The Crammed Cube is an internally geared twisty puzzle that expresses the Mathieu M11 group over its 12 edge pieces. The puzzle features a modifiedCompy Cube mechanism with 6 axes that would normally have 11 moving edge pieces. It is called the ‘Crammed Cube’ because, while the axis system would normally have 11 moving edges, I’ve crammed an extra one into the red/white edge and lifted the corner angle to make space. This edge duplication is necessary to make the group actions fit over cubic geometry.
Simulator
Construction
The puzzle’s six corner axes are divided into two groups of three that are mechanically linked. When a central corner is rotated clockwise, the diagonally opposite corners rotate counterclockwise simultaneously.
The mechanism combines gears and universal joints that pass through the puzzle’s core.
Video
Theory
The edge pieces express the M11 group over 12 points. In the natural 11-point representation of M11, stabilizing one point at a time produces a clear factorization:
- Degree 11 — sharply 4-transitive
- 7920 = 11×10×9×8 : M11 → M10 → M9 → M8 (Q8)
However, the 12-point representation used in this puzzle has a less intuitive point stabilizer series:
- Degree 12 — 3-transitive
- 7920 = 12×11×60 : M11 → PSL(2,11) → A5
The groups in this chain have unusual properties. PSL(2,11) over 11 points relates to the symmetries of the Paley biplane, while the Alternating group A5 over 10 points corresponds to the face symmetries of the projective icosahedron.
Puzzle Specifications:
- Total states: 71,280 (3×3×7920)
- God’s number: 20 (18 without corner stickers)
- Undistorted edge length: 70mm
- Weight: 240g
Gallery
