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[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is uncolored, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is blue, vertex 8 is blue, vertex 9 is blue, vertex 10 is red, vertex 11 is uncolored. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is red, vertex 8 is blue, vertex 9 is red, vertex 10 is blue, vertex 11 is blue. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is uncolored, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is blue, vertex 9 is red, vertex 10 is blue, vertex 11 is uncolored. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
6
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is uncolored, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue, vertex 9 is uncolored, vertex 10 is blue, vertex 11 is blue. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is red, vertex 9 is red, vertex 10 is blue, vertex 11 is red, vertex 12 is red. The 13-gon is then rotated clockwise by 10 positions, so that the vertex that was originally in position 0 is now in position 10. How many red vertices are now in positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is uncolored, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is uncolored, vertex 7 is uncolored, vertex 8 is red, vertex 9 is red, vertex 10 is red, vertex 11 is red. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue, vertex 9 is blue, vertex 10 is red, vertex 11 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is uncolored, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is blue, vertex 9 is uncolored, vertex 10 is blue, vertex 11 is uncolored. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is blue, vertex 10 is red, vertex 11 is blue, vertex 12 is red. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 13-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
5
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is blue, vertex 10 is red, vertex 11 is blue, vertex 12 is blue. The 13-gon is then rotated clockwise by 10 positions, so that the vertex that was originally in position 0 is now in position 10. How many blue vertices are now in positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is red, vertex 10 is blue, vertex 11 is blue. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, vertex 11 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is blue, vertex 9 is blue, vertex 10 is blue, vertex 11 is red, vertex 12 is red. The 13-gon is then rotated clockwise by 9 positions, so that the vertex that was originally in position 0 is now in position 9. How many red vertices are now in positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, vertex 11 is blue, vertex 12 is blue. The 13-gon is then rotated clockwise by 11 positions, so that the vertex that was originally in position 0 is now in position 11. How many red vertices are now in positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is blue, vertex 9 is blue, vertex 10 is red, vertex 11 is red. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
5
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is red, vertex 9 is blue, vertex 10 is red, vertex 11 is blue, vertex 12 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 13-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, vertex 11 is blue. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is uncolored, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is red, vertex 9 is red, vertex 10 is red, vertex 11 is red, vertex 12 is uncolored. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is blue, vertex 9 is blue, vertex 10 is blue, vertex 11 is blue, vertex 12 is blue. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 13-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
5
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is uncolored, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is uncolored, vertex 7 is blue, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, vertex 11 is red, vertex 12 is red. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, vertex 11 is blue, vertex 12 is red. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 13-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
5
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is blue, vertex 8 is red, vertex 9 is blue, vertex 10 is uncolored, vertex 11 is uncolored, vertex 12 is blue. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue, vertex 9 is blue, vertex 10 is blue, vertex 11 is blue. The 12-gon is then rotated clockwise by 4 positions, so that the vertex that was originally in position 0 is now in position 4. How many red vertices are now in positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is red, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is uncolored, vertex 9 is red, vertex 10 is blue, vertex 11 is blue. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is red, vertex 8 is red, vertex 9 is blue, vertex 10 is red, vertex 11 is red. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue, vertex 9 is blue, vertex 10 is red, vertex 11 is blue, vertex 12 is red. The 13-gon is then rotated clockwise by 10 positions, so that the vertex that was originally in position 0 is now in position 10. How many blue vertices are now in positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is uncolored, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue, vertex 9 is blue, vertex 10 is uncolored, vertex 11 is blue, vertex 12 is uncolored. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is uncolored, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is uncolored, vertex 10 is blue, vertex 11 is red. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue, vertex 9 is red, vertex 10 is red, vertex 11 is blue. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is uncolored, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is red, vertex 6 is red, vertex 7 is blue, vertex 8 is blue, vertex 9 is red, vertex 10 is uncolored, vertex 11 is red. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is red, vertex 9 is red, vertex 10 is red, vertex 11 is red, vertex 12 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 13-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is uncolored, vertex 7 is red, vertex 8 is uncolored, vertex 9 is uncolored, vertex 10 is blue, vertex 11 is red, vertex 12 is red. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is red, vertex 10 is blue, vertex 11 is blue. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is blue, vertex 9 is red, vertex 10 is red, vertex 11 is red. The 12-gon is then rotated clockwise by 4 positions, so that the vertex that was originally in position 0 is now in position 4. How many blue vertices are now in positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is red, vertex 9 is red, vertex 10 is red, vertex 11 is blue. The 12-gon is then rotated clockwise by 1 positions, so that the vertex that was originally in position 0 is now in position 1. How many red vertices are now in positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is red, vertex 2 is red, vertex 3 is uncolored, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is red, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, vertex 11 is uncolored. