砖块装箱
Bricking Box
题目详情
能否把 53 块尺寸为 的长方体砖块装入一个 的盒子中？要求砖块各面都与盒子各面平行。
提示：类比“多米诺覆盖棋盘”的染色不变量。
Can you pack 53 bricks of dimensions 1x1x4 into a 6x6x6 box? The faces of the bricks are parallel to the faces of the box
Hint
See its cousin called Domino Covering in which two opposite corners of a checkerboard are removed.
解析
不能。
把 盒子划分为 的小立方体，并按三维棋盘方式交替染成两色。可使两色小立方体数量为 14 与 13（每个小立方体体积为 8）。
任何 砖在这种染色下总会覆盖相同体积的两色区域（等量）。因此 52 块砖最多恰好覆盖掉较少的那一色的小立方体对应体积，无法再放第 53 块。
故不可装入。
Original Explanation
No!
Solution
Divide the volume of 6x6x6 box into 2x2x2 mini cubes. Imagine each of these mini cubes is either fully red or fully blue such that it forms a 3D checkerboard pattern on the box. This will restrict 14-13 restriction on cube colors, say 14 blue and 13 red. Now, putting bricks into box, parallel to faces, each brick will be half blue and half red, so 52 bricks fill all the red cubes and there is no way to place a 53rd brick.