Tokamak Puzzle

Oscar Van Devender did a varation of the Auzzle Puzzle and called it Tokamak:

Among the improvents Oscar mentions the following:

  1. The eight segments were given eight different colors. The segments were rather hard to keep track of in the original version.

  2. The magnets were placed such that the puzzle can only be solved when the top four segments are offset from the bottom ones.

While I like the shape and the looks of the colored Tokamak, and I would love to get the STL files to print one for myself (I find the original Auzzle too big for my hands), I have to admit I disagree with Oscar on both points:

  1. Not being able to easily identify segments at rest is the whole idea! When segments are colored or marked in any way, it stops being a magnetic puzzle. Magnets start to act merely as a decoration, they are not longer vital for a solution. The puzzle is reduced to a 2x2x2 cube with a few movemenet constraints, which is still trivial for any twisty puzzle enthusiast.

  2. Even in the non colored version, introduction of a shift between upper and lower layer does not add, but possibly (depending on an assembly process) reduces puzzle complexity even more. Here is a logical proof (you might need to refer my original Auzzle Analysis for some context):

    a) Lets assume that the way Tokamak is assembled, all 8 segments happend to still be unique (i.e all numbers from 0 to 7 are being represented: 000, 001, 010 .. 111). By definition then, that puzzle has also 96 non-shifted solutions in which segments are aligned, as described in my original analysis. So the shift during assembly was pretty much a useless gesture.

    b) To avoid that, we must ensure during assembly that segments are NOT unique and some encodings are not being represented. That means two or more segments must share the same binary encoding, as we still have 8 segments. That in turn means these identical segments can swap places and the puzzle would remain solved (all magnets would still attract). So total number of unique combinations is decreased by a factor of K! while the number of solutions increases by a factor of K! (where K is number of pieces sharing the same encoding).