I was pondering alternate mechanisms and terminology for the cheese and puck-type puzzles.
Cheese-type: those where an arbitrary diameter can be flipped without rotating the inner core. We can use the terminology MxN where M is the number of wedges and N is the number of layers. Known examples:
- Rubik's Cheese (6x1)
- Rubik's UFO (6x2)
- Masterball (6x4)
- ??? (6x6) (I've seen pictures, but don't know the name)
Puck-type: those where the inner core of the puzzle must be rotated to allow a diameter to flip. MxN terminology can also be used. Known examples:
- Saturn (6x1)
- Puck (12x2)
- Brain Ball - flips two opposing and unequal segments rather than half the puzzle
- Square-1 could be seen as a bandaged, Puck-type 12x2
- The puck mechanism allows for MxN where:
- M must be even or there are no diameters along which to flip the puzzle
- N can be any number; the puzzle must be symmetrical between the top and bottom, but this can either be odd (there's an equator layer) or even (there's no equator)
- The cheese mechanism seems to demand MxN where:
- M must be 2(2P+1) for any P; that is, twice an odd number. Here's my reasoning: the internal mechanism must be in the same state before and after a flip. The cheese mechanism uses pieces of alternating type. If there are an even number of pieces on each half of the puzzle, then when performing a flip you'll end up with pieces of the same type adjacent. Therefore, each half of the puzzle must have an odd number of pieces.
- N can be any number, for similar reasons to the puck.
This applies to the UFO mechanism as well as the Patent US5199711:
It's therefore easy to sketch a mechanism for a 6 (2*(2*1+1)) xN, 10 (2*(2*2+1)) xN or even 14xN (2*(2*3+1)) cheese-type puzzle. However, making 8xN or 12xN is not obvious.
(Bandaging a 12xN would give you a true Square-1 II, which is what made me think of all of this.)
Observation: bandaging a 2x2x2 cube gives you a 4x1 cheese which violates the above rule. (Or http://www.puzzle-shop.de/color-tonne.html
, which is actually 4x2 and lets you assume non-cylindrical shapes) Are there other mechanisms that allow for different values of M? The 2x2x2 cube's mechanism is internally a 3x3x3; can we follow this lead and co-opt other mechanisms?