[User]
Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 4, from vertex 10 to vertex 7, and  from vertex 5 to vertex 10. Then, rotate the entire setup, including the constructed diagonals, 3 vertices counterclockwise (so that vertex 1 ends up where vertex 4 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 10-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 9, from vertex 8 to vertex 10, and  from vertex 3 to vertex 9. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 6 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 5, from vertex 6 to vertex 4, and  from vertex 5 to vertex 2. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 9 to vertex 6, from vertex 10 to vertex 4, and  from vertex 4 to vertex 8. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 9 to vertex 6, from vertex 10 to vertex 4, and  from vertex 5 to vertex 11. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 1, from vertex 6 to vertex 1, and  from vertex 5 to vertex 2. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 2, from vertex 2 to vertex 7, and  from vertex 5 to vertex 1. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 5, from vertex 2 to vertex 4, and  from vertex 5 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 3 vertices counterclockwise (so that vertex 1 ends up where vertex 4 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 10, from vertex 1 to vertex 9, and  from vertex 3 to vertex 9. Then, rotate the entire setup, including the constructed diagonals, 10 vertices counterclockwise (so that vertex 1 ends up where vertex 11 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 3, from vertex 8 to vertex 2, and  from vertex 3 to vertex 5. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 5, from vertex 9 to vertex 2, and  from vertex 2 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 7 vertices counterclockwise (so that vertex 1 ends up where vertex 8 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 2, from vertex 3 to vertex 8, and  from vertex 1 to vertex 9. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 3, from vertex 3 to vertex 7, and  from vertex 6 to vertex 2. Then, rotate the entire setup, including the constructed diagonals, 6 vertices counterclockwise (so that vertex 1 ends up where vertex 7 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 1, from vertex 6 to vertex 1, and  from vertex 8 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 8, from vertex 4 to vertex 2, and  from vertex 7 to vertex 5. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 5, from vertex 5 to vertex 2, and  from vertex 5 to vertex 1. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 5, from vertex 2 to vertex 5, and  from vertex 5 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 2, from vertex 1 to vertex 5, and  from vertex 6 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 5 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 6, from vertex 3 to vertex 8, and  from vertex 1 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 6 vertices counterclockwise (so that vertex 1 ends up where vertex 7 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 9 to vertex 6, from vertex 5 to vertex 8, and  from vertex 1 to vertex 5. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 2, from vertex 5 to vertex 1, and  from vertex 2 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 3, from vertex 2 to vertex 5, and  from vertex 2 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 5 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 7, from vertex 1 to vertex 6, and  from vertex 1 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 3 vertices counterclockwise (so that vertex 1 ends up where vertex 4 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 8, from vertex 8 to vertex 1, and  from vertex 4 to vertex 9. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 10-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 5, from vertex 11 to vertex 2, and  from vertex 5 to vertex 11. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 8, from vertex 1 to vertex 5, and  from vertex 2 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 3 vertices counterclockwise (so that vertex 1 ends up where vertex 4 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 6, from vertex 4 to vertex 6, and  from vertex 8 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 6, from vertex 1 to vertex 4, and  from vertex 2 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 3, from vertex 6 to vertex 2, and  from vertex 3 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 7, from vertex 3 to vertex 6, and  from vertex 9 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 1, from vertex 3 to vertex 5, and  from vertex 2 to vertex 10. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 7 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 2, from vertex 10 to vertex 3, and  from vertex 2 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 10 vertices counterclockwise (so that vertex 1 ends up where vertex 11 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 2, from vertex 8 to vertex 2, and  from vertex 6 to vertex 1. Then, rotate the entire setup, including the constructed diagonals, 6 vertices counterclockwise (so that vertex 1 ends up where vertex 7 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 8, from vertex 7 to vertex 5, and  from vertex 2 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 7, from vertex 1 to vertex 7, and  from vertex 4 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 6 vertices counterclockwise (so that vertex 1 ends up where vertex 7 