The Maze Tiles

by Christoph Lohe

The idea of the Maze Tiles is to design different squares which carry lines on them, and to arrange these tiles to build a larger closed loop line.

For the Maze Cube, a center cuby just consisted of a single mini-surface. An edge cuby had two mini-surfaces, and the pair of mini-surfaces had build an interface to a corner cuby. The corner cuby was composed of three mini-surfaces, and any pair of these has build an interface towards the edges.

What about an arrangement of four mini-surfaces? You can arrange four mini-surfaces to a 2x2 square, and draw right-angles and straight lines on them:

Each side of such 2x2 Maze Tile connects to another tile, and the interface types can exactly be classified like the Maze Cube interfaces. You can consider all possibilities to arrange right-angles and straight lines on the tiles whilst avoiding any internal open line. Pay attention to any rotations which may lead to the same image. You will find 24 different arrangements for a 2x2 Maze Tile. Taking the same notation already known from the Maze Cube, the Maze Tiles are:

The tables show the different Maze Tiles sorted according to their properties. The first three tables show complementary pairs in their columns. The last table shows Maze Tiles that equal their complement.

If you mirror any tile of the first table then you get the original tile again. The mirrored tiles of the second table lead to new tiles. The third table shows tiles where the mirrored image equals the complement. Finally, the last table shows tiles which equal both, their complementary and their mirrored images.

The first two tables' first rows show tiles which have just a single open line on it. The very first tile can be considered having zero open lines as it builds a closed loop in itself. We name it the 'zero tile'. The complements in the second rows have three lines, but the zero tile's complement offers four lines. The third and fourth tables show all the tiles with two lines.

Those who have already investigated the properties of the Maze Cube cubies will not be surprised to realize that the number of all the right-angles and straight lines on the 24 Maze Tiles is 64:32, it is the ratio 2:1 again. However, against the Maze Cube, we don't get the same number for all interface types. For the Maze Tiles, you get 28 each of "0" type and "2" type interfaces, but only 20 each of "L" type and "R" type interfaces.

The goal is to arrange the 24 different Maze Tiles without any open ends. It is not possible to arrange them simply into a two-dimensional form (4x6 or 3x8 tiles wide) as there must be "0" type interfaces at all the borders in this case, and the properties of the Maze Tiles don't allow it. However, a 2x2x2 cube has 24 surfaces which can be equipped with the Maze Tiles!

Who is able to find a solution for such arrangement with no line ends left open?

Maze Tiles Printout Sheet

Maze Tiles Solution

Maze Cube

		updated: 09.01.2004