Track / Stage 5 / 252 of 400 #852 of 2000

Problem 852

AIME late

Geometry Difficulty 5.7 Prove it

Does there exist a hexagon (not necessarily convex) with side lengths 1, 2, 3, 4, 5, 6 (not necessarily in this order) that can be tiled with a) 31 b) 32 equilateral triangles with side length 1 ?

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solutions — 2

Solution 1

The adjoining figure shows that question a) can be answered positively.

For a negative answer to b), we show that the number of triangles has to be odd. Assume there are $x$ triangles in the triangulation. They hav altogether $3 x$ sides. Of these, $1+2+3+4+6=21$ are on the perimeter of the hexagon. The remaining $3 x-21$ sides are in the interior, and they touch each other pairwise. So $3 x-21$ has to be even, which is only possible, if $x$ is odd.

Solution 2

The adjoining figure shows that question a) can be answered positively.

For a negative answer to b), we show that the number of triangles has to be odd. Assume there are $x$ triangles in the triangulation. They have altogether $3 x$ sides. Of these, $1+2+3+4+6=21$ are on the perimeter of the hexagon. The remaining $3 x-21$ sides are in the interior, and they touch each other pairwise. So $3 x-21$ has to be even, which is only possible, if $x$ is odd.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.