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cardcoloringcounting_diff_5_0_f2a58ca7ca | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n8 4 3 7 6 5 1 2\n5 2 1 8 3 4 6 7\n2 6 1 3 4 5 7 8\n\nNow, you need to color these cards into **1** red cards, **1** green cards, and **6** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_1_d545784b3a | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 4 7 6 5 1 3 8\n\nNow, you need to color these cards into **2** red cards, **4** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_2_2821287b20 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **7** red cards, **1** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_3_af2169bece | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n6 5 4 7 1 3 8 2\n\nNow, you need to color these cards into **4** red cards, **3** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_4_8890cd2762 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n8 2 1 6 5 7 3 4\n\nNow, you need to color these cards into **3** red cards, **2** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_5_225db6660e | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n8 7 2 1 6 5 4 3\n4 8 5 1 2 6 3 7\n\nNow, you need to color these cards into **2** red cards, **4** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_6_887f9fa2b8 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **0** red cards, **1** green cards, and **7** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_7_fb084e80cc | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 7 5 1 3 4 6 8\n1 8 4 6 7 5 3 2\n\nNow, you need to color these cards into **2** red cards, **5** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_8_998f6fd6d3 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n4 6 1 7 2 5 3 8\n\nNow, you need to color these cards into **3** red cards, **3** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_9_43706c4405 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 7 6 1 8 4 2 5\n6 5 7 1 4 8 2 3\n3 2 7 1 5 4 8 6\n\nNow, you need to color these cards into **0** red cards, **7** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_10_6d82153206 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **2** red cards, **0** green cards, and **6** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_11_874acafaf0 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 7 5 4 2 1 6 8\n3 1 5 7 4 2 6 8\n2 1 7 8 5 3 6 4\n\nNow, you need to color these cards into **0** red cards, **7** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_12_72993293e0 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 4 1 8 7 2 5 6\n1 6 5 2 4 7 8 3\n4 1 5 2 7 3 6 8\n\nNow, you need to color these cards into **7** red cards, **1** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_13_46e03e58a0 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n4 6 2 7 5 1 3 8\n5 1 4 2 7 8 6 3\n\nNow, you need to color these cards into **2** red cards, **4** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_14_264bc7ada7 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **1** red cards, **4** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_15_b7ea9f44bf | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 2 1 6 7 8 4 5\n\nNow, you need to color these cards into **0** red cards, **3** green cards, and **5** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_16_098bee6fca | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n7 2 4 1 6 3 8 5\n4 1 3 6 5 8 7 2\n\nNow, you need to color these cards into **5** red cards, **0** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_17_a3794e5b76 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n5 2 6 8 3 1 4 7\n7 8 6 4 3 5 2 1\n3 5 1 2 7 6 8 4\n\nNow, you need to color these cards into **8** red cards, **0** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_18_f744427665 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n6 2 7 4 5 1 8 3\n\nNow, you need to color these cards into **2** red cards, **0** green cards, and **6** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_19_8b55ed9adc | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **8** red cards, **0** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_20_3e06f1ed4c | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n6 5 2 1 4 8 7 3\n\nNow, you need to color these cards into **2** red cards, **2** green cards, and **4** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_21_6383f5f8a4 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n5 4 2 7 6 3 1 8\n\nNow, you need to color these cards into **2** red cards, **3** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_22_1a7beaa946 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 6 1 8 7 4 2 5\n\nNow, you need to color these cards into **5** red cards, **1** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_23_bbef91deda | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n7 4 5 3 2 6 1 8\n8 1 2 5 4 7 3 6\n\nNow, you need to color these cards into **4** red cards, **0** green cards, and **4** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_24_e39a6acafc | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 2 7 1 6 8 4 5\n8 4 6 1 5 3 7 2\n8 6 3 4 1 7 2 5\n\nNow, you need to color these cards into **2** red cards, **0** green cards, and **6** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_25_ce9c893dc2 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n8 1 3 7 6 4 2 5\n2 5 6 8 3 1 7 4\n2 7 6 1 3 8 4 5\n\nNow, you need to color these cards into **2** red cards, **0** green cards, and **6** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_26_d40037789a | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 2 4 8 7 3 6 5\n\nNow, you need to color these cards into **0** red cards, **5** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_27_22d62e306f | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 1 3 5 6 8 4 7\n3 2 7 4 1 8 5 6\n7 6 1 4 3 8 2 5\n\nNow, you need to color these cards into **0** red cards, **7** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_28_19050155ec | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **2** red cards, **1** green cards, and **5** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_29_2084a3d971 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n4 3 1 8 5 7 6 2\n8 1 5 2 7 4 3 6\n\nNow, you need to color these cards into **0** red cards, **8** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_30_995492b697 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **1** red cards, **7** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_31_7cdea9e774 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **4** red cards, **3** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_32_08465e6166 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n8 6 5 7 2 3 4 1\n\nNow, you need to color these cards into **1** red cards, **5** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