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 7, from vertex 1 to vertex 3, and  from vertex 7 to vertex 9. Then, rotate the entire setup, including the constructed diagonals, 3 vertices counterclockwise (so that vertex 1 ends up where vertex 4 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 10-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 5, from vertex 7 to vertex 2, and  from vertex 1 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 9 to vertex 5, from vertex 2 to vertex 5, and  from vertex 7 to vertex 5. Then, rotate the entire setup, including the constructed diagonals, 6 vertices counterclockwise (so that vertex 1 ends up where vertex 7 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 5, from vertex 3 to vertex 6, and  from vertex 3 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 3 vertices counterclockwise (so that vertex 1 ends up where vertex 4 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 11 to vertex 9, from vertex 9 to vertex 1, and  from vertex 2 to vertex 10. Then, rotate the entire setup, including the constructed diagonals, 9 vertices counterclockwise (so that vertex 1 ends up where vertex 10 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 6, from vertex 3 to vertex 1, and  from vertex 7 to vertex 5. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 5 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 4, from vertex 4 to vertex 6, and  from vertex 6 to vertex 2. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 5 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 6, from vertex 5 to vertex 10, and  from vertex 1 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 10-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 5 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 1, from vertex 1 to vertex 6, and  from vertex 5 to vertex 1. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 3, from vertex 3 to vertex 8, and  from vertex 9 to vertex 11. Then, rotate the entire setup, including the constructed diagonals, 9 vertices counterclockwise (so that vertex 1 ends up where vertex 10 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 9 to vertex 1, from vertex 12 to vertex 9, and  from vertex 1 to vertex 10. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 8, from vertex 1 to vertex 4, and  from vertex 3 to vertex 5. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 3, from vertex 5 to vertex 9, and  from vertex 2 to vertex 12. Then, rotate the entire setup, including the constructed diagonals, 6 vertices counterclockwise (so that vertex 1 ends up where vertex 7 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 2, from vertex 4 to vertex 9, and  from vertex 6 to vertex 9. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 10-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 9, from vertex 1 to vertex 6, and  from vertex 9 to vertex 1. Then, rotate the entire setup, including the constructed diagonals, 3 vertices counterclockwise (so that vertex 1 ends up where vertex 4 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 10-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 6, from vertex 6 to vertex 8, and  from vertex 1 to vertex 8. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 6, from vertex 7 to vertex 4, and  from vertex 2 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 2, from vertex 11 to vertex 6, and  from vertex 10 to vertex 5. Then, rotate the entire setup, including the constructed diagonals, 10 vertices counterclockwise (so that vertex 1 ends up where vertex 11 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 1, from vertex 1 to vertex 6, and  from vertex 10 to vertex 5. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 7, from vertex 3 to vertex 9, and  from vertex 10 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 3 vertices counterclockwise (so that vertex 1 ends up where vertex 4 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 8, from vertex 6 to vertex 12, and  from vertex 11 to vertex 9. Then, rotate the entire setup, including the constructed diagonals, 7 vertices counterclockwise (so that vertex 1 ends up where vertex 8 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 5 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 8, from vertex 2 to vertex 5, and  from vertex 1 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 5, from vertex 5 to vertex 1, and  from vertex 1 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 5 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 3, from vertex 3 to vertex 1, and  from vertex 2 to vertex 10. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 10, from vertex 8 to vertex 6, and  from vertex 10 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 7, from vertex 10 to vertex 3, and  from vertex 11 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 10 vertices counterclockwise (so that vertex 1 ends up where vertex 11 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 9, from vertex 1 to vertex 9, and  from vertex 9 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 7 vertices counterclockwise (so that vertex 1 ends up where vertex 8 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 10-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 11, from vertex 3 to vertex 5, and  from vertex 3 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 5, from vertex 2 to vertex 5, and  from vertex 2 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 7, from vertex 3 to vertex 12, and  from vertex 3 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 8 vertices counterclockwise (so that vertex 1 ends up where vertex 9 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 12, from vertex 6 to vertex 10, and  from vertex 2 to vertex 9. