Last time I introduced the Sol LeWitt’s problem, devised by the eminent mathematical reporter Barry Cipra. The challenge was to take the tiles, as presented in the left image of Figure 1 and rearrange them such that all the lines form continuous rows, columns and diagonals across the grid. As I revealed there are many solutions, one such solution is presented in the right image of Figure 1.
|Figure 1. Left: the original Sol LeWitt tiles. Right: an arrangement in which all lines cross the entire grid.|
I also mentioned that there were some special relationships between certain solutions. For example rotating a solution through 90 degrees, reflecting it, or performing a combination of these two operations generates another, related solution. Furthermore, we can take the topmost row (or the leftmost column) and moving it all the way to the bottom (or to the right). Explicitly, a set of solutions can be drawn on the surface of a torus.
This leaves us with a new question. Are there any solutions which cannot be generated in such a way? Namely, starting from one solution are their other “distinct” solutions, which cannot be created through rotations, reflections or row/column operations. Each distinct solution will then generate a different solution set, which will lead to different to tori.
The famous mathematician John Conway demonstrated that there are actually three distinct solutions, from which all others can be derived. One has been given above. Can you find the other two possible distinct solutions?
As I was talking to Barry about this puzzle he told me a nice anecdote, where he had used this puzzle in a workshop involving maths teachers and maths researchers that had been paired together. He said that the teachers were constantly moving the pieces around, effectively using trial and error, whilst their researcher partner would sit back and think about the pieces. Eventually, one researcher claimed that the puzzle was impossible, not a moment later his partner produced a working solution! Let this be a lesson to any mathematician. Theory is all well and good, but practical intuition is invaluable.