The Square-1
The puzzle that changes shape
A cube whose layers turn in twelfths and stop being square the moment you slice it. The method has two halves: get the cube shape back, then sort the colours. Both halves are taught here in full, and both have been checked against every position they can meet.
A Square-1 has two layers of twelve slots, a middle sliver, and one cut straight through everything. Corners are wide pieces that fill two slots; edges fill one. Turning a layer moves its pieces round in twelfths of a circle, and the slash move swings half of the whole puzzle over the cut. Because corners and edges are different widths, a few slices leave the puzzle looking like nothing a cube should be.
That shape-shifting is the whole character of the event, and it splits the solve cleanly in two. First the shape has to come back: 170 shapes exist, and the method on the shape page reaches a cube from any of them. Then the colours: five algorithms, taught on the colours page, finish the puzzle from there.
8 + 8
Corners and edges
Eight of each. A corner is sixty degrees wide, an edge thirty, and that mismatch is what lets the puzzle leave cube shape.
twelfths
The turns
Layers turn in steps of one slot. A move is written (a,b): top layer a twelfths, bottom layer b, minus meaning anticlockwise.
/
The slice
The slash swaps the right half of both layers in one swing. It is the only move that changes which pieces share a layer.
Reading the notation
Square-1 algorithms are written as pairs and slashes. (3,0) turns the top layer three twelfths clockwise, a quarter turn, and leaves the bottom alone. (0,-3) turns the bottom a quarter anticlockwise. A bare slash is the slice. So / (0,3) / reads: slice, quarter-turn the bottom, slice again. A slice is only possible when no corner sits across the cut, which is why algorithms carry the exact turns they do.
The two halves of a solve
One. The shape. Four sentences of method, learned once: finish through the gate, grow the runs, know the ten shapes that need a fixed escape, finish from the landing. The worst any shape needs is 15 slices with the worst possible choices at every step; a perfect player needs 7. Every one of the 170 shapes has been swept to prove both numbers.
Two. The colours. Once the puzzle is a cube again, five algorithms finish it: one for the middle sliver, one nine-move sequence for the corners, one for the edges, and two short repairs for the positions a cube method cannot otherwise reach. The corner and edge sequences are each a single algorithm used at 16 angles, and at most 7 uses settle the hardest arrangement there is.
No animated player exists for this puzzle yet: the cube players on the other lanes cannot draw a piece two slots wide. The diagrams here are generated from the same engine the proofs ran on, and each one is a position the puzzle can really take. An animated Square-1 is on the build list, after the three-dimensional players for the corner-turning puzzles.