Toroidal sliding block puzzles


A new variation on the sliding block theme was recently introduced in the app store: sliding blocks on a torus! Of course, the marketing of the puzzles do not call it like that, but it actually is a nice further development, which uses the possibilities of the touch screen to produce a puzzle variation that is not easily reproduced in a physical puzzle.

So, what is a toroidal sliding block puzzle? Well, the basic idea is that the left and the right sides of the puzzle are connected, and likewise the top and the bottom sides. When you slide something off the top, it will reappear on the bottom, and likewise for the other sides. Mathematically speaking, this means that the pieces are sliding on a torus.

The actual puzzles using this idea are Nuzzle (shown to the right) and 15 Puzzle Pro (both links redirect to iTunes). Both puzzles decided to implement only one variation of sliding blocks on a torus, in which the whole tray is filled with 1×1 blocks, and they push each other around (as illustrated on the right). Unfortunately, this puzzle is extremely simple. The only slight problem is to rearrange the the last few blocks. To solve this, the basic move consists of a simple “row right, column up, row left, column down”-like sequence, which results in a circular permutation of three blocks

nuzzle15_16.PNGOne interesting aspect of these puzzles is that there are some eery situations that appear to be unsolvable. Do you notice something here to the left? This actually only happens with trays with even sides. The trick is that by sliding one row of four blocks for a distance of one block, you actually make three exchanges. By combining one slide with two triangular permutations this situation can be solved, for example by:

– slide: row4right
– triangular permutation: column2down, row4right, column2up, row4left
– triangular permutation: row4left, column2down, row4right, column2up

Now, just as I have argued before, the sliding block puzzles with only 1×1 blocks are trivial and not very interesting. However, it seems to be a really interesting possibility to make puzzles with larger blocks (1×2, 2×2, L-shaped, etc.) and open spaces to slide into (not the push-like approach as used in Nuzzle) on a torus. The traditional designs are of course no good. For example, Super-Century on a torus is solvable in about 25 movements. More interesting designs will probably need a more irregular tray (like Picnic).

And of course the crucial next step: who will make a sliding block puzzle on a real projective plane or on a klein bottle?

[Update] It turned out that I missed another instance of this idea, that is already in the app store since july: RotSquare. It even explains the toroidal nature of the puzzle!

[Update] And another one I missed: Rollo. The idea really seems to be in the air, as by now four different developers apparently independently took this step to develop toroidal sliding block puzzles.

[Update 20 November 2008] Another version just came available: slidePuzzle. The only new aspect is that you can choose your own photo from your photo library. Otherwise the same as the above. The fifth one in line … Still to Klein Bottles, no real projective plane, and no regular sliding block puzzles on any of those surfaces.

[Update 24 December 2008] If I rightfully interpret the description, then the game ‘lines’ in Gamelab is also a toroidal sliding block puzzle.

[Update 3 February 2009] The mentioned toroidal sliding block part of Gamelab is now also available as a standalone app: Lines.

[Update 14 February 2009] Another sliding block app with a toroidal option: PicPu.

[Update 28 February 2009] It won’t stop: Cube the Box is another basic toroidal sliding block puzzle.

[Update 6 June 2009] It has taken some time, but it seems like somebody else came across the same idea: Sly-Q.

[Update 22 June 2009] Another one claiming to have exactly the same completely new revelation in sliding puzzle games: Tile Slide.

[Update 25 June 2009] MatchMadnessLT “A whole new puzzle game idea.” Well, sorry … but it’s not.


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