I have a lot of extraneous, vaguely-mathematical toys in my office, but by far the most popular one is this calendar from DragonFjord Puzzles. It’s called “A-Puzzle-A-Day”, and one look at it is usually enough to communicate exactly what it is about.
The goal is to cover up every cell in the calendar-shaped grid with the tiles, except for today’s month and date. You can see above that I have so far been unable to solve for today, Halloween. It is very fun, and surprisingly has not gotten old in the months that I’ve had it. Probably because I only ever spend a couple minutes at a time with it, and sometimes I am able to solve it and thus forget about it for the rest of the day.
Aesthetically, I think it looks pretty nice, but from a math aesthetics point of view, there is a lot to be desired. For one, why is every piece a pentomino except for the \(2 \times 3\) rectangle? (I’m not sure which month we would have to cut to make this happen, but someone has to make the hard choices.)
Mathematically, I feel like there should be a nice method for solving it, or even proving that a solution always exists. Unfortunately, that too seems to be a bit ad hoc; the creators said they tried a bunch of shapes when prototyping, and finally checked all the possibilities with some code. Efficient, sure, but hardly satisfying. Some searching online led me to the exact cover problem, which does seem to generalize this puzzle pretty well. It’s a bit too general though; I would be interested in finding a way to study this type of problem from a more geometric point of view!