Malcom Jones: Bandaged Rubik's Cubes and their partial symmetries

Speaker: Malcom Jones (IMFM)

Abstract: Every move of a Rubik's Cube is a bijection of the tiles—a total symmetry. These moves generate the Rubik's Cube group. Choosing two tiles and bandaging them together changes the puzzle. There are fewer scrambles, but the position of the bandage restricts the legal moves at each stage. While a move of a bandaged Rubik's Cube is a bijection of the tiles, it does not generally preserve the bandage—a partial symmetry. Groupoids were introduced by Brandt in 1927 as generalisations of groups that model partial symmetry instead of total symmetry. We will discuss their widespread applications and build a groupoid that generalises the (unbandaged) Rubik's Cube group. Please bring your Rubik's Cube (or other twisty puzzle) if you have one.