I can solve the 3x3 and 4x4 Rubik's cubes.
I am right now writing a story about a world full of twisty puzzles where it starts out like a city with 3 friendship networks and becomes 3 empires with 1 being the cubes, 1 being the cuboids, and 1 being the other shapes which include dodecahedra and pentahedra and tetrahedra and lots of others.
In my story for both the cubes and cuboids the maximum dimension for any side is 17 and minimum is 1. For the other shapes it is similar but with their equivalents.
For the dodecahedra they follow the metric prefixes.
17x17: Kiloyottaminx(following the double prefix rules of smaller before larger)(kiloyotta- means 1000 * yotta)
If something new is cooked or discovered there is a rotation of jobs in the empire of cubes.
It started like this:
Animal care: 16x16
Guard (1st degree): 10x10
Guard (2nd degree): 11x11
Guard (3rd degree): 12x12
Guard (4th degree): 13x13
Guard (5th degree): 14x14
Guard (6th degree): 15x15
The 3x3 is always the emperor
As for determining the cube that will take the job after a rotation here is the math:
If N > 17 then N+1
If N = 17 then 1x1
Here is how it was after the first batch of cheddar cheese was made:
Animal care: 17x17
Guard (1st degree): 11x11
Guard (2nd degree): 12x12
Guard (3rd degree): 13x13
Guard (4th degree): 14x14
Guard (5th degree): 15x15
Guard (6th degree): 16x16
And here is how it is now after the pizza was made:
Animal care: 1x1
Guard (1st degree): 12x12
Guard (2nd degree): 13x13
Guard (3rd degree): 14x14
Guard (4th degree): 15x15
Guard (5th degree): 16x16
Guard (6th degree): 17x17
The cubes have genes that determine what cubes they will get given a cube they are mating with and external factors like temperature and sterility as in no viruses and no bacteria or other infectious agents.
The determination of what it should be is larger cube - smaller cube so that 3x3 - 2x2 = 1x1 etc.
If they are in the cold it is the larger of the middle values between the larger cube and the smaller cube and high water gives lower of the middle values. High food gives the sum meaning that a 17x17 mating with a 16x16 and eating a lot of food will give birth to a 33x33 cube. In any case where there was a high food condition surgery needs to be done.
If they are in a sterile environment they stay as little 1x1s proving that immunity, not sterility is needed. In a hot environment the baby dies during the pregnancy and is never born.
These cubes are special in that if 1 or more pieces come off the cube that area of the cube is automatically cauterized so that there is little or no bleeding. This is nice.