Rubik

This repository contains an implementation of Thistlethwaite's algorithm for solving the Rubik's cube. A good first read can be found [here], although an explanation on tetrad orbits is lacking. The algorithm considers five subgroups G0, G1, G2, G3 and G4, of the cube's complete group G0. G0 is generated by quarter turns on the up, down, front, back, left and right faces. G4 only contains the identity element. The other subgroups are detailed below. The algorithm foresees a look-up table for reducing resp. G0 to G1, G1 to G2, G2 to G3, and G3 to G4.

Below, the conventions used in this repository are provided, as well as a breakdown of the group reductions. By numerical exploration, I figured out what the tetrad orbits mean, which is also explained below. For each of the stages in the algorithm, a uint32_t key is associated with the cube's state, which allows to look up an action and the next key, i.e. the key of the cube's state after performing the action. Once key 0 is found, that particular stage of the algorithm is solved.

Test solve.cpp allows to track a full execution of the algorithm.

Conventions

Permutation state

             +--------+
             | 0  1  2|
             | 3  U  4|
             | 5  6  7|
    +--------+--------+--------+--------+
    | 8  9 10|16 17 18|24 25 26|32 33 34|
    |11  L 12|19  F 20|27  R 28|35  B 36|
    |13 14 15|21 22 23|29 30 31|37 38 39|
    +--------+--------+--------+--------+
             |40 41 42|
             |43  D 44|
             |45 46 47|
             +--------+

Compact state

           C3--E3--C7
            |   Z   |
           E10  U  E11
            |   Z   |
   C3-E10--C6--E2--C2-E11--C7--E3--C3
    | W Z W |       | W Z W |       |
   E6 Z L Z E4  F  E5 Z R Z E7  B  E6
    | W Z W |       | W Z W |       |
   C4--E8--C1--E0--C5--E9--C0--E1--C4
            |   Z   |
           E8   D   E9
            |   Z   |
           C4--E1--C0

• W are the corner parity references (fixed in G2 = { U2, D2, F2, B2, L, R })
• Z are the edge parity references (fixed in G1 = { U2, D2, F, B, L, R })

Subgroups

G0 = { U, D, F, B, L, R }

• sum of corner parities mod 3 is always zero (see compact_state.cpp : corner_generators):
  • corner occupants: 8! states
  • corner parities: 3^7 states
• sum of edge parities mod 2 is always zero (see compact_state.cpp : edge_generators):
  • edge occupants: 12! states
  • edge parities: 2^11 states
• corner and edge moves are not independent:
  • each edge and corner occupant permutation is odd
  • e.g. corner occupant permutation from the solution with U1: { C6 : 6, C3 : 3, C7 : 7, C2 : 2 } => { C6 : 3, C3 : 7, C7 : 2, C2 : 6 }: 3 swaps
  • the global corner occupant permutation and global edge occupant permutation are hence either both even or both odd
• the total number of states is hence upper bounded by (8! * 3^7 * 12! * 2^11) / 2 = 43252003274489856000
• it turns out that the upper bound is also the actual number of states

G1 = { U2, D2, F, B, L, R }

• only half twists are allowed for up and down
• edge parities (Z) cannot change
• going from G0 to G1 reduces the number of allowed states by 2048:
  • edge parities: 2^11 states

G2 = { U2, D2, F2, B2, L, R }

• only half twists are allowed for up, down, front and back
• corner parities (W) cannot change
• edge occupants in slice { E0, E1, E2, E3 } remain within this slice
• going from G1 to G2 reduces the number of allowed states by 1082565:
  • corner parities: 3^7 states
  • edge slice occupants: 12! / (8! * 4!) states

G3 = { U2, D2, F2, B2, L2, R2 }

• only half twists are allowed
• edge occupants in slices { E4, E5, E6, E7 } and { E8, E9, E10, E11 } remain within their slice
• corner occupants in tetrad groups { C0, C1, C2, C3 } and { C4, C5, C6, C7 } remain within their tetrad group
• the tetrad orbits are restricted to 4! * 4 states:
  • suppose the first tetrad occupation is { C0 : a, C1 : b, C2 : c, C3 : d } with 0 <= a, b, c, d <= 3
  • the second tetrad occupation is any of:
    • { C4 : a + 4, C5 : b + 4, C6 : c + 4, C7 : d + 4 }
    • { C4 : b + 4, C5 : a + 4, C6 : d + 4, C7 : c + 4 }
    • { C4 : c + 4, C5 : d + 4, C6 : a + 4, C7 : b + 4 }
    • { C4 : d + 4, C5 : c + 4, C6 : b + 4, C7 : a + 4 }
  • the global corner occupant permutation is hence even, and therefore also the global edge occupant permutation
• going from G2 to G3 reduces the number of allowed states by 29400:
  • tetrad groups: 8! / (4! * 4!) states
  • tetrad orbits: 6 states (out of 4! * 4! permutations 4! * 4 are allowed)
  • remaining edge slice occupants: 8! / (4! * 4!) states

G4 = { U0, D0, F0, B0, L0, R0 }

• the solution
• going from G3 to G4 reduces the allowed states by 663552:
  • tetrad orbits: 4! * 4 states
  • (even) edge occupant permutations within their slice: (4!)^3 / 2 states