Change the state.
Face names refer to Up, Down, Left, Right, Front, and Back. A turn is clockwise when looking directly at that face. A prime (′) reverses it.
Mithilesh / interactive experiments
Small puzzles.
Big state spaces.
Turn a face. Watch a permutation. Follow an idea.
A small playground for the mathematics behind the next move.
3×3 / permutation engineSolved
Drag to inspect arrow keys rotate the view
Moves: 0Last: —54 stickers
The classic
3×3×3 cube
Real sticker permutations. Quarter turns, inverse turns, and the option to retrace every move back to the start.
Outer face turns
prime = counterclockwiseWide turns
two layers togetherTry a four-move sequence
Repeat six times to return every sticker to its starting position.
120° × 3 = 360°
A geometric illustration of tetrahedral symmetry.
Session / move history
Your next move goes here.
01How to explore
A few useful moves.
Find the way back.
Undo reverses the last move. Reverse moves retraces this session in reverse order; it is a history reversal, rather than a general-purpose solver.
Use the keyboard.
Focus the canvas to use U, D, L, R, F, B. Hold Shift for a prime turn. Arrow keys rotate the view; Home restores the camera. All controls also work with Tab and Enter.
02The mathematics underneath
Look for the invariant.
Permutations & constraints.
The 3×3 state space combines corner and edge permutations and orientations. Legal moves preserve constraints — a useful lens for thinking about policy spaces.
Scale the structure.
2×2, 3×3, 4×4, and 5×5 share the same geometric engine. Bigger cubes add centers and edge pieces; wide turns move two layers at once.
Change the rules.
Pyraminx explores tetrahedral symmetry. Clock explores modular time through a coupled-dial illustration. Their kinetic previews complement the playable cube models.