Fifty-one ants and a small square card

Question

Fifty-one ants are sitting still on a square table of side 1. You hold a square card of side 1/5, which you may place anywhere on the table with its sides parallel to the table's edges. Show that, wherever the ants are, you can place the card so that it covers at least three of them. Would 50 ants be enough for the same guarantee?

Accepted Answer

Yes: divide the table into a 5 by 5 grid of cells the size of the card. Fifty-one ants in 25 cells force some cell to hold at least three, and the card placed on that cell covers them. The counting. Twenty-five cells, each a square of side 1/5. If every cell contained at most 2 ants, the table would hold at most 25 × 2 = 50 ants. There are 51, so some cell contains at least 3. Place the card exactly over that cell. Ants on grid lines. An ant sitting exactly on a boundary between cells can be counted in either cell; assign each such ant to one cell of your choice. The card, placed on the chose…

Fifty-one ants and a small square card

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