Problem:
Three players play tic-tac-toe together. In other words, the three players take turns placing an "A", "B", and "C", respectively, in one of the free spots of a grid, and the first player to have three of their label in a row, column, or diagonal wins. How many possible final boards are there where the player who goes third wins the game? (Rotations and reflections are considered different boards, but the order of placement does not matter.)
Solution:
In all winning cases for the third player, every spot in the grid must be filled. There are two ways that player wins along a diagonal, and six ways that player wins along a row or column. In the former case, any arrangement of the s and s is a valid board, since every other row, column, and diagonal is blocked. So there are different finishing boards each for this case. However, in the latter case, we must make sure players and do not complete a row or column of their own, so only of the finishing boards are valid. The final answer is .