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Tiling a Hexagon with Diamonds

A regular hexagon is divided into a triangular grid, and completely tiled with diamonds (two triangles glued together). Diamonds can be placed in one of three orientations. Prove that, no matter how the board is tiled, there will be the same number of diamonds in each orientation.

Here is an example of such a tiling. Though this hexagon has 5 triangles to a side, the problem asks you to prove this for any size hexagon, and any tiling of it.

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This is one of those puzzles that has many solutions, so I'm very curious to see what people's favorite approaches are.