# Sunday Times Teaser 2762 – What a Gem!

*by Michael Fletcher*

#### Published: 30 August 2015 (link)

A friend showed me a beautiful gem with shiny flat faces and lots of planes of symmetry. After a quick examination I was able to declare that it was “perfectly square”. This puzzled my friend because none of the faces had four edges. So I explained by pointing out that the gem’s number of faces was a perfect square, its number of edges was a perfect square, and its number of vertices was a perfect square.

How many faces did it have, and how many of those were triangular?

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The key to this one is recognising a shape that might match the given constraints. This turns out to be a pyramid consisting of a regular N sided polygon bases with N identical triangles rising from its edges to a common vertex above its centre. The rest is easy.

if 2n = e = y^2

and n+1 = v = f = x^2

then the Pell-like equation x^2 – 2y^2 = -2 also produces the solutions.

Nice one Arthur, I should have thought of that!

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