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cardcoloringcounting_diff_5_33_75081a8972 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n4 1 8 3 7 2 5 6\n2 5 7 8 1 6 3 4\n\nNow, you need to color these cards into **0** red cards, **8** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_34_3ebf43c47a | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 8 4 7 1 5 2 6\n2 3 1 5 6 4 8 7\n\nNow, you need to color these cards into **0** red cards, **0** green cards, and **8** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_35_b24bdd9963 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 1 5 8 6 4 3 7\n5 8 1 7 4 2 3 6\n\nNow, you need to color these cards into **3** red cards, **0** green cards, and **5** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_36_d4b3d75ded | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **1** red cards, **0** green cards, and **7** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_37_3106800a32 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 3 6 4 7 1 5 8\n6 3 8 7 5 2 4 1\n2 5 4 8 6 3 1 7\n\nNow, you need to color these cards into **1** red cards, **6** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_38_44656d6be6 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n4 2 8 5 1 3 7 6\n\nNow, you need to color these cards into **3** red cards, **3** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_39_ebc10d28a6 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n4 2 3 6 8 1 5 7\n4 5 6 8 1 7 2 3\n5 3 2 8 1 6 4 7\n\nNow, you need to color these cards into **1** red cards, **0** green cards, and **7** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_40_dc6cb74bda | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n7 5 6 3 8 2 1 4\n1 4 2 6 3 8 5 7\n2 7 5 1 4 8 6 3\n\nNow, you need to color these cards into **1** red cards, **1** green cards, and **6** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_41_29e09d747d | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **4** red cards, **0** green cards, and **4** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_42_aa73d3d08c | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 4 2 8 1 7 5 6\n2 8 5 3 4 6 7 1\n6 1 3 8 2 7 5 4\n\nNow, you need to color these cards into **2** red cards, **6** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_43_a0ecba8351 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 3 7 1 5 8 4 6\n5 7 1 3 6 8 4 2\n\nNow, you need to color these cards into **4** red cards, **1** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_44_123bcf826c | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **6** red cards, **1** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_45_2491d98068 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n6 3 5 4 8 7 1 2\n1 8 7 3 4 5 6 2\n\nNow, you need to color these cards into **5** red cards, **3** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_46_e76b97437e | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n4 6 8 2 5 3 1 7\n4 2 5 6 7 8 1 3\n\nNow, you need to color these cards into **0** red cards, **0** green cards, and **8** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_47_5df114c7f2 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n8 7 3 1 6 5 4 2\n6 7 1 3 5 8 4 2\n6 5 7 4 1 8 2 3\n\nNow, you need to color these cards into **0** red cards, **7** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_48_c3c7b9ed04 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n4 7 2 5 3 6 8 1\n\nNow, you need to color these cards into **4** red cards, **3** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_49_aa59a96bbe | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n6 4 1 2 5 7 8 3\n8 6 1 7 4 2 5 3\n\nNow, you need to color these cards into **4** red cards, **0** green cards, and **4** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_50_d89946f6f0 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n6 8 4 1 5 7 2 3\n\nNow, you need to color these cards into **3** red cards, **3** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_51_072b963c8d | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n8 4 7 2 6 5 3 1\n\nNow, you need to color these cards into **2** red cards, **3** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_52_8508a3421e | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **5** red cards, **0** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_53_48873ae2f1 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 2 4 6 3 7 5 8\n\nNow, you need to color these cards into **1** red cards, **3** green cards, and **4** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_54_61c43b4a63 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 2 3 8 5 7 6 4\n\nNow, you need to color these cards into **3** red cards, **0** green cards, and **5** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_55_b196ef8713 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n5 2 4 6 3 1 8 7\n3 4 1 2 6 5 7 8\n\nNow, you need to color these cards into **2** red cards, **4** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_56_8d39b4600c | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n6 3 7 1 4 2 8 5\n7 4 3 2 6 8 1 5\n3 8 1 4 2 6 5 7\n\nNow, you need to color these cards into **0** red cards, **7** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_57_591eb6fad4 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n5 6 3 2 8 1 7 4\n\nNow, you need to color these cards into **6** red cards, **0** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_5_58_39908e8286 | [
{
"content": "You have **8** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 8 integers: X_1, X_2, ..., X_8. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n8 6 1 4 7 5 3 2\n2 7 6 3 1 4 5 8\n4 3 5 1 6 7 8 2\n\nNow, you need to color these cards into **7** red cards, **0** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_0_8841947198 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 2 1\n2 3 1\n3 1 2\n\nNow, you need to color these cards into **0** red cards, **0** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_1_a01510a373 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 2 3\n\nNow, you need to color these cards into **0** red cards, **2** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_2_8bc3f53d47 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **3** red cards, **0** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_3_e8b1cfaad3 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 3 1\n\nNow, you need to color these cards into **2** red cards, **1** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_4_0ad4001c00 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 1 3\n\nNow, you need to color these cards into **1** red cards, **1** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_5_e129faf823 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 1 3\n1 3 2\n\nNow, you need to color these cards into **1** red cards, **2** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_6_2f53c2e096 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **0** red cards, **0** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_7_e6b77ce456 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 3 2\n3 1 2\n\nNow, you need to color these cards into **1** red cards, **2** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_8_cd7ef4f61f | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 