Thread: "Twist types and achievable puzzle states"

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Nelson and I were talking a little because he is looking into possibly
altering Magic120Cell to make a 4D Pyraminx
Crystalpuzzle. Since
that puzzle would have no 2C pieces, there is an issue
(somewhat similar to in MC4D for the 2^4) of how to perform the twists
normally made by clicking those pieces. I'm sure he'd appreciate any
feedback or ideas on that, but for this post, it is the background for
something else I found cool.

I suggested perhaps simply not allowing those kinds of rotations (classic
lazy programmer approach), and made the comment that the scramble
functionality would need to be altered so as to not apply 2C-twists when
scrambling. My incorrect supposition was that if he didn't do this, the
puzzle could get into a state unsolvable with the limited set of 3C and 4C
twists. But Nelson then pointed out that in M120C you can get the result of
a 2C twist with two 3C twists. I was surprised since this is in contrast to
MC4D, where such a substitution is not possible.

But it makes sense in the context of the recent parity discussions here. No
M120C twist types produce any odd parities (for pieces or stickers), so any
subset of twist types can be used to generate all possible puzzle states.
You could solve any fully scrambled M120C using only 4C-twists, or using
only 3C-twists. Neat!

Besides the cube and 120-cell, the only other of the 6 regular convex 4D
polytopes where four faces meet at a vertex is the 4-simplex. For this
puzzle with tetrahedral faces, 2C-twists and 4C-twists are identical in
effect, and lead to even permutation parities for all piece types.
3C-twists are different, but also lead to even parities for all piece types
since the 2-cycles generated all come in pairs.

So it looks like M120C and MagicSimplex4D (or whatever one might want to
call it) are both puzzles that enjoy this symmetry between differing twist
types. MC4D is the only odd one out, where 2C-twists can combine to form a
3C or 4C twist but not visa versa. Also, 3C twists can combine to form a 4C
one, but not the reverse. If I did my counting right, 4C-twists are the
only twists on MC4D that are fully even.

Take care all,
Roice

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