messages
listlengths 1
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values | dataset
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{
"content": "Consider a regular 11-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'blue' vertices. For each k from 0 to 11, determine how many ways we can color the vertices such that there exists at least one rotation of the 11-gon where every 'blue' vertex ends up at a position originally occupied by a 'red' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=11.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
[1, 11, 55, 165, 220, 55, 0, 0, 0, 0, 0, 0]
|
FINEGRAINED
|
[
{
"content": "Consider a regular 10-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'blue' vertices. For each k from 0 to 10, determine how many ways we can color the vertices such that there exists at least one rotation of the 10-gon where every 'blue' vertex ends up at a position originally occupied by a 'red' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
[1, 10, 45, 120, 130, 32, 0, 0, 0, 0, 0]
|
FINEGRAINED
|
[
{
"content": "Consider a regular 11-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'blue' vertices. For each k from 0 to 11, determine how many ways we can color the vertices such that there exists at least one rotation of the 11-gon where every 'blue' vertex ends up at a position originally occupied by a 'red' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=11.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
[1, 11, 55, 165, 220, 55, 0, 0, 0, 0, 0, 0]
|
FINEGRAINED
|
[
{
"content": "Consider a regular 11-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'blue' vertices. For each k from 0 to 11, determine how many ways we can color the vertices such that there exists at least one rotation of the 11-gon where every 'blue' vertex ends up at a position originally occupied by a 'red' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=11.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
[1, 11, 55, 165, 220, 55, 0, 0, 0, 0, 0, 0]
|
FINEGRAINED
|
[
{
"content": "Consider a regular 11-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'red' vertices. For each k from 0 to 11, determine how many ways we can color the vertices such that there exists at least one rotation of the 11-gon where every 'red' vertex ends up at a position originally occupied by a 'blue' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=11.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
[1, 11, 55, 165, 220, 55, 0, 0, 0, 0, 0, 0]
|
FINEGRAINED
|
[
{
"content": "Consider a regular 10-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'blue' vertices. For each k from 0 to 10, determine how many ways we can color the vertices such that there exists at least one rotation of the 10-gon where every 'blue' vertex ends up at a position originally occupied by a 'red' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
[1, 10, 45, 120, 130, 32, 0, 0, 0, 0, 0]
|
FINEGRAINED
|
[
{
"content": "Consider a regular 10-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'red' vertices. For each k from 0 to 10, determine how many ways we can color the vertices such that there exists at least one rotation of the 10-gon where every 'red' vertex ends up at a position originally occupied by a 'blue' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
[1, 10, 45, 120, 130, 32, 0, 0, 0, 0, 0]
|
FINEGRAINED
|
[
{
"content": "Consider a regular 11-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'red' vertices. For each k from 0 to 11, determine how many ways we can color the vertices such that there exists at least one rotation of the 11-gon where every 'red' vertex ends up at a position originally occupied by a 'blue' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=11.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
[1, 11, 55, 165, 220, 55, 0, 0, 0, 0, 0, 0]
|
FINEGRAINED
|
[
{
"content": "Consider a regular 11-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'red' vertices. For each k from 0 to 11, determine how many ways we can color the vertices such that there exists at least one rotation of the 11-gon where every 'red' vertex ends up at a position originally occupied by a 'blue' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=11.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
[1, 11, 55, 165, 220, 55, 0, 0, 0, 0, 0, 0]
|
FINEGRAINED
|
[
{
"content": "Consider a regular 10-gon where every vertex is uncolored. We want to color every vertex either Red or Blue. Let k be the number of 'blue' vertices. For each k from 0 to 10, determine how many ways we can color the vertices such that there exists at least one rotation of the 10-gon where every 'blue' vertex ends up at a position originally occupied by a 'red' vertex. What is the total number of such valid colorings across all possible values of k?Please return your results as a Python list of counts for k=0 to k=10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
[1, 10, 45, 120, 130, 32, 0, 0, 0, 0, 0]
|
FINEGRAINED
|
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