Archive for January, 2017

Finally, a Magic Magic 45-omino

January 14th, 2017

In the figure below, the numbers in each row, column, and main diagonal sum to 115:

Quite a long time ago, I came up with the idea of representing the lo shu (3×3 magic square) as a set of squares in a 9×9 grid, partitioned into nine 3×3 cells. The number of squares in each cell would correspond to a number in the lo shu. The most “magic” way to arrange the cells would seem to be to have 5 squares in the set in each row, column, and main diagonal. (This can be done because the lo shu’s magic sum of 15 can be divided among three rows or columns.) Although it doesn’t affect the “magicality” of a figure, I thought it aesthetically desirable for such a figure to be connected (i.e., a single polyomino) and hole-free. There are 12 hole-free magic 45-ominoes, if my code for discovering them is correct.

A figure with the same number of squares in each row, column, and main diagonal makes an ideal canvas for a sparse magic square. But with 45 numbers to place, and 20 constraints to meet, we start to push on the edge of what’s computationally feasible. The solver I wrote (which, I admit, might not have been very good) could not find a solution. Bryce Herdt manually tweaked the output of my solver to make a semimagic solution, that is, one where the rows and columns add to the magic sum, but the diagonals still didn’t work.

When I discovered that the Numberjack constraint engine could easily be used to code solvers for magic figures, I tried it on this problem, but got nowhere. The solver would run for an arbitrarily long period of time without spitting out any solutions. Recently I tried it again, and this time I got solutions. Paradoxically, what made the problem easier to solve was that I added more constraints. I manually placed the numbers 1 through 9 in the 3×3 cells that they correspond to. This seems to have made the search space small enough that the solver would not be able to spend an inordinate amount of time stuck in a barren zone.

Complete combination colorings on the torus

January 9th, 2017

I posted previously about my talk at Gathering for Gardner 12 on colorings of pentomino tilings. One unexpected consequence of that is that my work has now been cited in a very prestigious… um… coloring book. Alex Bellos and Edmund Harriss previously collaborated on Patterns of the Universe, a mathematical coloring book for adults, and were looking for material for the sequel. They were attending G4G12, and saw my talk, and thought that they had found some. They wanted to use something like the strict complete combination 3,4-coloring of the pentomino tiling that I showed, but for the purpose of a coloring book page, they needed something with more shapes to color. Could I come up with such?

It seemed to me that the problem called for a pentomino tiling of a torus, which they could use as a wallpaper-like pattern, repeated as many times as they needed. The choice of the particular torus to use is a matter of taste, but I thought it would be nice to maximize the minimum distance between two images of the same point. (I haven’t proven that I succeeded, but it’s close.) In coding the solver for this, I used a shortcut: instead of directly checking whether a given tiling had a coloring of the correct type, I checked whether each pentomino bordered exactly six others. This turns out to be a necessary condition, but not a sufficient one, so I manually checked a few such tilings until I found one that worked. This is the pattern that appears, in user-colorable form, in Visions of The Universe by Bellos and Harriss.

The hexiamonds were the other obvious set of 12 polyforms to try to tile with this coloring scheme. Here, there is one torus with maximal symmetry. Amazingly, my solver found just two tilings where every piece bordered 6 others, of which exactly one had the right coloring properties. Recall that the solution for the pentominoes on the 6×10 rectangle was also unique. It seems incredible to me that this problem type has yielded two instances that were so finely balanced as to be solvable, but only by the barest of margins.