Tiling a hexagon with lozenges

Question

A regular hexagon with side length n is divided into unit equilateral triangles (6 n squared of them) and then tiled completely with lozenges, each lozenge being two adjacent triangles glued along an edge. A lozenge can lie in one of three orientations, depending on which pair of the hexagon's sides it is parallel to.

Prove that any such tiling contains exactly the same number of lozenges in each of the three orientations, and find that number.

Accepted Answer

Every tiling contains exactly n squared lozenges of each orientation. The three-dimensional picture. Shade the three orientations in three colours. The tiling then looks exactly like a drawing of unit cubes stacked in the corner of a room, viewed along the room's long diagonal: each lozenge of the first orientation is the top face of some cube (or of the floor), each lozenge of the second orientation is a left-hand face (or of the left wall), and each lozenge of the third orientation is a right-hand face (or of the right wall). This is not just a resemblance: every lozenge tiling of the hexag…

Tiling a hexagon with lozenges

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