by Christoph Lohe
There are certainly very different approaches to find a solution. Some people may just try to arrange the tiles using the Maze Tiles Printout Sheet. Others may write a computer program. I am not a programming wizard, so my approach was try and error. However, as a first starting point, I have reduced the problem to just the 'L' and 'R' interfaces. I briefly describe my approach here as it may also be helpful to others, and for programmers.
If you completely neglect the '0' type and '2' type interfaces on the tiles then you just have to focus on 20 'L' type interfaces to match to 20 'R' type interfaces. That sounds a lot easier than the original Maze Tiles problem.
Seventeen of the 24 tiles offer 'L' and/or 'R' interfaces, and there are seven tiles remaining with only '0' and '2' type interfaces. If you neglect the '0' and '2' interfaces then the 17 tiles are no longer all different. Let us note an 'x' or 'y' for any '0' or '2' type interface. Then the 17 tiles can be classified into:
The last three tiles only appear once, there are four tiles of type 'LxRy', and five of each type 'LLxy' or 'RRxy'. That's nice! We can arrange the 17 tiles on the 24 fields of the 2x2x2 cube, and we have a high degree of freedom as it is our decision which seven fields are left empty for the moment. These fields will get the seven tiles with only '0' and '2' interfaces later on. Further, there are only six different 'tile designs' to care about for the moment.
I had drawn 17 extra images of the seventeen tiles, just indicating the 'L' interfaces in red, and the 'R' interfaces in blue, and found the following arrangement to match them:
Notice that - for example - the four 'LxRy' tiles are still interchangeable as long as we do not fill the seven empty fields. The same is true for the 'LLxy' and 'RRxy' tiles. All the 'L' and 'R' interfaces are matched in the arrangement. There are interfaces between the 17 tiles as well as towards the seven empty fields which are not defined so far. All we can say is that these open interfaces will either get an '0' or a '2', but no 'L' or 'R' interface.
The second step is to care about the '0' and '2' interfaces on the first 17 tiles, and finally to add the seven tiles which only have '0' and '2' interfaces.
Due to this approach, all the Maze Tiles solutions shown so far are of the same 'family' as their 'L'-'R' interface arrangement is the same.