 3, 4, 6, 7, 9, 10, 12, 14, 16], [0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 17], [0, 1, 2, 4, 5, 7, 8, 9, 11, 12, 15, 16, 17], [0, 1, 2, 5, 6, 7, 9, 10, 12, 13, 14, 15, 16, 17], [0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17], [1, 2, 3, 4, 5, 7, 11, 12, 14, 15, 16, 17], [1, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17], [0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 16, 17], [1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15], [0, 1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15]], 'chromatic_number': 9} |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[6, 0, 0, 0, 0, 0, 1, 0, 8], [0, 0, 4, 8, 7, 0, 0, 2, 5], [0, 0, 3, 1, 0, 4, 0, 0, 0], [0, 0, 1, 9, 0, 2, 8, 0, 0], [0, 0, 0, 0, 0, 3, 0, 5, 0], [3, 2, 0, 0, 0, 0, 0, 6, 1], [0, 0, 6, 0, 0, 8, 5, 0, 0], [0, 1, 2, 0, 9, 0, 0, 0, 7], [0, 5, 7, 6, 2, 0, 9, 3, 0]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[6, 7, 5, 2, 3, 9, 1, 4, 8], [1, 9, 4, 8, 7, 6, 3, 2, 5], [2, 8, 3, 1, 5, 4, 7, 9, 6], [5, 4, 1, 9, 6, 2, 8, 7, 3], [7, 6, 8, 4, 1, 3, 2, 5, 9], [3, 2, 9, 5, 8, 7, 4, 6, 1], [9, 3, 6, 7, 4, 8, 5, 0, 0], [0, 1, 2, 0, 9, 0, 0, 0, 7], [0, 5, 7, 6, 2, 0, 9, 3, 0]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[6, 7, 5, 2, 3, 9, 1, 4, 8], [1, 9, 4, 8, 7, 6, 3, 2, 5], [2, 8, 3, 1, 5, 4, 7, 9, 6], [5, 4, 1, 9, 6, 2, 8, 7, 3], [7, 6, 8, 4, 1, 3, 2, 5, 9], [3, 2, 9, 5, 8, 7, 4, 6, 1], [9, 3, 6, 7, 4, 8, 5, 1, 0], [0, 1, 2, 0, 9, 0, 0, 0, 7], [0, 5, 7, 6, 2, 0, 9, 3, 0]]
L(i-1) = Solvable
**Current state:**
S(i) = [[6, 7, 5, 2, 3, 9, 1, 4, 8], [1, 9, 4, 8, 7, 6, 3, 2, 5], [2, 8, 3, 1, 5, 4, 7, 9, 6], [5, 4, 1, 9, 6, 2, 8, 7, 3], [7, 6, 8, 4, 1, 3, 2, 5, 9], [3, 2, 9, 5, 8, 7, 4, 6, 1], [9, 3, 6, 7, 4, 8, 5, 1, 2], [0, 1, 2, 0, 9, 0, 0, 0, 7], [0, 5, 7, 6, 2, 0, 9, 3, 0]]
L(i) = Solvable
**Explored next state:**
S(i+1) = [[6, 7, 5, 2, 3, 9, 1, 4, 8], [1, 9, 4, 8, 7, 6, 3, 2, 5], [2, 8, 3, 1, 5, 4, 7, 9, 6], [5, 4, 1, 9, 6, 2, 8, 7, 3], [7, 6, 8, 4, 1, 3, 2, 5, 9], [3, 2, 9, 5, 8, 7, 4, 6, 1], [9, 3, 6, 7, 4, 8, 5, 1, 2], [8, 1, 2, 0, 9, 0, 0, 0, 7], [0, 5, 7, 6, 2, 0, 9, 3, 0]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[6, 