Twentyseven

Rubik's cube solver in Haskell.

Inspired by Herbert Kociemba's Cube Explorer.

The main idea is to precompute, for every configuration, the number of moves required to put certain subsets of the 27 cubies composing the 3x3 Rubik's cube in their right place and/or in the right orientation. This gives lower bounds used for an A⋆-like search in the graph of scrambled cubes.

By default, a suboptimal "two-phase" solver is used, as it runs rather quickly. It currently solves 1000 random cubes (uniformly distributed) in about one minute. The optimal solver is quite slow however, taking between five minutes and two hours to solve a random cube (18 moves in average).

The solver must precompute a certain number of lookup tables, which can be stored in files. These tables take fifteen seconds to compute and weigh 13MB for the two-phase solver, compare that to about 8 hours and 2GB for the optimal one!

You may check the produced files with the checksums in ts-tables.sha256. A compressed archive ts-tables.zip (723MB) of all precomputed tables is available in the branch fetch-tables via git-lfs. Unzip it in $HOME/.27/, or wherever (see usage below).

Usage summary

twentyseven [-p] [--strict] [-d DIR] [--optimal]

• For the first invocation, use -p to precompute nonexistent lookup tables, otherwise an error is thrown when twentyseven tries to load them;
• --strict loads tables immediately, otherwise they are loaded "by need" (so you can also send it a cube to solve);
• -d DIR specifies the directory where the tables should be read and written (default: $HOME/.27/).

The input is read line by line.

Input format

A line can be one of:

• A string of 54 characters (ignoring spaces) from a set of (almost any) 6 characters. Each character then corresponds to the color of one facelet, in the order illustrated below.

  Output: a sequence of moves to unscramble it.

  Facelets are numbered in base 9. Faces 0,1,2,3,4,5 correspond to U,L,F,R,B,D.

              00 01 02
              03 04 05
              06 07 08
  
    10 11 12  20 21 22  30 31 32  40 41 42
    13 14 15  23 24 25  33 34 35  43 44 45
    16 17 18  26 27 28  36 37 38  46 47 48
  
              50 51 52
              53 54 55
              56 57 58
  

• A dot . followed by a sequence of moves to scramble the cube.

  The basic moves are given by a letter in [ULFRBD], or their lowercase counterparts. Each letter corresponds to a clockwise quarter turn of the given face (up, left, front, right, back, down). The orientation is determined when looking directly at the turning face.

  For every basic move, an optional suffix [23'] allows to specify a half turn (e.g., U2), equivalent to a sequence of two quarter turns (UU), or a counterclockwise quarter turn (e.g., U3 or U') equivalent to a sequence of three clockwise (UUU).

  Output: a description of the resulting cube if the moves are applied starting from the solved cube (in the format above, with letters ULFRBD as colors).

• The keyword random.

  Output: a random solvable cube with uniform distribution.

• The keyword quit (or an end-of-file) terminates the interactive session.

Example

Initialization

$ echo quit|twentyseven -p --strict

Example

examples.txt:

qwqwqwqwq erererere tytytytyt rerererer ytytytyty wqwqwqwqw
qwqwqwqwq erqrerere tytytytyt rerererer ytytytyty wqwqwqwqw
BBBBUBBBB UUUULUUUU RRRRFRRRR DDDDRDDDD LLLLBLLLL FFFFDFFFF
DDDFUDLRB FUFDLLLRR UBLBFDFUD ULBFRULLB RRRLBBRUB UBFFDFDRU
111121111 333313333 222232222 444454444 666646666 555565555
111111214 223222222 131333333 344444444 555555555 666666666
.udddlrrrbfffuddd
random

The output then looks like this:

$ twentyseven < examples.txt
U2 D2 L2 R2 F2 B2
Facelets [6,18,11] ("qtq") do not match any regular cubie.
U D F B L R U2 R2 F2 R2 U2 L2 B2 U' D' B2
U L B' L R2 D R U2 F U2 L2 B2 U B2 D' B2 U' R2 U L2 R2 U
U D L R F B U2 B2 L2 F2 D2 B2 R2 U' D' L2
L U' F2 U F2 U L U' L2 D F2 D' F2
BBBBUBBBB UUUULUUUU RRRRFRRRR DDDDRDDDD LLLLBLLLL FFFFDFFFF
BDLLUFBUD LBUBLURFL RLBFFBFRU RLFURULRR UBDRBRDDU DFBDDDFLF

Detail of current heuristics

The distance estimations are based on cosets corresponding to the following elements.

Two-phase

Phase 1

• Corner Orientation × UD Slice
• Edge Orientation × UD Slice

It is possible to store the actual distances to the goal set in phase 1 but the current speed seems good enough for now.

Phase 2

• Corner Permutation × UD Slice Permutation (Phase 2)
• UD Edge Permutation (Phase 2) × UD SlicePermutation (Phase 2)

Optimal

• Corner Orientation × Edge Orientation × XY Slice Permutation, for XY in {UD, LR, FB}
• Corner Orientation × Corner Permutation