CSC4006

Solving Standard and Non-Standard Rubik’s Cubes using Group Theory

This project implements Rubik’s Cube models and solving algorithms in GAP using permutation group theory.

The project includes:

• A 3×3 cube representation and solver based on Thistlethwaite’s algorithm
• Reduction methods for 4×4 and 5×5 cubes
• Cube state manipulation through permutation generators
• Validation and reduction utilities

Requirements:

• GAP 4.15 or later
• Standard GAP libraries

Files:

• Models/n3RubiksCube.g 3x3 cube model and generators
• Models/n4RubiksCube.g 4x4 cube model and generators
• Models/n5RubiksCube.g 5x5 cube model and generators
• Models/MegaminxCube.g Megaminx model and generators
• Models/PyraminxCube.g Pryaminx model and generators
• Reduction/CollapseCube4.g 4x4 → 3x3 reduction
• Reduction/CollapseCube5.g 5x5 → 3x3 reduction
• Thistlethwaite/Thistlethwaiten3.g 3x3 solver implementation
• Thistlethwaite/Thistlethwaiten4.g 4x4 solver implementation
• Thistlethwaite/Thistlethwaiten5.g 5x5 solver implementation
• Thistlethwaite/TSolven3.g Driver script for solving 3x3 cubes
• Thistlethwaite/SolveReducedn4.g Driver script for solving reduced 4x4 cubes
• Thistlethwaite/SolveReducedn5.g Driver script for solving reduced 5x5 cubes
• Subgroups/SubgroupSolven3.g 3x3 subgroup-based solver
• Subgroups/SubgroupSolven4.g 4x4 subgroup-based solver

File locations must be changed in:

• all test*.g validation scripts
• CollapseCuben*.g (4 & 5) reduction scripts
• SolveReducedn*.g (4 & 5) and TSolven3.g solution drivers
• Thistlethwaiten*.g (4 & 5) solution scripts
• SubgroupSolven*.g (3 & 4) solution scripts

Known Limitations:

• Certain reduction states may produce invalid 3×3 configurations due to parity inconsistencies
• The reduction process does not fully resolve all even-cube parity cases
• Solver performance decreases significantly for deeper scrambles

Aditional commands for puzzle model manipluation: 3x3

• Cube3.DoTurn(Cube3.*, **); Applies a turn *(F,B,L,R,U,D) **(-1,1,2)
• Cube3.Rot*(); Applies a rotation *(X,Y,Z)
• Cube3.Do*(); Applies a middle slice turn *(M,E,S)
• Cube3.ResetCube(); Resets cube to identity state
• Print(Cube3.Cube); Prints permutation list
• Print(Cube3.GetLayout()); Prints cube layout as list
• Print(Size(Cube3.G)); Prints group size

4x4

• Cube4.DoTurn(Cube4.*, **); Applies a turn *(F,B,L,R,U,D,f,b,l,r,u,d,Ff,Bb,Ll,Rr,Uu,Dd) **(-1,1,2)
• Cube4.Rot*(); Applies a rotation *(X,Y,Z)
• Cube4.ResetCube(); Resets cube to identity state
• Print(Cube4.Cube); Prints permutation list
• Print(Cube4.GetLayout()); Prints cube layout as list
• Print(Size(Cube4.G)); Prints group size

5x5

• Cube5.DoTurn(Cube5.*, **); Applies a turn *(F,B,L,R,U,D,f,b,l,r,u,d,Ff,Bb,Ll,Rr,Uu,Dd) **(-1,1,2)
• Cube5.Rot*(); Applies a rotation *(X,Y,Z)
• Cube5.Do*(); Applies a middle slice *(M,E,S)
• Cube5.ResetCube(); Resets cube to identity state
• Print(Cube5.Cube); Prints permutation list
• Print(Cube5.GetLayout()); Prints cube layout as list
• Print(Size(Cube5.G)); Prints group size

Pyraminx

• CubeP.DoTurn(Cube3.*, **); Applies a turn *(A,B,C,D,a,b,c,d) **(-1,1)
• CubeP.ResetCube(); Resets cube to identity state
• Print(CubeP.Cube); Prints permutation list
• Print(CubeP.GetLayout()); Prints cube layout as list
• Print(Size(CubeP.G)); Prints group size

Megaminx

• CubeM.DoTurn(Cube3.*, **); Applies a turn *(Af,Bf,Cf,Df,Ef,Ff,Gf,Hf,If,Jf,Kf,Lf) **(-1,1)
• Print(CubeM.Cube); Prints permutation list
• Print(CubeM.GetLayout()); Prints cube layout as list
• CubeM.ResetCube(); Resets cube to identity state
• Print(Size(CubeM.G)); Prints group size