缩放正方形（硬币镜像移动）
Scaling a Square
题目详情
桌面上有 4 枚硬币位于一个 1×1 正方形的四个顶点（两两相距 1）。
一次合法操作：任选两枚硬币，其中一枚为镜子 M，一枚为跳子 J。把 J 移动到关于 M 的对称点（即以 M 为圆心、J 为圆周上一点，把 J 移到圆周上与之对径的位置）。
问：经过若干次合法操作，能否得到一个 2×2 的正方形？若不能，为什么？
提示：不变量。
On a table you have a square made of 4 coins at the corner at distance 1. So, the square is of size 1×1. In a valid move, you can choose any two coin let’s call them mirror and jumper. Now, you move the jumper in a new position which is its mirror image with respect to mirror. That is, imagine that mirror is a centre of a circle and the jumper is on the periphery. You move the jumper to a diagonally opposite point on that circle. With any number of valid moves, can you form a square of size 2×2? If yes, how? If no, why not?
Hint
Invariance
解析
不能。
初始 4 个硬币位于整数格点（例如 (0,0),(0,1),(1,0),(1,1)）。每次“关于某点对称”会保持坐标仍为整数，因此所有硬币始终在整数格点上。
整数格点上两点的最小非零距离是 1，因此任意两枚硬币之间距离永远不可能小于 1。
若能从 1×1 变成 2×2，则由于操作可逆，应当也能缩小成 等更小正方形，这会要求出现距离小于 1 的硬币对，矛盾。
Original Explanation
No!
Solution
Note, that every valid move is reversible. That is if you make a move in one direction, the other direction is also a possibility. So, if you can scale a square of size 1×1 into a square of size 2×2, then you should also be able to shrink it into arbitrary small square. Say 1/2 x 1/2 or 1/1024 x 1/1024.
So, now we need to show that you cannot shrink the square. If we show that distance between any two coins is always greater than 1 unit, we are done. And here is the simplest part!
Imagine a grid on the 2-d plane. Let the coins be at (0,0) (0,1) (1,0) (1,1) forming a square of size 1×1. Our grid cells are of size 1×1. Note, that no matter how many moves you make, the coins will always have integer co-ordinates ( basically stays on the grid ). Since, the shortest distance on the grid is 1 unit, no two coins can have distance shorter than 1 unit. And we are done!