A friend introduced me to FridgeIQ and she planted an idea in my head.
FridgeIQ is a geometric cutting puzzle composed of 16 polygonal tiles, as shown in the terrible picture at the bottom of this article.
The tile shapes are based on a square grid with an occasional diagonal on one of the grid's fields. The total area of all tiles is 40 grid units. Each tile is identified by a single letter.
The box contains a set of challenges, each giving a subset of the tiles and a shape to create. For example, a challenge is: "build a square using BDQSYJN tiles". What I did in this picture and I still have tiles that were not part of this particular challenge.
In addition to the standard challenges, my friend has introduced an additional set of target forms that are ultimately the goal of this issue. I have attached another photo showing one of them in solved state.
These additional challenges all use the complete mosaic game, so their total area is 40 units and they all exhibit quad mirror symmetry. Not everyone is as simple as the photo, some have a hole or holes, but they are all connected.
I'll call these forms "snowflakes" because they look somehow like decorative snowflakes cut out of folded paper to generate the symmetries. Especially those with holes.
I asked myself the following question: how many forms of this type, subject to these constraints, can actually be made from the given tiles?
Snowflakes per se, there is infinitely man. But I believe that all bound elements whose finite area consists of half-unit triangles must be finite. Properly huge, but finished. Solvents are a subset of these.
What would be a good algorithm to generate them? I do not expect that there is an effective way, and I do not believe that the universe has enough time to create them all. But I would like to find at least some new ones.
Here are some unfinished reflections up to now:
At first, I built a simple solver, which reproduces my manual solutions (and finds many alternative solutions). So, even if it would be terribly inefficient, I could just generate snowflake shapes at will and use my solver to check them.
I thought it would be enough to generate an eighth of the form, a half-quadrant, with a total area of 10 units. Symmetry provides the rest. Note that the solution does not have the same symmetry when taking into account individual mosaics, only when the complete form is taken into account.
To generate candidate forms, I could simply insert 20 triangles of half-units in the half-quadrant. Let's hope in a systematic way. Most of these arrangements will not be compact and insoluble enough. And without a good idea to eliminate them, their number is absolutely intractable.
Another approach is simply to generate combinations of tiles and to check their symmetry. Again, the algorithm should focus on symmetrical arrangements or explosive combinatorics means that I can never hope to find a single new provision.
Ideas? Keywords for searches and / or pointers in the literature are also appreciated.
Thank you so much.