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, vertex 11 is red. The 12-gon is then rotated clockwise by 4 positions, so that the vertex that was originally in position 0 is now in position 4. How many blue vertices are now in positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is uncolored, vertex 5 is red, vertex 6 is red, vertex 7 is red, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, vertex 11 is uncolored. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is uncolored, vertex 2 is blue, vertex 3 is red, vertex 4 is uncolored, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is blue, vertex 9 is blue, vertex 10 is red, vertex 11 is uncolored, vertex 12 is blue. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue, vertex 9 is red, vertex 10 is red, vertex 11 is blue, vertex 12 is red. The 13-gon is then rotated clockwise by 3 positions, so that the vertex that was originally in position 0 is now in position 3. How many blue vertices are now in positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is red, vertex 8 is blue, vertex 9 is blue, vertex 10 is red, vertex 11 is red, vertex 12 is red. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 13-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is red, vertex 8 is red, vertex 9 is uncolored, vertex 10 is uncolored, vertex 11 is uncolored. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is red, vertex 10 is red, vertex 11 is blue. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is red, vertex 7 is blue, vertex 8 is blue, vertex 9 is red, vertex 10 is red, vertex 11 is blue, vertex 12 is blue. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 13-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is blue, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, vertex 11 is red, vertex 12 is blue. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 13-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
5
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue, vertex 9 is red, vertex 10 is red, vertex 11 is red. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is red, vertex 7 is red, vertex 8 is blue, vertex 9 is blue, vertex 10 is red, vertex 11 is red, vertex 12 is blue. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 13-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, vertex 11 is blue. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is blue, vertex 9 is red, vertex 10 is red, vertex 11 is blue. The 12-gon is then rotated clockwise by 4 positions, so that the vertex that was originally in position 0 is now in position 4. How many blue vertices are now in positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is uncolored, vertex 2 is uncolored, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is red, vertex 9 is red, vertex 10 is red, vertex 11 is red. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is blue, vertex 9 is blue, vertex 10 is blue, vertex 11 is blue, vertex 12 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 13-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
5
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue, vertex 9 is blue, vertex 10 is red, vertex 11 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is red, vertex 9 is red, vertex 10 is uncolored, vertex 11 is red, vertex 12 is blue. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is blue, vertex 8 is blue, vertex 9 is red, vertex 10 is blue, vertex 11 is blue. The 12-gon is then rotated clockwise by 11 positions, so that the vertex that was originally in position 0 is now in position 11. How many blue vertices are now in positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, vertex 11 is blue, vertex 12 is red. The 13-gon is then rotated clockwise by 12 positions, so that the vertex that was originally in position 0 is now in position 12. How many red vertices are now in positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is uncolored, vertex 3 is uncolored, vertex 4 is uncolored, vertex 5 is red, vertex 6 is red, vertex 7 is red, vertex 8 is red, vertex 9 is red, vertex 10 is blue, vertex 11 is blue, vertex 12 is blue. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is red, vertex 8 is uncolored, vertex 9 is uncolored, vertex 10 is blue, vertex 11 is red. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is blue, vertex 8 is blue, vertex 9 is red, vertex 10 is blue, vertex 11 is blue, vertex 12 is blue. The 13-gon is then rotated clockwise by 1 positions, so that the vertex that was originally in position 0 is now in position 1. How many blue vertices are now in positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is red, vertex 8 is blue, vertex 9 is red, vertex 10 is blue, vertex 11 is blue, vertex 12 is blue. The 13-gon is then rotated clockwise by 3 positions, so that the vertex that was originally in position 0 is now in position 3. How many red vertices are now in positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
5
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is uncolored, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is uncolored, vertex 7 is blue, vertex 8 is red, vertex 9 is red, vertex 10 is uncolored, vertex 11 is blue. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is uncolored, vertex 2 is red, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is red, vertex 7 is red, vertex 8 is blue, vertex 9 is blue, vertex 10 is red, vertex 11 is blue. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is blue, vertex 7 is red, vertex 8 is blue, vertex 9 is red, vertex 10 is red, vertex 11 is red, vertex 12 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 13-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is uncolored, vertex 7 is uncolored, vertex 8 is red, vertex 9 is red, vertex 10 is uncolored, vertex 11 is blue. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is red, vertex 10 is red, vertex 11 is red. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is red, vertex 8 is blue, vertex 9 is red, vertex 10 is red, vertex 11 is uncolored, vertex 12 is uncolored. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, vertex 11 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is red, vertex 7 is red, vertex 8 is blue, vertex 9 is blue, vertex 10 is blue, vertex 11 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is uncolored, vertex 3 is blue, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue, vertex 9 is uncolored, vertex 10 is blue, vertex 11 is blue. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is uncolored, vertex 2 is red, vertex 3 is blue, vertex 4 is uncolored, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue, vertex 9 is blue, vertex 10 is blue, vertex 11 is blue, vertex 12 is blue. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is red, vertex 9 is red, vertex 10 is blue, vertex 11 is blue. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is blue, vertex 2 is blue, vertex 3 is uncolored, vertex 4 is red, vertex 5 is uncolored, vertex 6 is red, vertex 7 is red, vertex 8 is red, vertex 9 is blue, vertex 10 is red, vertex 11 is blue. