Perhaps not surprisingly, a variation of a recreational math puzzle (a so-called edge-matching puzzle or scramble square) is equivalent to a combinatorics question of interest (in this case, about quantum knot mosaics, question #9). In a traditional edge-matching puzzle, you are given $n^2$ tiles, each tile square in shape and bearing a design, with the goal of arranging the tiles in an $n\times n$ grid so that the designs on the side of adjacent tiles "match". For instance, here's a puzzle with 24 possible tiles (4 sides, 3 colors, 2 halves) of which at most 9 actually appear (link here if the image is broken):

For what it's worth, solving a general edge-matching puzzle is NP-complete (see article of Demaine). The combinatorics problem is phrased in a slightly different fashion. You begin with a finite collection of designs (such as quadruples of colored halves of butterflies) and for each design an ample supply of square tiles bearing that design. The problem is to calculate the number of arrangements of these tiles in an $n\times n$ grid so that, as in the game described above, the designs on sides of adjacent tiles match. The number of arrangements should be in terms of the size of the grid and the collection of designs on the tiles.

Is anyone aware of results along these lines or, even better, able to provide a quick calculation of the number of arrangements of tiles into an edge-matched grid?

My suspicion is that the number of arrangements goes like $\lambda^{n^2}$ where $\lambda$ is determined from the collection of designs on the tiles.