messages listlengths 1 1 | ground_truth stringclasses 9 values | dataset stringclasses 1 value |
|---|---|---|
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{
"content": "A 9-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 blue, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is blue. The 9-gon is then rotated clockwise by 5 positions, so ... | 2 | MATH |
[
{
"content": "A 6-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 blue, vertex 5 is blue. The 6-gon is then rotated clockwise by 1 positions, so that the vertex that was originally in position 0 is n... | 2 | MATH |
[
{
"content": "A 8-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 blue, vertex 6 is blue, vertex 7 is red. The 8-gon is then rotated clockwise by 6 positions, so that the vertex tha... | 2 | MATH |
[
{
"content": "A 9-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 red, vertex 3 is uncolored, vertex 4 is red, vertex 5 is uncolored, vertex 6 is red, vertex 7 is blue, vertex 8 is blue. How many valid colorings of the uncolore... | 2 | MATH |
[
{
"content": "A 7-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 red. What is the maximum number of blue vertices that can be made to occupy a position where ther... | 3 | MATH |
[
{
"content": "A 7-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 red, vertex 4 is red, vertex 5 is uncolored, vertex 6 is uncolored. How many valid colorings of the uncolored vertices exist such that the 7-gon... | 5 | MATH |
[
{
"content": "A 6-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. What is the maximum number of blue vertices that can be made to occupy a position where there were originall... | 2 | MATH |
[
{
"content": "A 9-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 uncolored, vertex 4 is blue, vertex 5 is red, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is blue. How many valid colorings of the uncolor... | 0 | MATH |
[
{
"content": "A 8-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. The 8-gon is then rotated clockwise by 6 positions, so that the vertex t... | 2 | MATH |
[
{
"content": "A 6-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 red, vertex 5 is uncolored. How many valid colorings of the uncolored vertices exist such that the 6-gon can then be rota... | 2 | MATH |
[
{
"content": "A 6-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. What is the maximum number of blue vertices that can be made to occupy a position where there were originally ... | 2 | MATH |
[
{
"content": "A 7-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 uncolored, vertex 3 is uncolored, vertex 4 is red, vertex 5 is blue, vertex 6 is red. How many valid colorings of the uncolored vertices exist such that the 7-go... | 3 | MATH |
[
{
"content": "A 8-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 blue. The 8-gon is then rotated clockwise by 3 positions, so that the vertex tha... | 2 | MATH |
[
{
"content": "A 7-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 uncolored, vertex 4 is red, vertex 5 is red, vertex 6 is uncolored. How many valid colorings of the uncolored vertices exist such that the 7-gon... | 5 | MATH |
[
{
"content": "A 8-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 red, vertex 7 is blue. What is the maximum number of blue vertices that can be made to occupy a... | 2 | MATH |
[
{
"content": "A 8-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 blue, vertex 6 is blue, vertex 7 is blue. What is the maximum number of blue vertices that can be made to occupy a p... | 3 | MATH |
[
{
"content": "A 9-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 blue. What is the maximum number of red vertices that can be... | 2 | MATH |
[
{
"content": "A 9-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 uncolored, vertex 6 is red, vertex 7 is red, vertex 8 is uncolored. How many valid colorings of the uncolored... | 0 | MATH |
[
{
"content": "A 6-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. How many valid colorings of the uncolored vertices exist such that the 6-gon can then be rot... | 2 | MATH |
[
{
"content": "A 6-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 blue. What is the maximum number of red vertices that can be made to occupy a position where there were originally b... | 2 | MATH |
[
{
"content": "A 7-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 uncolored, vertex 3 is blue, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is red. How many valid colorings of the uncolored vertices exist such that the 7-go... | 2 | MATH |
[
{
"content": "A 7-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 blue, vertex 5 is blue, vertex 6 is red. What is the maximum number of blue vertices that can be made to occupy a position where the... | 3 | MATH |
[
{
"content": "A 8-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 red, vertex 4 is uncolored, vertex 5 is red, vertex 6 is uncolored, vertex 7 is blue. How many valid colorings of the uncolored vertices exist su... | 0 | MATH |
[
{
"content": "A 9-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 red, vertex 8 is blue. What is the maximum number of blue vertices that can be ma... | 3 | MATH |
[
{
"content": "A 9-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 blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue, vertex 8 is red. The 9-gon is then rotated clockwise by 4 positions, so ... | 3 | MATH |