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 5 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 6, from vertex 2 to vertex 5, and  from vertex 3 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 12, from vertex 12 to vertex 2, and  from vertex 1 to vertex 8. Then, rotate the entire setup, including the constructed diagonals, 3 vertices counterclockwise (so that vertex 1 ends up where vertex 4 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 5 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 7, from vertex 6 to vertex 1, and  from vertex 6 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 2, from vertex 3 to vertex 1, and  from vertex 9 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 6 vertices counterclockwise (so that vertex 1 ends up where vertex 7 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 10, from vertex 7 to vertex 3, and  from vertex 11 to vertex 2. Then, rotate the entire setup, including the constructed diagonals, 10 vertices counterclockwise (so that vertex 1 ends up where vertex 11 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 7, from vertex 3 to vertex 5, and  from vertex 8 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 3 vertices counterclockwise (so that vertex 1 ends up where vertex 4 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 10-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 5 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 1, from vertex 2 to vertex 7, and  from vertex 1 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 4, from vertex 4 to vertex 8, and  from vertex 7 to vertex 5. Then, rotate the entire setup, including the constructed diagonals, 6 vertices counterclockwise (so that vertex 1 ends up where vertex 7 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 8, from vertex 7 to vertex 2, and  from vertex 5 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 1, from vertex 2 to vertex 6, and  from vertex 5 to vertex 1. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 12 to vertex 5, from vertex 2 to vertex 5, and  from vertex 5 to vertex 11. Then, rotate the entire setup, including the constructed diagonals, 10 vertices counterclockwise (so that vertex 1 ends up where vertex 11 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 6, from vertex 3 to vertex 5, and  from vertex 6 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 2, from vertex 10 to vertex 6, and  from vertex 3 to vertex 8. Then, rotate the entire setup, including the constructed diagonals, 8 vertices counterclockwise (so that vertex 1 ends up where vertex 9 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 10-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 12 to vertex 10, from vertex 3 to vertex 5, and  from vertex 1 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 8 vertices counterclockwise (so that vertex 1 ends up where vertex 9 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 9 to vertex 3, from vertex 6 to vertex 2, and  from vertex 5 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 7 vertices counterclockwise (so that vertex 1 ends up where vertex 8 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 4, from vertex 4 to vertex 1, and  from vertex 7 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 5, from vertex 3 to vertex 7, and  from vertex 5 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 6, from vertex 7 to vertex 9, and  from vertex 5 to vertex 2. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 4, from vertex 5 to vertex 2, and  from vertex 1 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 3 vertices counterclockwise (so that vertex 1 ends up where vertex 4 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 5, from vertex 3 to vertex 1, and  from vertex 4 to vertex 1. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 11, from vertex 4 to vertex 1, and  from vertex 11 to vertex 9. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 6, from vertex 2 to vertex 8, and  from vertex 7 to vertex 2. Then, rotate the entire setup, including the constructed diagonals, 7 vertices counterclockwise (so that vertex 1 ends up where vertex 8 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 6, from vertex 5 to vertex 3, and  from vertex 3 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 2, from vertex 10 to vertex 1, and  from vertex 7 to vertex 1. Then, rotate the entire setup, including the constructed diagonals, 7 vertices counterclockwise (so that vertex 1 ends up where vertex 8 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 9, from vertex 9 to vertex 2, and  from vertex 1 to vertex 8. Then, rotate the entire setup, including the constructed diagonals, 3 vertices counterclockwise (so that vertex 1 ends up where vertex 4 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 5 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 3, from vertex 8 to vertex 10, and  from vertex 12 to vertex 9. Then, rotate the entire setup, including the constructed diagonals, 10 vertices counterclockwise (so that vertex 1 ends up where vertex 11 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 5, from vertex 4 to vertex 2, and  from vertex 2 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 7, from vertex 6 to vertex 4, and  from vertex 2 to vertex 7. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 7, from vertex 6 to vertex 10, and  from vertex 3 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 10-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 6, from vertex 2 to vertex 5, and  from vertex 1 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 5, from vertex 10 to vertex 7, and  from vertex 6 to vertex 10. