HMMT 十一月 2020 · 冲刺赛 · 第 16 题

HMMT November 2020 — Guts Round — Problem 16

[10] 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 3 × 3 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.)

解析

[10] 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 3 × 3 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.) Proposed by: Andrew Lin Answer: 148 Solution: In all winning cases for the third player, every spot in the grid must be filled. There are two ways that player C wins along a diagonal, and six ways that player C wins along a row or col- umn. In the former case, any arrangement of the As and Bs is a valid board, since every other row, ( ) 6 column, and diagonal is blocked. So there are = 20 different finishing boards each for this case. 3 However, in the latter case, we must make sure players A and B do not complete a row or column of their own, so only 20 − 2 = 18 of the finishing boards are valid. The final answer is 2 · 20+6 · 18 = 148.