Visual Math, Puzzles, and Dice by Alexandre Muñiz
Previously, we looked at this set of dihexes and trihexes with half-weight hexes affixed, which we used in a generalized type of tiling where fractional cells may overlap if the weights on those cells sum to one. This notion of generalized tiling turns out to be powerful enough to describe all kinds of problems involving … Continue reading Notation Notions: Actions on Fractions
Marking external edges of polyforms leads naturally to restrictions on tilings that make for good puzzles. We can simply match marked edges, and if two edges on each piece are marked, we can draw a path between them and make challenges based on properties of the paths formed. Internal cell edges on pieces can also … Continue reading Internal Edge Markings
1. Previously, we looked at what we might mean by such things as “The 3+2-ominoes”, and distilled the beginning of a notation system from that. But before we get very far, we might find some instances where our polyform set addition notation is unsatisfactory. Here are the 2+1+1-ominoes, or 2+1+1■ for short: It strikes me … Continue reading Notation Notions: Addition Addendum
In planning out blog posts, I typically use the rule of thumb that a post should contain between two and four images. If I have five or more images, I try to break the material up into multiple posts, and if I have only one I will usually wait until I have more material. A … Continue reading Four Square Sequels
A few months ago, I noticed that a king can tour a pentomino tiling so that every cell in each pentomino is visited exactly once in an uninterrupted sequence, and the tour forms a closed loop. Not every pentomino tiling admits such a tour, but it’s not hard to find one that does. A more … Continue reading Touring Tilings
I posted last year about a path puzzle using polyiamond tiles. Those tiles were marked with a complete set of paths between cell edges on the perimeters of diamonds and triamonds. Recently I’ve been exploring a variation on tiles with marked paths. In these tile sets, the paths are constrained to straight lines aligned with … Continue reading Stripe Club
Welcome to the 236th Carnival of Mathematics! 236 is the number of total partitions of 5. I drew up a graphic of the 26 total partitions of 4: The total partition sequence is A000311 in the OEIS. It also describes the number of possible phylogenetic trees of n species in evolutionary biology. The 2025th Carnival … Continue reading Carnival of Mathematics #236
Early last year I became aware of a paper by Jamie Tucker-Foltz on Locked Polyomino Tilings. It defines a recombination move on a tiling of n-ominoes as creating a new tiling by removing an adjacent pair of tiles and replacing them with a pair of tiles in different positions. Locked tilings are those that admit … Continue reading Moves in Tilings
Here’s a tiling of the 3+2-ominoes: This use of a plus sign seems natural enough, but we might want to think a bit more about what it implies. We have established an operation on polyform sets, and a notation for that operation. This raises some questions: what other operations might we want to use? How … Continue reading Notation Notions: Operations on Ominoes
In 2022, Jacques Ferroul sent some notes on a remarkable exploration in polyominoes to Kate Jones, who shared them with George Sicherman, who in turn forwarded them to me. I quickly saw that there was quite a lot of potential there, and exchanged a few emails with Ferroul, where we shared ideas riffing off of … Continue reading Fuzzyominoes: Weighty equivalence