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 6, from vertex 5 to vertex 2, and  from vertex 5 to vertex 1. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 9, from vertex 1 to vertex 3, and  from vertex 8 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 5, from vertex 7 to vertex 5, and  from vertex 3 to vertex 5. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 7, from vertex 5 to vertex 1, and  from vertex 2 to vertex 8. Then, rotate the entire setup, including the constructed diagonals, 6 vertices counterclockwise (so that vertex 1 ends up where vertex 7 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 11 to vertex 7, from vertex 5 to vertex 2, and  from vertex 8 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 8 vertices counterclockwise (so that vertex 1 ends up where vertex 9 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 2, from vertex 4 to vertex 1, and  from vertex 5 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 5, from vertex 7 to vertex 4, and  from vertex 7 to vertex 5. Then, rotate the entire setup, including the constructed diagonals, 6 vertices counterclockwise (so that vertex 1 ends up where vertex 7 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 1, from vertex 3 to vertex 7, and  from vertex 4 to vertex 2. Then, rotate the entire setup, including the constructed diagonals, 6 vertices counterclockwise (so that vertex 1 ends up where vertex 7 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 4 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 3, from vertex 2 to vertex 10, and  from vertex 5 to vertex 10. Then, rotate the entire setup, including the constructed diagonals, 6 vertices counterclockwise (so that vertex 1 ends up where vertex 7 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 3, from vertex 6 to vertex 9, and  from vertex 7 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 3, from vertex 2 to vertex 4, and  from vertex 1 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 2 vertices counterclockwise (so that vertex 1 ends up where vertex 3 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 12 to vertex 7, from vertex 3 to vertex 10, and  from vertex 2 to vertex 8. Then, rotate the entire setup, including the constructed diagonals, 10 vertices counterclockwise (so that vertex 1 ends up where vertex 11 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 12-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 5, from vertex 6 to vertex 3, and  from vertex 6 to vertex 2. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 1, from vertex 6 to vertex 4, and  from vertex 1 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 8, from vertex 6 to vertex 4, and  from vertex 6 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 6 vertices counterclockwise (so that vertex 1 ends up where vertex 7 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 1, from vertex 4 to vertex 6, and  from vertex 2 to vertex 8. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 10-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 1 to vertex 4, from vertex 2 to vertex 4, and  from vertex 5 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 8 to vertex 4, from vertex 4 to vertex 2, and  from vertex 2 to vertex 9. Then, rotate the entire setup, including the constructed diagonals, 3 vertices counterclockwise (so that vertex 1 ends up where vertex 4 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 9-gon, with vertices numbered 1 through 9 in counterclockwise order. Draw the diagonal from vertex 9 to vertex 3, from vertex 5 to vertex 3, and  from vertex 7 to vertex 9. Then, rotate the entire setup, including the constructed diagonals, 7 vertices counterclockwise (so that vertex 1 ends up where vertex 8 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 9-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 5, from vertex 3 to vertex 8, and  from vertex 10 to vertex 1. Then, rotate the entire setup, including the constructed diagonals, 10 vertices counterclockwise (so that vertex 1 ends up where vertex 11 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 7, from vertex 5 to vertex 9, and  from vertex 1 to vertex 5. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 10-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 7-gon, with vertices numbered 1 through 7 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 2, from vertex 5 to vertex 3, and  from vertex 2 to vertex 5. Then, rotate the entire setup, including the constructed diagonals, 6 vertices counterclockwise (so that vertex 1 ends up where vertex 7 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 7-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 5 sides?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 4, from vertex 3 to vertex 5, and  from vertex 1 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 7 to vertex 2, from vertex 8 to vertex 6, and  from vertex 3 to vertex 1. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 6 to vertex 4, from vertex 3 to vertex 7, and  from vertex 6 to vertex 3. Then, rotate the entire setup, including the constructed diagonals, 1 vertices counterclockwise (so that vertex 1 ends up where vertex 2 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 8-gon will be partitioned into a collection of smaller polygons. (Only count the "smallest" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 5 sides?

Present the answer in LaTex format: \boxed{Your answer}