HMMT 二月 2012 · COMB 赛 · 第 1 题

HMMT February 2012 — COMB Round — Problem 1

In the game of Minesweeper, a number on a square denotes the number of mines that share at least one vertex with that square. A square with a number may not have a mine, and the blank squares are undetermined. How many ways can the mines be placed in this configuration? 2 1 2

解析

In the game of Minesweeper, a number on a square denotes the number of mines that share at least one vertex with that square. A square with a number may not have a mine, and the blank squares are undetermined. How many ways can the mines be placed in this configuration? 2 1 2 Answer: 95 Let A be the number of mines in the first two columns. Let B, C, D, E be the number of mines in the third, fourth, fifth, and sixth columns, respectively. We need to have A + B = 2, B + C + D = 1, and D + E = 2. This can happen in three ways, which are ( A, B, C, D, E ) = (2 , 0 , 1 , 0 , 2) , (2 , 0 , 0 , 1 , 1) , (1 , 1 , 0 , 0 , 2). This gives (10)(2)(1) + (10)(3)(2) + (5)(3)(1) = 95 possible configurations.