# 51 Points in Unit Square

In any set of 51 points inside a unit square, there are always three points that can be covered by a circle of radius 1/7.

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Copyright © 1996-2018 Alexander BogomolnyIn any set of 51 points inside a unit square, there are always three points that can be covered by a circle of radius 1/7.

### Solution

Cut the square into 25 equal subsquares. By the pigeonhole principle, at least one of the squares contains at least 3 = ⌈51/25⌉ points. (To remind, ⌈x⌉ denotes the *ceiling function*: ⌈x⌉ is the least integer not less than x.) The diagonal of a subsquare equals

(Instead of using approximations, we might have squared the two numbers:

### Reference

- C. Alsina, R. B. Nelsen,
*Charming Proofs*, MAA, 2010, p. 46

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Copyright © 1996-2018 Alexander Bogomolny