by Christoph Lohe
There are a number of solutions to the Maze Tiles problem. One is shown here:
The solution is shown in a 2D projection of a 2x2x2 cube. Navigate to the Maze Tiles Printout Sheet to look which sides of the projection must fit to each other.
The solution shown here is a special one. Beside the 'zero' tile which builds a closed loop in itself, the tiles build a closed loop line dividing the 2x2x2 cube's surface into just two areas. Let us name these type of solutions the 'perfect' one.
There are much more solutions building more than one closed loop, and so generating more than two areas:
The first example for a non-perfect solution shows a closed loop on the 2x2x2 surface which is embedded in another loop, or one closed area surrounding another one.
The second example demonstrates to pay attention if you are looking for a perfect solution. The solution shown here is not perfect. As soon as you fold the figure to a 2x2x2 cube, there appears one little additional loop which was not very obvious in the 2D projection.
The last example shows a Maze Tiles solution with just two very small changes compared to the perfect solution at the top of this document. The 'LLLL' tile and the 'RRRR' tile were rotated by a quarter turn, and all the other 22 tiles remain exactly the same. Rotating the 'LLLL' or 'RRRR' tiles has no effect on matching to the neighboring tiles as you don't change the 'LLLL' or 'RRRR' configuration by rotation. Nevertheless, there is a small difference: these tiles do not look the same after a quarter turn. The perfect Maze Tiles solution divided the 2x2x2 cube's surface into two areas, but there are two more closed areas - colored in green and blue - as soon as you rotate the 'LLLL' and 'RRRR' tiles.