Version 1 (modified by kali, 16 years ago) (diff)

first draft on cube constraints

Overview

Once the color reduction issue is fixed, we have several other interesting problems still at hand.

  • computing configurations of cubes : from images to finished cubes.
  • implementing these configuration in real life on six hundred cubes : from defined computer known cubes configurations to physical world.

Computing cubes

Constraints on Rubik's cubes

The pixels displayed with facelets on each side of the cube can not adopt any combination.

Some constraints are imposed by the cubies themselves:

  • the two center facelets have two colours from opposites sides of the cube
  • no more than four corner facelets (out of height, four on each face) or four edge facelets can show the same colour
  • there are more constraints, but they are more difficult to express. Let's assume white is opposite yellow ; if we consider only corner facelets, it is not valid to display four white pixels and three blue on the same cube, as there are four corner cubies with a blue facelet, but two of them are required to show white pixels... We may want to be more specific here, and get an exact expression of which combination are permited.

And there are also "parity" constraints imposed by the cube permutations : if one break a cube apart, using a screwdriver, and put it back together randomly, eleven time out of twelve, the cube will reach a state that can not be "solved". More specifically, if a cube is in a "valid" configuration:

  1. there is always an even number of edges that are flipped,
  2. the sum of elementary rotation on corner cubies is always a multiple of three,
  3. the total number of permutation of cubies (including both corners and edges) is even.

As we will have four hidden edge cubies, we can accomodate constraints 1 and 3 can be arranged with no difficulty. Obviously the second one remains an issue.

Implementing configuration

Links