Solving Rubik's Magic: Master Edition, Part 0: Counting configurations

This is old, old news to everyone but me. Growing up, I loved my Rubik's Magic, but I never played with the Master Edition. Moreover, I never tried to analyse it mathematically. So, I decided to finally play the thing, and try to understand it.

Solving it wasn't difficult, but getting around to writing it up has been a right pain. :) I thought I'd start with something very simple: Counting the number of configurations. Specifically, the number of 2x6 flat, rectangular configurations (like the starting position).

First a couple of fairly obvious constraints: Whenever the puzzle is laid flat, it's always with the same set of pieces facing up - there are two "sides", and pieces don't move between the sides. Moreover, each piece is always connected to the same other two pieces - the sequence of pieces in the loop is always the same.

So, we lay it flat in front of us. We can choose either side to face up. There are twelve possible pieces that can go in the top left. This piece can be one of four rotations. That fully describes the configuration of the top-left piece.

What about everything else? The neighbouring pieces will always be the same, due to the limitations of the "loop". However, this loop could be running clockwise or anti-clockwise. However, once that's determined, everything else is determined. The orientation of all the other pieces are full determined by the orientation of the top-left piece - as it rotates, all the others rotate, like gears.

This gives us 12 x 4 x 2 = 96 configurations. However, the puzzle itself has a rotational symmetry of a half-turn, so really that's just 48 configurations. This is far fewer than I was expecting! By similar logic, there should be 32 configurations for the vanilla magic.

This assumes, of course, that all the configurations are reachable - this is an upper bound, but perhaps some configurations can't actually be achieved. As it turns out, all the configurations are reachable, and that's what I plan to demonstrate next time...

Posted 2015-06-28.