The gathering and solvability-testing took place together. Next, I gathered all of the layouts into equivalence classes, determining which layouts were related to each other by legal sliding moves (different states of the same puzzle), and tested them for their solvability. Unsurprisingly, the 5x4 layouts were the vast majority of those (12,573 of them). At this point I had a pool of 13,961 different board layouts which potentially represented good transposition puzzles, but many of them would not be solvable, and of course many of them were rearrangements of one another - different states of the same puzzle.
The rectangular boards allow horizontal and vertical reflections, as well as 180- and (in the 4x4 case) 90-degree rotations.įrom these, I selected layouts with a “distinguished pair” of blocks - two blocks of different shape from all the others in the puzzle, which would be the target pieces to swap. So: In creating this collection, I began by enumerating all symmetric arrangements of blocks that are possible on 4x3, 4x4, and 5x4 grids (subject to a few constraints for making reasonable puzzles, for example, enough blank space for all the pieces to move). And especially when the goal of the puzzle is a transposition, it's aesthetically pleasing if the layout has some symmetry as well. Transposition puzzles seem a little more natural and satisfying. I enjoy sliding block puzzles, but the “Move Block A to Position X” type of puzzle feels arbitrary. You may attempt the Perfect Solve as a further challenge once you have achieved the basic swap, but beware: there is one puzzle where it is not possible. In almost all of the puzzles, you can swap the colored blocks and restore all of the gray obstacles to their original position as well: a “Perfect Solve,” as in the example above. Note: You've solved the puzzle as soon as the colored blocks have been swapped.
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