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is blue, vertex 8 is blue, vertex 9 is red, vertex 10 is uncolored, vertex 11 is uncolored, vertex 12 is red. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is uncolored, vertex 9 is red, vertex 10 is uncolored, vertex 11 is red. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is uncolored, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is uncolored, vertex 7 is blue, vertex 8 is red, vertex 9 is red, vertex 10 is blue, vertex 11 is blue, vertex 12 is blue. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, vertex 11 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is red, vertex 9 is blue, vertex 10 is red, vertex 11 is blue, vertex 12 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 13-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue, vertex 9 is red, vertex 10 is red, vertex 11 is blue. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is red, vertex 8 is blue, vertex 9 is red, vertex 10 is red, vertex 11 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is uncolored, vertex 2 is red, vertex 3 is blue, vertex 4 is uncolored, vertex 5 is red, vertex 6 is red, vertex 7 is uncolored, vertex 8 is red, vertex 9 is blue, vertex 10 is blue, vertex 11 is blue, vertex 12 is blue. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is red, vertex 7 is red, vertex 8 is red, vertex 9 is red, vertex 10 is red, vertex 11 is blue, vertex 12 is red. The 13-gon is then rotated clockwise by 5 positions, so that the vertex that was originally in position 0 is now in position 5. How many blue vertices are now in positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is uncolored, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is red, vertex 7 is red, vertex 8 is uncolored, vertex 9 is red, vertex 10 is blue, vertex 11 is blue, vertex 12 is red. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is blue, vertex 9 is red, vertex 10 is blue, vertex 11 is red. The 12-gon is then rotated clockwise by 7 positions, so that the vertex that was originally in position 0 is now in position 7. How many blue vertices are now in positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is uncolored, vertex 2 is blue, vertex 3 is blue, vertex 4 is uncolored, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is blue, vertex 9 is red, vertex 10 is uncolored, vertex 11 is red, vertex 12 is blue. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is uncolored, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is red, vertex 9 is blue, vertex 10 is uncolored, vertex 11 is red. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is uncolored, vertex 3 is red, vertex 4 is red, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is red, vertex 8 is red, vertex 9 is blue, vertex 10 is red, vertex 11 is uncolored. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is blue, vertex 5 is blue, vertex 6 is red, vertex 7 is red, vertex 8 is blue, vertex 9 is blue, vertex 10 is red, vertex 11 is red. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is blue, vertex 4 is uncolored, vertex 5 is red, vertex 6 is red, vertex 7 is uncolored, vertex 8 is uncolored, vertex 9 is red, vertex 10 is blue, vertex 11 is red, vertex 12 is red. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is blue, vertex 8 is uncolored, vertex 9 is blue, vertex 10 is red, vertex 11 is uncolored. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is uncolored, vertex 2 is red, vertex 3 is red, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is red, vertex 8 is blue, vertex 9 is red, vertex 10 is red, vertex 11 is uncolored. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is red, vertex 7 is red, vertex 8 is blue, vertex 9 is blue, vertex 10 is blue, vertex 11 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 12-gon?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is red, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is red, vertex 7 is blue, vertex 8 is red, vertex 9 is blue, vertex 10 is red, vertex 11 is blue, vertex 12 is red. The 13-gon is then rotated clockwise by 5 positions, so that the vertex that was originally in position 0 is now in position 5. How many blue vertices are now in positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is uncolored, vertex 2 is red, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is blue, vertex 8 is blue, vertex 9 is blue, vertex 10 is blue, vertex 11 is uncolored, vertex 12 is red. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is uncolored, vertex 9 is uncolored, vertex 10 is red, vertex 11 is red, vertex 12 is uncolored. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is red, vertex 2 is blue, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is red, vertex 6 is uncolored, vertex 7 is blue, vertex 8 is red, vertex 9 is uncolored, vertex 10 is red, vertex 11 is blue, vertex 12 is blue. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is blue, vertex 3 is uncolored, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is blue, vertex 9 is blue, vertex 10 is blue, vertex 11 is uncolored, vertex 12 is blue. How many valid colorings of the uncolored vertices exist such that the 13-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is blue, vertex 2 is red, vertex 3 is uncolored, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is red, vertex 8 is red, vertex 9 is red, vertex 10 is blue, vertex 11 is red. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
5
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is blue, vertex 3 is red, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is blue, vertex 9 is red, vertex 10 is uncolored, vertex 11 is blue. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 12-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is blue, vertex 1 is blue, vertex 2 is uncolored, vertex 3 is blue, vertex 4 is red, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is red, vertex 8 is red, vertex 9 is red, vertex 10 is uncolored, vertex 11 is blue. How many valid colorings of the uncolored vertices exist such that the 12-gon can then be rotated in such a way that all of the blue vertices end up at positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is blue, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is red, vertex 8 is blue, vertex 9 is red, vertex 10 is red, vertex 11 is blue, vertex 12 is blue. The 13-gon is then rotated clockwise by 6 positions, so that the vertex that was originally in position 0 is now in position 6. How many blue vertices are now in positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
MATH
|
[
{
"content": "A 13-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is red, vertex 3 is red, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is blue, vertex 8 is red, vertex 9 is red, vertex 10 is blue, vertex 11 is blue, vertex 12 is red. The 13-gon is then rotated clockwise by 9 positions, so that the vertex that was originally in position 0 is now in position 9. How many blue vertices are now in positions where there were originally red vertices?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
MATH
|
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