[
{
"content": "A 9-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 red, vertex 8 is red. What is the maximum number of blue vertices that can be m... | 3 | MATH |
[
{
"content": "A 8-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 red, vertex 7 is red. What is the maximum number of blue vertices that can be made to occupy a po... | 3 | MATH |
[
{
"content": "A 6-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 uncolored, vertex 4 is uncolored, vertex 5 is red. How many valid colorings of the uncolored vertices exist such that the 6-gon can then be rot... | 2 | MATH |
[
{
"content": "A 8-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 uncolored, vertex 6 is blue, vertex 7 is blue. How many valid colorings of the uncolored vertices exist... | 8 | MATH |
[
{
"content": "A 7-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 uncolored, vertex 4 is uncolored, vertex 5 is red, vertex 6 is uncolored. How many valid colorings of the uncolored vertices exist such that the 7-gon ... | 7 | MATH |
[
{
"content": "A 9-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 blue, vertex 5 is blue, vertex 6 is red, vertex 7 is blue, vertex 8 is red. What is the maximum number of blue vertices that can be ... | 3 | MATH |
[
{
"content": "A 8-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 blue, vertex 7 is blue. What is the maximum number of red vertices that can be made to occupy a... | 1 | MATH |
[
{
"content": "A 8-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 blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is red. What is the maximum number of blue vertices that can be made to occupy a ... | 3 | MATH |
[
{
"content": "A 7-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 red, vertex 6 is blue. What is the maximum number of red vertices that can be made to occupy a position where the... | 2 | MATH |
[
{
"content": "A 9-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 red, vertex 8 is blue. What is the maximum number of red vertices that can be ... | 3 | MATH |
[
{
"content": "A 8-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 red, vertex 7 is red. What is the maximum number of red vertices that can be made to occupy a pos... | 3 | MATH |
[
{
"content": "A 8-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 blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is blue. The 8-gon is then rotated clockwise by 6 positions, so that the vertex th... | 2 | MATH |
[
{
"content": "A 7-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 blue. The 7-gon is then rotated clockwise by 2 positions, so that the vertex that was originally ... | 2 | MATH |
[
{
"content": "A 6-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 blue. What is the maximum number of red vertices that can be made to occupy a position where there were originally b... | 2 | MATH |
[
{
"content": "A 8-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 blue, vertex 7 is red. What is the maximum number of red vertices that can be made to occupy a posi... | 2 | MATH |
[
{
"content": "A 6-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 red, vertex 5 is blue. What is the maximum number of blue vertices that can be made to occupy a position where there were original... | 1 | MATH |
[
{
"content": "A 7-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 red. What is the maximum number of blue vertices that can be made to occupy a position where there... | 2 | MATH |
[
{
"content": "A 8-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 red, vertex 3 is uncolored, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is uncolored. How many valid colorings of the uncolored vertices exist su... | 7 | MATH |
[
{
"content": "A 7-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. What is the maximum number of blue vertices that can be made to occupy a position where there ... | 1 | MATH |
[
{
"content": "A 7-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 uncolored, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is blue, vertex 6 is red. How many valid colorings of the uncolored vertices exist such that the 7-go... | 3 | MATH |
[
{
"content": "A 9-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 blue, vertex 8 is blue. The 9-gon is then rotated clockwise by 7 positions, so th... | 2 | MATH |
[
{
"content": "A 8-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 uncolored, vertex 4 is uncolored, vertex 5 is blue, vertex 6 is blue, vertex 7 is red. How many valid colorings of the uncolored vertices exist ... | 1 | MATH |
[
{
"content": "A 8-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 blue, vertex 3 is uncolored, vertex 4 is red, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is red. How many valid colorings of the uncolored vertices exist s... | 5 | MATH |
[
{
"content": "A 7-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. What is the maximum number of blue vertices that can be made to occupy a position where th... | 2 | MATH |
[
{
"content": "A 6-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 uncolored, vertex 4 is blue, vertex 5 is uncolored. How many valid colorings of the uncolored vertices exist such that the 6-gon can then be rot... | 3 | MATH |
