| medium | discrete |
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)
Pigeonhole principle
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!