抓蚂蚁

Catching Ants

专题: Discrete Math / 离散数学
难度: L4
来源: Brainstellar

题目详情

边长为 1 的正方形桌面上有 51 只蚂蚁。

你有一张边长为 $1/5$ 的正方形卡片。问：你是否总能把卡片放在桌面某个位置，使其覆盖至少 3 只蚂蚁？

提示：抽屉原理。

There are 51 ants sitting on top of a square table with side length of 1. If you have a square card with side 1/5, can you put your card at a position on the table to guarantee that the card encompasses at least 3 ants?

(updated: square card was originally disk of radius 1/7)

Hint

Pigeonhole principle

解析

可以。

把桌面划分成 25 个 $1/5\times 1/5$ 的小正方形格子。

51 只蚂蚁落在 25 个格子里，根据抽屉原理，至少有一个格子包含至少 $\lceil 51/25\rceil=3$ 只蚂蚁。

卡片大小正好覆盖任一格子，因此可覆盖至少 3 只。

Original Explanation

Solution

To guarantee that the card encompasses at least 3 ants, we can separate the square into 25 smaller areas (squares of side 1/5 each). Applying the generalized Pigeon Hole Principle, we can show that at least one of the areas must have at least 3 ants. The card is large enough to cover any of the 25 smaller areas. Done!