正方形桌面覆盖至少 3 只蚂蚁
Ants on a Square
题目详情
边长为 1 的正方形桌面上有 51 只蚂蚁。你有一个半径为 的圆形玻璃杯。
证明：总能把杯子放在某个位置，使其覆盖至少 3 只蚂蚁。
There is a 1×1 square with 51 ants on it. You have a circular glass of radius 1/7. Show that you can place the glass in a position so that it covers at least 3 ants.
解析
用抽屉原理。
把正方形划分成足够多的小区域，使得每个小区域都能被半径 的圆覆盖（例如划成边长不超过 的小正方形网格）。
这些小区域数量有限。51 只蚂蚁落入这些区域中，根据抽屉原理，至少有一个区域包含至少 3 只蚂蚁。
把杯子覆盖该区域即可覆盖至少 3 只。
Original Explanation
Divide the square into a grid of small squares (or use pigeonhole). A suitable approach is to show that you can partition so that at least one subregion must have 3 or more ants. Then note that a circle of radius 1/7 can cover that subregion. One can do a more precise argument involving -sized squares, or an equivalent geometric argument, ensuring at least 3 ants are within the circle.