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I started to think if this hat tile shape could be used to make some sort of tile-laying game. In figure 1.1 you can see sort of hexagonal overlay grid, so maybe using that somehow

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It's hard to make aperiodicity matter in a board game: as players place tiles, they either extend an aperiodic layout to a bigger one (unexciting) or they are unable to place a tile because someone, possibly much earlier, deviated from the "correct" construction.

Moreover, with a single einstein there is no tile unpredictability or choice whatsoever, eliminating two of the main strategic dimensions of tile-laying games.

One idea could be to have the tiles have different colorings, then your game goal could be to make some patterns or continuous areas. Example tile coloring scheme https://ibb.co/n0JJXF3

Could make for more interesting "terrain" of tiles used to form hex-based playfields.

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