7, 5, 2, 3, 9, 1, 4, 8], [1, 9, 4, 8, 7, 6, 3, 2, 5], [2, 8, 3, 1, 5, 4, 7, 9, 6], [5, 4, 1, 9, 6, 2, 8, 7, 3], [7, 6, 8, 4, 1, 3, 2, 5, 9], [3, 2, 9, 5, 8, 7, 4, 6, 1], [9, 3, 6, 7, 4, 8, 5, 1, 0], [0, 1, 2, 0, 9, 0, 0, 0, 7], [0, 5, 7, 6, 2, 0, 9, 3, 0]], 'current_state': [[6, 7, 5, 2, 3, 9, 1, 4, 8], [1, 9, 4, 8, 7, 6, 3, 2, 5], [2, 8, 3, 1, 5, 4, 7, 9, 6], [5, 4, 1, 9, 6, 2, 8, 7, 3], [7, 6, 8, 4, 1, 3, 2, 5, 9], [3, 2, 9, 5, 8, 7, 4, 6, 1], [9, 3, 6, 7, 4, 8, 5, 1, 2], [0, 1, 2, 0, 9, 0, 0, 0, 7], [0, 5, 7, 6, 2, 0, 9, 3, 0]], 'current_status': 'Solvable', 'unsolvable_child': [[6, 7, 5, 2, 3, 9, 1, 4, 8], [1, 9, 4, 8, 7, 6, 3, 2, 5], [2, 8, 3, 1, 5, 4, 7, 9, 6], [5, 4, 1, 9, 6, 2, 8, 7, 3], [7, 6, 8, 4, 1, 3, 2, 5, 9], [3, 2, 9, 5, 8, 7, 4, 6, 1], [9, 3, 6, 7, 4, 8, 5, 1, 2], [8, 1, 2, 0, 9, 0, 0, 0, 7], [0, 5, 7, 6, 2, 0, 9, 3, 0]]} |
sudoku_statetransition | You are given an initial Sudoku puzzle S(0), followed by a sequence of progressive states leading to the current state S(i). Alongside each state, its solvability status L(*) is given. Your task is to determine the next state by making exactly one move, ensuring progress toward a valid solution. A valid Sudoku solution requires that each row, column, and 3x3 subgrid contains the numbers 1 to 9 without repetition.
Additionally, you are provided with a previously explored next state that has been proven to be unsolvable. Use this information to avoid revisiting this failed path.
**A move is defined as either:**
1. Filling: Replacing a 0 in exactly one empty cell with a value from 1 to 9.
2. Removing: Replacing a value in exactly one filled cell with 0.
**Initial puzzle:**
S(0) = [[1, 6, 0, 0, 3, 8, 0, 0, 5], [0, 9, 0, 5, 0, 0, 0, 7, 0], [5, 4, 0, 0, 0, 0, 2, 6, 0], [3, 0, 1, 0, 6, 2, 0, 0, 7], [0, 0, 0, 7, 0, 1, 0, 8, 0], [0, 5, 0, 9, 0, 0, 0, 0, 4], [0, 0, 4, 0, 0, 0, 0, 3, 2], [0, 0, 2, 6, 1, 0, 4, 5, 0], [9, 7, 0, 0, 0, 0, 0, 0, 8]]
L(0) = Solvable
**Two moves ago:**
S(i-2) = [[1, 6, 7, 2, 3, 8, 9, 4, 5], [2, 9, 3, 5, 4, 6, 8, 7, 1], [5, 4, 8, 1, 7, 9, 2, 6, 3], [3, 8, 1, 4, 6, 2, 5, 9, 7], [0, 0, 0, 7, 0, 1, 0, 8, 0], [0, 5, 0, 9, 0, 0, 0, 0, 4], [0, 0, 4, 0, 0, 0, 0, 3, 2], [0, 0, 2, 6, 1, 0, 4, 5, 0], [9, 7, 0, 0, 0, 0, 0, 0, 8]]
