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Tiling a Hexagon with Diamonds

tiling
Published

January 6, 2023

Source: the back cover of Mathematical Puzzles: A Connoisseur’s Collection Paperback, by Peter Winkler; and h/t to D. Sivakumar for the pointer!

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.

Here’s the image demonstrating the grid and the individual tiles from the book cover:

Showing the triangular grid

One attempt is to see if we can peel off layers using induction: the innermost hexagon in the partially color-coded image below corresponds to a base case of sorts; but it’s not clear how one might extend this approach — at least working with a ccc layers at a time for some constant ccc — since it’s not clear that every tiling can be split up as a combination of a complete tiling of ccc outermost layers and the rest.

Example tiling

Turns out that the image on the book’s front cover is a pretty cool hint!

Hint


© 2022 • Neeldhara Misra • Credits •

 

Corrections? Please leave a comment here or a PR in this repository, thanks!

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