# Three More Crossed Stick Puzzles

March 28th, 2013 by munizao

Since I last posted about crossed stick puzzles, I’ve come up with three new ones. Here’s one of them:

This one nicely fills one of the spots in the binary word table. There are eight pieces with three slots apiece; if we use binary shallow/deep slots, there are exactly eight possible pieces, because the pieces won’t be flippable vertically or horizontally within the puzzle.

The principle behind this piece configuration can be used to make other configurations. You can take any line segment, together with a center point, reflect the segment across an axis passing through the center, and create images of the original and reflected segments over successive rotations around the center to make configurations with rotational (indeed, dihedral) symmetry. However, because the pieces wouldn’t generally be horizontally flippable, (are there exceptions?) this only produces puzzles with combinatorially complete piece sets when the number of pieces is a power of two, and only 8 and 16 make reasonable numbers of puzzle pieces. In fact, I’d argue that the ideal range of numbers of pieces for a puzzle of this type is between 8 and 12, with a penumbra extending to about 6 and 16 in which puzzles will still be interesting to some solvers, but not as many. This necessarily constrains the quantity of viable puzzle configurations to a number that is finite, and may indeed be quite small.

But if the math keeps the number of possible puzzles down, inventing new ways to skirt the constraints we’ve set for ourselves can create new possibilities. Witness the following configuration:

Here the slots are ternary: intersection points can have slots that are ⅔ of the width of the piece (pointing up or down) or they can have two slots ⅓ of the width of the piece, pointing both up and down. Where three pieces intersect, one of each of these three slot types must be present. Three piece intersections are harder to work with than regular intersections. I’m pretty sure there is no finite configuration on the triangular grid containing all pieces of the same size with all intersections being three piece intersections.

So we have to fudge. This configuration looks like it has two kinds of three slot pieces, because three of the three slot pieces intersect at a two piece intersection. The trick is that the slot on the two slot pieces at the two slot intersection has a special, single ⅓ width slot, so that it can match the regular slots on the three slot pieces. Nicely, there are exactly three possible two slot pieces with one slot of this type and one regular slot.

In this scheme there are ten three slot ternary pieces, when both horizontal and vertical flipping is allowed. Since there are only nine three slot pieces in the configuration, one piece must be excluded. In fact, our hand is forced: the excluded piece must be the one that has all three slots of the double ⅓ width type. This is because the total number of double ⅓ width slots must equal the number of three piece intersections, which is nine. Before excluding that piece, there is one double ⅓ width slot in the two slot pieces, and 11 such slots in the three slot pieces. Well, if we can’t have a combinatiorially complete set of pieces, the next best thing is a combinatorially complete set minus the most exceptional member, which we could argue that this piece is, on account of it being the only piece with both horizontal and vertical reflection symmetry.

Finally, two configurations for the same piece set:

The configuration on the left was one that I implied in this post. If we use shallow/deep binary slots, we have pieces that can be flipped horizontally but not vertically. There are six of these and twelve pieces in the configuration, so we can use a double set of these as our piece set. As an extra bit of good fortune, there is a second configuration that uses these pieces.

I’m currently getting prototypes of the last two of these lasercut, so I hope to be able to show off the results soon. (I came up with the first one after I put in the order, so it will be a while before I get around to making a prototype of it.)