[CareerCup] 6.2 Dominos on Chess Board 棋盘上的多米诺

 

6.2 There is an 8x8 chess board in which two diagonally opposite corners have been cut off. You are given 31 dominos, and a single domino can cover exactly two squares. Can you use the 31 dominos to cover the entire board? Prove your answer (by providing an example or showing why it's impossible).

 

这道题给我们了一个8x8的国际象棋棋盘，从一个对角线挖去两个格子，应为同一颜色，然后给了我们31个多米诺，每一多米诺可以覆盖两个格子，问我们能不能覆盖整个棋盘。

这题乍看去去好像可以，因为64个格子挖去两个剩62个，31个多米诺正好覆盖62个格子。实际上是不对的，整个棋盘原本有32个黑格子，32个白格子，挖去两个黑格子，还剩30个黑格子，32个白格子。而一个多米诺只能覆盖一个黑格子和一个白格子，31个多米诺只能覆盖31个黑格子和31个白格子，所以总会有1个白格子无法覆盖，所以是不可能的。