[
{
"content": "A 9-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 blue, vertex 8 is red. What is the maximum number of red vertices that can be... | 2 | MATH |
[
{
"content": "A 6-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 red. What is the maximum number of red vertices that can be made to occupy a position where there were originally b... | 2 | MATH |
[
{
"content": "A 9-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 red, vertex 8 is blue. What is the maximum number of blue vertices that can be ... | 3 | MATH |
[
{
"content": "A 9-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 blue, vertex 3 is red, vertex 4 is uncolored, vertex 5 is red, vertex 6 is red, vertex 7 is red, vertex 8 is uncolored. How many valid colorings of the uncolored ... | 0 | MATH |
[
{
"content": "A 8-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 red, vertex 7 is red. What is the maximum number of red vertices that can be made to occupy a po... | 4 | MATH |
[
{
"content": "A 6-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 blue. What is the maximum number of blue vertices that can be made to occupy a position where there were originally r... | 1 | MATH |
[
{
"content": "A 7-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 red, vertex 5 is red, vertex 6 is blue. What is the maximum number of red vertices that can be made to occupy a position where ther... | 2 | MATH |
[
{
"content": "A 7-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 red. What is the maximum number of blue vertices that can be made to occupy a position where ther... | 2 | MATH |
[
{
"content": "A 6-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 blue, vertex 5 is blue. What is the maximum number of blue vertices that can be made to occupy a position where there were originall... | 2 | MATH |
[
{
"content": "A 8-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 red, vertex 5 is uncolored, vertex 6 is uncolored, vertex 7 is blue. How many valid colorings of the uncolored vertices exist s... | 5 | MATH |
[
{
"content": "A 8-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 red, vertex 5 is red, vertex 6 is uncolored, vertex 7 is uncolored. How many valid colorings of the uncolored vertices exist ... | 6 | MATH |
[
{
"content": "A 7-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 uncolored, vertex 5 is uncolored, vertex 6 is red. How many valid colorings of the uncolored vertices exist such that the 7-g... | 7 | MATH |
[
{
"content": "A 7-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 red. The 7-gon is then rotated clockwise by 1 positions, so that the vertex that was originally... | 1 | MATH |
[
{
"content": "A 9-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 uncolored, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is red, vertex 7 is uncolored, vertex 8 is red. How many valid colorings of the uncolored... | 0 | MATH |
[
{
"content": "A 8-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 uncolored, vertex 3 is red, vertex 4 is blue, vertex 5 is blue, vertex 6 is uncolored, vertex 7 is red. How many valid colorings of the uncolored vertices exist s... | 1 | MATH |
[
{
"content": "A 8-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 uncolored, vertex 4 is blue, vertex 5 is uncolored, vertex 6 is uncolored, vertex 7 is blue. How many valid colorings of the uncolored vertices exist s... | 2 | MATH |
[
{
"content": "A 6-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 uncolored, vertex 3 is uncolored, vertex 4 is red, vertex 5 is red. How many valid colorings of the uncolored vertices exist such that the 6-gon can then be rota... | 3 | MATH |
[
{
"content": "A 6-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 red, vertex 4 is red, vertex 5 is blue. How many valid colorings of the uncolored vertices exist such that the 6-gon can then be rota... | 5 | MATH |
[
{
"content": "A 9-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 blue, vertex 6 is red, vertex 7 is blue, vertex 8 is blue. The 9-gon is then rotated clockwise by 1 positions, so... | 2 | MATH |
[
{
"content": "A 8-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 blue, vertex 5 is blue, vertex 6 is red, vertex 7 is red. What is the maximum number of red vertices that can be made to occupy a po... | 4 | MATH |
[
{
"content": "A 6-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 red, vertex 5 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally... | 2 | MATH |
[
{
"content": "A 7-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 blue, vertex 5 is red, vertex 6 is red. What is the maximum number of blue vertices that can be made to occupy a position where ther... | 3 | MATH |
[
{
"content": "A 8-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 red, vertex 6 is red, vertex 7 is red. What is the maximum number of red vertices that can be made to occupy a posi... | 2 | MATH |
[
{
"content": "A 6-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. The 6-gon is then rotated clockwise by 5 positions, so that the vertex that was originally in position 0 is no... | 2 | MATH |
[
{
"content": "A 7-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 red. What is the maximum number of red vertices that can be made to occupy a position where there... | 2 | MATH |
