A while back, I was finalizing my drawing files for laser-cut frames for my Cross Products puzzles, which I was intending to sell at Gathering for Gardner 13. I wasn’t happy with just wasting the material in the center of the frames, so I looked for a simple idea that would make use of it. The shape that was being cut out was a rectangle with a 3:4 aspect ratio. I could cut that into 12 squares, and then engrave something on each of them. Now what might there be 12 of?
It will probably not surprise anyone here that my mind went straight to the pentominoes. Tiles with pentominoes could be useful for choosing one at random. (Sure, I already owned a 12-sided die with the pentominoes on it, which I bought from Eric Harshbarger, but with tiles one could select without replacement, or even effectively shuffle an ordering of the entire set.)
The tiles reminded me of rune stones, with the pentominoes forming a cryptic alphabet. I thought it would be amusing to make sets of them my exchange gift for G4G13, along with a slip that instructed the user on how to use the arcane power of the pentominoes to divine the future. It would be the kind of playful deadpan jab at ungrounded mysticism that Gardner’s alter ego, Dr. Matrix, might have enjoyed making. But to really justify the effort, it couldn’t be just that. I’d need to include some activity using the pieces that would have genuine recreational math interest. Perhaps a puzzle.
What I found was a variation on the common superform framework that incorporated squares with pentomino runes. The basic common superform problem is to find a figure into which any of the polyforms in a set can be placed. Usually, the object is to minimize the area of such a figure, but in this case, the area will be set by each particular challenge using the pieces. We add a couple of restrictions:
- Each set of five tiles that forms a pentomino must contain the corresponding rune.
- Each rune must be contained in at least one set of tiles that forms that pentomino.
The above figure was included as an example on the instruction sheet. The challenges provided were to find a valid tile arrangement using nine of the tiles, to do so with all of the tiles but one, and to do so with the complete set. Remarkably, this is possible!
I’ve since looked for other sets of polyforms that are able to make valid rune configurations with a complete set. Here are the tri-diamonds:
And here are the tetrahexes:
Can you find others?
Incidentally, at G4G13, I gave a few pentomino readings to fellow conference goers, one of whom reacted with cheerful amusement, and another with stony skepticism. Honestly, I could not have hoped for anything better.
I love this idea! I can see developing this into a 2-player game. I’ve been searching for a way to make a game with the appeal of set, but based on polyforms, and this might work!
I’d love to see what you come up with!
I could be wrong, since I did a lot of the sorting manually, but I’m afraid the hexiamond runes don’t work.
I’ve considered a variant. I haven’t found a hexiamond solution (see above), but perhaps if there were walls or slits between triangles (maybe including the hexagon hexiamond), that would restrict placements enough to enable a consistent matching.
(If the hexagon isn’t slit, that means my search of single-hexagon superforms, if accurate under the first set of rules, remains eliminated under the variation. Either way, superform candidates with a 1-1-2-1-1-2 hexagon are a good place to start looking. There are ten such 12-iamonds that act as superforms.)