Hi there, I'm new, smite me if I'm posting in the wrong place XD

I'm too lazy/busy at the moment to implement this as a program, but I've been thinking about the possibility of an infinite twisty puzzle and how it might be made into a playable game. I know "Magic Tile" exists but I can't get it to work on my computer and it looks to me like it doesn't do things the way I was thinking. Attached is a little gif, which should be easier than trying to describe the idea. (sorry it's not fully colored at infinity )

...If for whatever reason it's not animating, the idea is to tile the hyperbolic plane, put a vertex in the center, color each quadrant of the tiling differently, and then allow the puzzle to be "twisted" along geodesics (which will look like circles). Messing around a bit I seemed to find that though I think "twisting" along the center (straight) lines would be sufficient, this feels pretty limiting and it would be far better to implement it along at least the largest circles as well.

The only major problem with this puzzle would be that it would be possible for pieces to wander arbitrarily far away from the center, and thus become invisible to the solver. An interesting solution to this would be to have little arrows outside the edge of the circle, telling what colors lie in those directions.

What do people think of this puzzle? Anyone have better ideas for the coloring? I probably won't get around to programming it until the semester is over, and even then it might be harder than I think because I'll have to come up with a way of handling the fact that it's infinite (shouldn't be too bad but who knows).

cool but first solve

This post would be better in "General Puzzle Topics" [Admin: Done!] because building and modding is mostly for the physical world. If you wrote a program to actually allow solving it could also go in "New Puzzles".

Your idea reminds me a little bit of TomZ's infinite unbandaging simulator:
http://twistypuzzles.com/forum/viewtopic.php?f=1&t=22644

My problem is that you have an infinite number of red, white, brown, and green pieces. There is really no such thing as a true scramble that doesn't just pick a random color for each piece uniformly. You don't have to worry about parity restrictions and the like because as far as I can imagine, "parity" should be an undefined concept on an infinite group.

Even if you only did a scramble of say, 100 moves, it's sorta like herding cats. The pieces would go in every direction and getting them back to the center would just be annoying, not fun.

Magic tile solves the infinite problem by using a repeating coloring. For example, all of the tiles colored "red" all turn at the same time. Although the projection is infinite, the objects you are manipulating are finite because there are a finite number of colors. This allows for very strange (but finite) objects like Klein's Quartic to be simulated.

Scrambling 100 pieces or so was exactly what I had in mind; or rather, 100 moves or so, which is probably much different gameplay-wise. It's appealing to me for a couple of reasons. Firstly the number of scrambling moves sets difficulty level/game length, which is kind of neat. Secondly, an infinite puzzle scrambled a finite number of moves can be subjected to rigorous analysis in terms of algorithmic complexity; does game length double with twice the scrambling moves, or increase exponentially? Any finite puzzle has a 'constant time' solution and so is relatively hard to analyze algorithmically.

It's possible nothing interesting happens with increased scrambling and it is just 'like herding cats'; but there might not have such an obvious solution (so might be more algorithmically complex). I'll try programming it eventually.

Since your puzzle has an infinite number of pieces, every move will shuffle pieces differently, so the only solution for solving the puzzle without infinitesimal aberrations of the pieces, is to solve the puzzle using the same sequence of moves used to scramble it, but in reverse.

Even if you build an accurate simulator, with each move, the number of disturbed pieces grows exponentially, so even in an era of 64-bit supercomputing with virtually unlimited RAM, in order to track all pieces, the memory required for the simulation to run would grow exponentially with each move, so after a brief amount of manipulation, the computer would likely run out of memory.

Another great way to simulate a Rubik's Cube type puzzle would be to use hexagonal faces in an infinite tiling of hexagons. The combinatorial rules for edges and corners would still remain valid for the entire plane, and you would only have to solve as much of the infinite tiling as you scramble. Centers will be hexagons with one color, corners will be triangles with three colors, and edges will be rectangles with two colors.

Cube -> Megaminx (dodecahedron) -> hexagonal tiling

You could also make a cluster of seven hexagons with seven colors, and tile the entire cluster across the infinite plane, so that all like-colored faces rotate in tandem. Each face will border one face of every other color, and it would solve similar to a slice megaminx.


It would operate like this, but the puzzle plane is infinite:

I would hope it operates better than the Mad Triad-Mine is terrible turning!

You can analyze finite puzzles by considering infinite families of them. For example, analyze the complexity of various ways of solving NxNxN cubes, either as a function of N, or a function of the theoretical shortest solution.