by Oscar Ivarsson and Pär Eriksson

Cubiks Rube

a Rubik's Cube AI solver

 - Background

 - Algorithm 1: Layer-By-Layer

 - Algorithm 2: Thistlewaite

 - Result

 - Demo

Introduction

HOW TO SOLVE RUBIK'S CUBE

1. Shuffle the cube.

2. Rotate the sides into correct positions

Background

Invented in the 70s by Erno Rubik

43 * 10^18 possible combinations

Rubik's Cube Group: { L, R, F, B, U, D }

Background

God's Number

The maximum number of moves to solve any instance of the cube. 

1981 Jan - 140 moves

1981 Jul - 104 moves

1992 May - 56 moves

1995 Jan - 42 moves

2005 Nov - 40 moves

2007 Jul - 34 moves

2009 Jun - 29 moves

2014 Aug - 26 moves

Background

Solvability

Only 1/12 cubes are solvable

Algorithm 1: Layber-By-Layer

Algorithm 2: Thistlethwaite

Dr. Morwen Thistlethwaite

Thistlethwaite's Algorithm, works by restricting the positions of the cubes into groups of cube positions, that can be solved using a certain set of moves.

Maximum moves: 52 (Half-turn metric)

Algorithm 2: Thistlethwaite

G0 = < L, R, F, B, U, D >

G1 = < L, R, F, B, U2, D2 >

G2 = < L, R, F2, B2, U2, D2 >

G3 = < L2, R2, F2, B2, U2, D2 >

Moves of each subgroup

The plan is to manoeuvre down through the chain:

G=G0> G1> G2> G3> 1

One gets from Gi to Gi+1 by using moves in Gi only!

G4 = { I }   solved state

Edge orientation example

Algorithm overview

Algorithm 2: Thistlethwaite

Optimizing the moves in each phase

The optimal theoretical solution would be to store all states of the cube, but as mentioned before this is practically impossible.

Instead we can store possible position for each phase. (G1, G2, G3)

?

!

Algorithm 2: Thistlethwaite

Iterative-Deepening A*    (IDA*)

​depth = 1

​depth = 2

​depth = 3

​depth = 4

Result

 

 Shuffle Count                   100        200        300        400        500        600        700        800        900 
      
 Move Count: Layer-By-Layer      252        244        156        225        230        281        229        380        183       
 Move Count: Thistlewaites       54         48         50         45         55         44         41         48         55        
 
 Time: Layer-By-Layer            00:00:11   00:00:12   00:00:05   00:00:03   00:00:05   00:00:09   00:00:07   00:00:08   00:00:06  
 Time: Thistlewaites             00:11:71   00:28:96   01:14:59   01:01:79   00:19:46   00:14:38   00:24:21   01:18:67   02:40:86  

Demo

http://cubiksrube.com