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tiling holey triangles with diamonds

Source: IMO Shortlist 2006, Combinatorics 6

June 28, 2007
rhombuscombinatoricstilingsIMO ShortlistHall s marriage theorem

Problem Statement

A holey triangle is an upward equilateral triangle of side length nn with nn upward unit triangular holes cut out. A diamond is a 60∘−120∘60^\circ-120^\circ unit rhombus. Prove that a holey triangle TT can be tiled with diamonds if and only if the following condition holds: Every upward equilateral triangle of side length kk in TT contains at most kk holes, for 1≤k≤n1\leq k\leq n.
Proposed by Federico Ardila, Colombia
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