# Pigeon principle question: Nine points in a diamond

A diamond (a parallelogram with equal sides) is given, and its sides are 2 cm long. The sharp angels are 60 degrees. If there are nine points inside the diamond, prove that there must be two of them so that the distance between them is at most 1 cm.

Ideas where to start from?

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Nine points and the pigeonhole principle suggests trying to find 8 "pigeonholes" for nine points. –  Mark Bennet Jun 9 '12 at 17:25
Your diamond is a two-dimensional figure, I think? –  Mark Bennet Jun 9 '12 at 17:27
I believe that “a parallelogram with equal sides” is usually called a “rhombus”. –  MJD Jan 16 '14 at 17:51

## 1 Answer

Hint: Split the diamond into eight congruent equilateral triangles. The triangles are your pigeon holes.

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The diamond may be a regular tetrahedron of side $2$. –  André Nicolas Jun 9 '12 at 19:00
@André, that possibility did not occur to me. Well spotted! But in that case there would be no need to specify the acute angle? I'm sticking to the interpretation: a diamond = a parallelogram with equal sides, but let's wait for the OP to comment. –  Jyrki Lahtonen Jun 9 '12 at 19:13
@André, thanks for the thought, but it is two-dimensional diamond.Therefore, this hint will do the job. –  adamco Jun 10 '12 at 5:30