Observations on the Magic Dodecahedron (Megaminx)
-------------------------------------------------

I've never seen anything on patterns for the megaminx, with the
sole exception of Kurt Endl's book "Megaminx". Unfortunately there are
no detailed examples, only vague references to "many possible dot
patterns" and "star patterns". A pattern similar to the 6 X order 3
of the cube is on the cover, but only part of the dodecahedron is
visible.

Using the solving skills I developed myself, I deliberately solved
the megaminx with the centres not matching the surrounding face.
Techniques like mono-twists and mono-flips carried over well from
the cube.

My conclusion: A 10-dot pattern is possible!

Here is a description....

One pair of opposite faces is completely solid. The 5 faces adjacent
to solid face A are spotted, also the 5 faces adjacent to solid face
B (opposite to A) are spotted. If we look at one set of 5 faces we
can observe that in this particular 10-spot that the 5 centres
appear rotated to the left, or (since the centres don't really move
in position) that the rest of the face is moved to the right.

Similarly, in the lower tier of 5 faces, we can observe that 5 centres
appear rotated to the left also.

Let's try a small thought experiment. Imagine a skeleton, a
disassembled megaminx. Grab the top and bottom with thumb and
forefinger. Now, while keeping the top and bottom centres immobile,
rotate the rest of the puzzle. What happens? The 10 other centres
rotate in the same direction! If we do this on a cube skeleton the
same thing happens, but on a fleshed out cube this would become a
4 cycle of centres, which is in the swap orbit and can't be reached by
face turns. On the megaminx we have 2 five cycles of centres, and this
is legal.

There are 6 opposite pairs of faces on the megaminx. There are 4 ways
to rotate the centres for each pair to generate a 10 spot. I'll
speculate that there are 6*4 = 24 possible 10-spots.

I suspect various 12-spots are possible. I have no idea how to
easily permute centre pieces on the megaminx.

-> Mark <-