# The pigeonhole principle question: diamond

Given diamond which every side length is 2 cm. the sharp angels are 60 degrees.

Given nine points inside the diamond. Prove that between two of which the distance between them is at most 1 cm.

ideas where to start from?

thanks.

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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
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## 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
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