[
{
"content": "A 7-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 uncolored, vertex 3 is red, vertex 4 is red, vertex 5 is blue, vertex 6 is uncolored. How many valid colorings of the uncolored vertices exist such that the 7-go... | 3 | MATH |
[
{
"content": "A 7-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 red, vertex 5 is blue, vertex 6 is blue. What is the maximum number of blue vertices that can be made to occupy a position where th... | 2 | MATH |
[
{
"content": "A 8-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 blue. The 8-gon is then rotated clockwise by 2 positions, so that the vertex t... | 2 | MATH |
[
{
"content": "A 6-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. What is the maximum number of red vertices that can be made to occupy a position where there were originally... | 2 | MATH |
[
{
"content": "A 8-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. What is the maximum number of blue vertices that can be made to occupy a po... | 3 | MATH |
[
{
"content": "A 9-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 red, vertex 5 is blue, vertex 6 is red, vertex 7 is red, vertex 8 is blue. The 9-gon is then rotated clockwise by 1 positions, so t... | 3 | MATH |
[
{
"content": "A 9-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 blue, vertex 5 is blue, vertex 6 is blue, vertex 7 is red, vertex 8 is blue. What is the maximum number of red vertices that can be m... | 3 | MATH |
[
{
"content": "A 7-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 blue, vertex 6 is blue. The 7-gon is then rotated clockwise by 2 positions, so that the vertex that was originally ... | 1 | MATH |
[
{
"content": "A 7-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. What is the maximum number of blue vertices that can be made to occupy a position where th... | 2 | MATH |
[
{
"content": "A 6-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 blue. What is the maximum number of blue vertices that can be made to occupy a position where there were originally... | 3 | MATH |
[
{
"content": "A 7-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 red, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is red, vertex 6 is uncolored. How many valid colorings of the uncolored vertices exist such that the 7-go... | 3 | MATH |
[
{
"content": "A 8-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 uncolored, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is uncolored. How many valid colorings of the uncolored vertices exist su... | 7 | MATH |
[
{
"content": "A 7-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. What is the maximum number of red vertices that can be made to occupy a position where th... | 1 | MATH |
[
{
"content": "A 8-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 uncolored, vertex 4 is red, vertex 5 is red, vertex 6 is red, vertex 7 is uncolored. How many valid colorings of the uncolored vertices exist s... | 3 | MATH |
[
{
"content": "A 6-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. The 6-gon is then rotated clockwise by 4 positions, so that the vertex that was originally in position 0 is no... | 2 | MATH |
[
{
"content": "A 8-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 red. What is the maximum number of blue vertices that can be made to occupy a p... | 4 | MATH |
[
{
"content": "A 6-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. What is the maximum number of blue vertices that can be made to occupy a position where there were originally ... | 2 | MATH |
[
{
"content": "A 9-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 blue, vertex 4 is red, vertex 5 is blue, vertex 6 is red, vertex 7 is uncolored, vertex 8 is uncolored. How many valid colorings of the uncolore... | 2 | MATH |
[
{
"content": "A 8-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 blue, vertex 6 is blue, vertex 7 is red. What is the maximum number of blue vertices that can be made to occupy a po... | 3 | MATH |
[
{
"content": "A 9-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 uncolored, vertex 5 is blue, vertex 6 is blue, vertex 7 is uncolored, vertex 8 is uncolored. How many valid colorings of the uncolor... | 4 | MATH |
[
{
"content": "A 8-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 red, vertex 7 is red. What is the maximum number of red vertices that can be made to occupy a posit... | 1 | MATH |
[
{
"content": "A 7-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 uncolored, vertex 5 is blue, vertex 6 is blue. How many valid colorings of the uncolored vertices exist such that the 7-go... | 3 | MATH |
[
{
"content": "A 8-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 uncolored, vertex 3 is red, vertex 4 is red, vertex 5 is uncolored, vertex 6 is blue, vertex 7 is blue. How many valid colorings of the uncolored vertices exist ... | 3 | MATH |
[
{
"content": "A 7-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 blue. The 7-gon is then rotated clockwise by 5 positions, so that the vertex that was originally i... | 2 | MATH |
[
{
"content": "A 6-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 red. What is the maximum number of red vertices that can be made to occupy a position where there were originally b... | 2 | MATH |
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