# Posts Tagged ‘gathering for gardner’

## Binary System, Decimal Star

April 19th, 2012

If you participated in the gift exchange at the 10th Gathering for Gardner, which was held recently in Atlanta, you would have received one of these in your bag of exchange gifts:

It is common, (but by no means required) for participants to use the number of the conference as a theme in their exchange gift in some way. I considered a ten pointed star with pieces that slot together as a promising shape for a puzzle, and I recalled that there were ten distinct reversible binary sequences of length four. (In this scheme 0011 and 1100, for example, are considered equivalent because they reverse to each other.) This meant that with four slots at intersection points, if there were two possible positions for each slot, (like up and down) there would be exactly ten possible pieces, which would make an elegant puzzle set if I used one of each. Conveniently, the pieces could be flipped horizontally to physically realize the reversal of the string. Inconveniently, the pieces could also be flipped vertically, which would invert the 1’s and 0’s, and lower the number of distinct piece shapes to six. Another problem is that some configurations of pieces could not be physically assembled. If there was a triangle of pieces where each had an up slot followed by a down going around the triangle, there would be no way to fit the third piece in, because it would simultaneously need to be slotted in from above and below.

I solved both of these problems at once by changing the inner slots to all face the same direction, and to have shallow vs. deep as their two possible states instead of up and down. Now the ten pieces can be divided into two pentagonal configurations that are connected by their outer slots, and connect to each other by the inner slots. Because every triangle in the star contains the two inner slots of a piece, the triangles are all assemblable. The pentagons must also be assemblable, because there are only four pieces with up and down outer slots, so one side of the pentagon must have two slots pointing the same direction, and that side may be placed last. And because the direction of the inner slots is forced, only horizontal flipping is allowable. Here’s a photo of an assembled puzzle, along with an unassembled set of pieces:

Mathematical niftiness aside, is this a good puzzle? I think so. It has a fair number of solutions, but neither so many that you can easily stumble upon one without applying any strategy to solving the puzzle, nor so few that you have to spend a lot of time engaged in trial and error. Let me know if you have one of these and need hints for solving it.

In an upcoming post, I’ll discuss some variations on this type of puzzle.