In a (somewhat) recent post, I discussed my decagram puzzle that I used for the gift exchange at Gathering for Gardner 10. Of course, I wondered what other puzzles of this type are out there waiting to be found. I’m particularly interested in mathematically elegant puzzles, which for me means ones that use combinatorially complete sets of pieces, that is, sets of pieces where every possible piece for a given scheme is actually present. Rob Stegmann has a page with a listing of similar puzzles, which he calls crossed stick puzzles.
In a crossed stick puzzle like my decagram puzzle, each piece has a number of slots, which are at the same position on every piece, and each slot can be one of a number (typically two) of types. This means that you can assign a code to every piece, which will be a binary string if there are two slot types.
Stegmann’s page also notes the possibility of different slot compatibility schemes: instead of 1 and 0 being compatible with each other and not with themselves, the opposite is possible, although it seems harder to realize physically. (Perhaps magnets might lend themselves to such a scheme.) With three or more slot types, more compatibility schemes become possible.
Thus there are essentially four attributes that describe a puzzle of this type. These are the configuration of pieces in the completed puzzle, the allowable actions that can be performed upon a piece in a given position, (rotation, flipping) the compatibility scheme, and the piece codes that are present.
What viable piece configurations exist? For our purposes, a legal puzzle configuration consists of a number of congruent pieces, which can be represented as line segments marked at their intersection points. Exactly two pieces must meet at each intersection point. (Relaxing that requirement may bear fruit, but it also complicates the physical design of slots in a puzzle.) If there are two intersection points per piece, the situation is not terribly interesting: the pieces must topologically form a loop, and there are an infinite number of ways they can do this. So I’ll restrict myself to configurations where the number of intersections per piece is three or greater.
As my decagram puzzle might suggest, one class of puzzle configurations consists of star polygons. Oskar van Deventer designed a heptagram puzzle, which uses flexible foam pieces to get around the assemblability problem I mentioned in the previous post. It not have a combinatorially complete set of pieces; making up for this, the flexible material does allow the pieces to form a number of different patterns.
Another class of crossed stick puzzle configuration consists of square grid patterns; most of the examples on Stegmann’s page are of this type. Few of them use complete sets of pieces, although “The Fence”, designed by Jean Claude Constantin, has a complete set from which the 1111 and 0000 piece have been removed.
In addition to grids and stars, we can make “broken stars”, where small segments are excised from the center of each of the pieces in a star configuration. Are other puzzle configurations possible?
As you can see, at least one is. In this configuration there are eight pieces. Taking a note from “The Fence”, I used all of the possible pieces except the two where all of the slots are of the same type. (As in the decagram puzzle, the pieces can be flipped horizontally, which means that there are ten different piece types possible before we exclude those two.)
A complicating aspect of this configuration is that intersections may occur at different angles depending on whether the piece is part of the outer square or the inner square. The slot shape I used allows the same slot position to work for both 45° and 90° angles. An unintended consequence of the design of this puzzle is that the pieces can also form an octagram.
I’d certainly be interested in hearing about other piece configurations that could be used in a crossed stick puzzle, if you can come up with any.