L(i-2) = Solvable
**One move ago:**
S(i-1) = [[1, 6, 7, 2, 3, 8, 9, 4, 5], [2, 9, 3, 5, 4, 6, 8, 7, 1], [5, 4, 8, 1, 7, 9, 2, 6, 3], [3, 8, 1, 4, 6, 2, 5, 9, 7], [4, 0, 0, 7, 0, 1, 0, 8, 0], [0, 5, 0, 9, 0, 0, 0, 0, 4], [0, 0, 4, 0, 0, 0, 0, 3, 2], [0, 0, 2, 6, 1, 0, 4, 5, 0], [9, 7, 0, 0, 0, 0, 0, 0, 8]]
L(i-1) = Solvable
**Current state:**
S(i) = [[1, 6, 7, 2, 3, 8, 9, 4, 5], [2, 9, 3, 5, 4, 6, 8, 7, 1], [5, 4, 8, 1, 7, 9, 2, 6, 3], [3, 8, 1, 4, 6, 2, 5, 9, 7], [4, 2, 0, 7, 0, 1, 0, 8, 0], [0, 5, 0, 9, 0, 0, 0, 0, 4], [0, 0, 4, 0, 0, 0, 0, 3, 2], [0, 0, 2, 6, 1, 0, 4, 5, 0], [9, 7, 0, 0, 0, 0, 0, 0, 8]]
L(i) = Solvable
**Explored next state:**
S(i+1) = [[1, 6, 7, 2, 3, 8, 9, 4, 5], [2, 9, 3, 5, 4, 6, 8, 7, 1], [5, 4, 8, 1, 7, 9, 2, 6, 3], [3, 8, 1, 4, 6, 2, 5, 9, 7], [4, 2, 6, 7, 0, 1, 0, 8, 0], [0, 5, 0, 9, 0, 0, 0, 0, 4], [0, 0, 4, 0, 0, 0, 0, 3, 2], [0, 0, 2, 6, 1, 0, 4, 5, 0], [9, 7, 0, 0, 0, 0, 0, 0, 8]]
L(i+1) = Unsolvable
Let's think step by step. Analyze the progress made so far and determine the immediate next move. End your answer with "Next state: \{grid\}", where \{grid\} is in the same python list format as the previous states. | null | {'parent_state': [[1, 6, 7, 2, 3, 8, 9, 4, 5], [2, 9, 3, 5, 4, 6, 8, 7, 1], [5, 4, 8, 1, 7, 9, 2, 6, 3], [3, 8, 1, 4, 6, 2, 5, 9, 7], [4, 0, 0, 7, 0, 1, 0, 8, 0], [0, 5, 0, 9, 0, 0, 0, 0, 4], [0, 0, 4, 0, 0, 0, 0, 3, 2], [0, 0, 2, 6, 1, 0, 4, 5, 0], [9, 7, 0, 0, 0, 0, 0, 0, 8]], 'current_state': [[1, 6, 7, 2, 3, 8, 9, 4, 5], [2, 9, 3, 5, 4, 6, 8, 7, 1], [5, 4, 8, 1, 7, 9, 2, 6, 3], [3, 8, 1, 4, 6, 2, 5, 9, 7], [4, 2, 0, 7, 0, 1, 0, 8, 0], [0, 5, 0, 9, 0, 0, 0, 0, 4], [0, 0, 4, 0, 0, 0, 0, 3, 2], [0, 0, 2, 6, 1, 0, 4, 5, 0], [9, 7, 0, 0, 0, 0, 0, 0, 8]], 'current_status': 'Solvable', 'unsolvable_child': [[1, 6, 7, 2, 3, 8, 9, 4, 5], [2, 9, 3, 5, 4, 6, 8, 7, 1], [5, 4, 8, 1, 7, 9, 2, 6, 3], [3, 8, 1, 4, 6, 2, 5, 9, 7], [4, 2, 6, 7, 0, 1, 0, 8, 0], [0, 5, 0, 9, 0, 0, 0, 0, 4], [0, 0, 4, 0, 0, 0, 0, 3, 2], [0, 0, 2, 6, 1, 0, 4, 5, 0], [9, 7, 0, 0, 0, 0, 0, 0, 8]]} |
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