3 2\n\nNow, you need to color these cards into **1** red cards, **1** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_9_e1adcfb33a | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 3 2\n3 2 1\n2 1 3\n\nNow, you need to color these cards into **0** red cards, **3** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_10_db6c8bd200 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 2 3\n1 2 3\n1 3 2\n\nNow, you need to color these cards into **0** red cards, **3** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_11_f469ad6277 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 2 3\n2 3 1\n3 2 1\n\nNow, you need to color these cards into **3** red cards, **0** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_12_4a5eadffb9 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 1 2\n1 3 2\n\nNow, you need to color these cards into **1** red cards, **2** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_13_014b602a20 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **0** red cards, **2** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_14_bc95b3a21f | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 3 2\n\nNow, you need to color these cards into **0** red cards, **1** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_15_3486545773 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 1 3\n1 2 3\n\nNow, you need to color these cards into **2** red cards, **0** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_16_2d308f9e45 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 2 3\n1 3 2\n3 2 1\n\nNow, you need to color these cards into **3** red cards, **0** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_17_03c20943a3 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 1 3\n\nNow, you need to color these cards into **1** red cards, **0** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_18_056cae38d5 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 2 1\n\nNow, you need to color these cards into **1** red cards, **1** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_19_67e158fb30 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 3 2\n\nNow, you need to color these cards into **2** red cards, **0** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_20_417ef0815d | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 3 1\n1 3 2\n\nNow, you need to color these cards into **1** red cards, **0** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_21_3bcc303217 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 3 2\n3 1 2\n3 1 2\n\nNow, you need to color these cards into **0** red cards, **0** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_22_3c43d410ae | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 2 1\n2 1 3\n3 2 1\n\nNow, you need to color these cards into **0** red cards, **0** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_23_249714d64c | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 1 2\n\nNow, you need to color these cards into **0** red cards, **2** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_24_c9901ab51a | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 3 2\n3 2 1\n2 3 1\n\nNow, you need to color these cards into **0** red cards, **3** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_25_fc36e18945 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **1** red cards, **0** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_26_bdda12514f | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 1 3\n2 1 3\n\nNow, you need to color these cards into **0** red cards, **2** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_27_ccb756322d | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **0** red cards, **3** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_28_ea94ec5336 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **2** red cards, **1** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_29_304cf19767 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 1 3\n\nNow, you need to color these cards into **0** red cards, **2** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_30_52bafc4962 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 3 1\n2 3 1\n\nNow, you need to color these cards into **0** red cards, **1** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_31_e5864425ea | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 2 3\n2 1 3\n\nNow, you need to color these cards into **2** red cards, **0** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_32_7d49f50c7b | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 3 2\n1 2 3\n\nNow, you need to color these cards into **1** red cards, **0** green cards, and **2** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_33_87fdb824fe | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 2 3\n2 1 3\n3 2 1\n\nNow, you need to color these cards into **0** red cards, **3** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_34_dd73b45094 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 1 2\n2 3 1\n1 3 2\n\nNow, you need to color these cards into **0** red cards, **0** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_35_c49a1fca29 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n2 3 1\n2 1 3\n3 1 2\n\nNow, you need to color these cards into **0** red cards, **0** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_36_83fdcecd11 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n\n\nNow, you need to color these cards into **2** red cards, **0** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_37_dcce9b55f8 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 2 3\n1 3 2\n1 3 2\n\nNow, you need to color these cards into **0** red cards, **0** green cards, and **3** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_38_3384fc205b | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n1 2 3\n1 3 2\n\nNow, you need to color these cards into **2** red cards, **0** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_39_ccecd14c7f | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 2 1\n3 2 1\n\nNow, you need to color these cards into **2** red cards, **1** green cards, and **0** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
cardcoloringcounting_diff_0_40_8d8bc7a651 | [
{
"content": "You have **3** cards, and there are some card shuffling methods. Each shuffle method is described by a row of 3 integers: X_1, X_2, ..., X_3. After applying a shuffle method, the card at position i moves to the position X_i. The shuffling methods are as follows:\n3 1 2\n1 3 2\n\nNow, you need to color these cards into **2** red cards, **0** green cards, and **1** blue cards.\nDetermine how many distinct coloring schemes exist, where two colorings are considered the same if and only if one can be transformed into the other by any combination of the available shuffling methods (shuffles may be applied multiple times and in any order).\n\n**Output Format:** Your final answer should be a single integer — the total number of distinct coloring schemes, considering equivalence under the shuffling methods.",
"role": "user"
}
]
|
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