Back to a cube

The shape method

Four rules cover all 170 shapes this puzzle can take. The worst case is 15 slices even choosing badly, and that number comes from sweeping every shape, not from practice.

Shape is about where the wide pieces sit, and nothing else. Colours play no part in this half, so the pictures below show corners as wide tinted wedges and edges as narrow pale ones. A layer of a cube-shaped Square-1 has its four corners spread evenly; the method is the route back to that from anywhere.

One word does most of the work below: a run is a stretch of edges sitting together in one layer with no corner between them. Count the longest run on each layer and you have read the position.

cube shape: what square layers look like as wedges

Rule one: the gate

Exactly one shape in the whole puzzle is a single slice from the cube: both layers in the kite shape, a corner pair at the cut and the rest spread. Every solve on earth finishes through this gate, because nothing else is one slice out. Learn to see it, line the two kites up, and cut.

the gate: both layers kites

Rule two: gather the edges

From any other shape, make a slice after which the puzzle's longest run of edges is longer, or as long as before with the other layer's best run longer. Any slice that manages this is correct; there is always at least one, except at the ten shapes below. Keep gathering until one layer holds all eight edges. The count settles it: gathering cannot loop, because the runs only ever grow.

Rule three: the ten escapes

the only shapes with no gathering slice

Ten shapes offer no slice that grows a run. Each has a fixed route of at most three slices to safety, found by sweeping the puzzle, and the route is followed to its end before the gathering rule starts again. Stopping half way is how a solver gets sent in circles, and the sweep that fixed these routes found that failure before any reader could.

2 fixed slices to safety

3 fixed slices to safety

2 fixed slices to safety

3 fixed slices to safety

2 fixed slices to safety

3 fixed slices to safety

Rule four: the landings

Gathering ends with all eight edges in one layer, which forces six corners into the other. The two stray corners in the edge layer sit in one of five arrangements, and each has a fixed finish: four slices when the strays touch, six at the worst separations. The finishes were read off the full table, so they are shortest possible.

corners at gaps (0, 8) — 4 slices to the cube

corners at gaps (1, 7) — 6 slices to the cube

corners at gaps (2, 6) — 4 slices to the cube

A whole solve, from the worst shape there is

square + kite: 7 slices from a cube, played perfectly

The farthest shape from a cube looks nearly finished: one layer already square, the other a kite. This is the method's own route from it, 11 slices, every picture the exact alignment before the knife.

the start: square on top, kite below

slice 2

slice 3

slice 4

slice 5

slice 6

slice 7

slice 8

slice 9

slice 10

slice 11

Each picture is the moment before a slice: turn your layers until they match, then cut. After the last slice the puzzle is a cube.

All of this has been proved by sweep: a gathering slice exists at every shape except the ten; the escapes and finishes replay to where they claim to go; and a solver making the worst allowed choice at every step still reaches the cube within 15 slices. Next: the colours.