Tag: G4G13

The Rune Where It Happens

~ munizao ~ 4 Comments

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 I made for my G4G13 exchange gift didn’t have these fancy swirled colors.

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!

Show Solution

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.

The Devil’s in the Angles

~ munizao ~ Leave a comment

Recently, while I was considering possible designs for a puzzle for my exchange gift for the next Gathering for Gardner, I thought about doing something with multiple layers of clear plastic, where interactions of markings on the layers define the puzzle. When you’re going to lasercut a large quantity of puzzles, keeping down the cost, and therefore the cut length, is paramount. So I wanted to be able to use the simplest possible markings on the pieces.

A straight line segment looked like a pretty good candidate, and it leads to an obvious puzzle goal: make the segments on two layers perpendicular. I still needed to choose pieces for these markings, but after a little trial and error, I landed on dominoes, with a segment centered in each square. For these, given some reasonable restriction on the allowable angles of the segments, the number of different pieces possible would land somewhere in the range of what would make for a good puzzle.

I ended up using segments that were turned either 15° or 45° off from the edges of the pieces. These admit exactly 12 different pieces, which can tile two layers of a 3×4 rectangle:


What makes this set particularly nice is that you can get two more puzzle challenges by changing the goal angle for the crossing segments. In addition to making them all perpendicular, you can make them all cross at 30° or 60°. These challenges should be easier, as there are two ways for an angle to differ from another one by 30° or 60°, but only one way to be perpendicular.

I also found a related puzzle that uses 10 dihexes. There are 13 pieces possible in this scheme, but I’ve omitted the ones with a lengthwise axis of symmetry from the puzzle:

In the end, I decided not to make either of these my exchange gift. I had a couple of prototypes made of the first puzzle, and it was clear to me that it needed to be larger than I could afford to make it and give away a few hundred copies. It also works best with a frame to hold the pieces and keep them neatly aligned, which adds considerably to the time and expense per copy. But even though I won’t be able to give this away at G4G13, I hope to be able to be able to sell a few copies at my vendor table there!