多米诺覆盖棋盘
Domino Covering
题目详情
8×8 国际象棋棋盘可用 32 个 2×1 多米诺骨牌完全覆盖。
若把棋盘两个对角角格切掉（它们同色），证明剩余棋盘无法用 31 个 2×1 多米诺完全覆盖。
An 8x8 chessboard can be entirely covered by 32 dominoes of size 2x1. Suppose we cut off two opposite corners of chess (i.e. two white blocks or two black blocks). Prove that now it is impossible to cover the remaining chessboard with 31 dominoes.
Hint
The two diagonally opposite corners are of the same color.
解析
多米诺每次覆盖相邻两格，必定覆盖 1 黑 1 白。
原棋盘黑白各 32。切掉两个对角同色角格后，黑白数量变为 30 与 32（不再相等）。
但 31 个多米诺覆盖黑白各 31 格，矛盾，因此不可能覆盖。
Original Explanation
Solution
The two diagonally opposite corners are of the same color. A domino covers adjacent faces & hence a domino always covers 1 black and 1 white square. The 31 dominoes will cover 31 blacks and 31 whites. The chess has 30 & 32 square instead. Hence this can't be done.