A large regular hexagon is cut out of a triangular grid and tiled with diamonds (pairs of triangles glued together along an edge). Diamonds come in three varieties, depending on orientation; prove that precisely the same number of each variety must appear in the tiling.
This puzzle is from Gurmeet.net's Tiling with Calissons puzzle page. The "Proof without Words" solution is accompanied by this image:
My approach: I tried reading through the solution but it simply blows my mind. The proof without words, how did they even come up with that? Any clue how to start/proceed will help.