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Solving the Flat Cube - mklabgr - 08-28-2026
Solving the Flat Cube — James Propp
Published: August 19, 2026
James Propp introduces the Flat Cube, a planar puzzle inspired by the Rubik’s Cube. Instead of small cubes, it consists of lozenges (rhombi) tiling a hexagon. Whenever three lozenges form a small hexagon, they may be rotated around its center in multiples of $60^\circ$. The goal is to restore a scrambled configuration using the fewest possible twists. This leads to an analogue of the Rubik’s Cube’s “God’s number”: the maximum, over all possible scrambled states, of the minimum number of moves required to solve the puzzle. Propp initially proves that this number must be at least 27.
He gives two elegant visual proofs. The first adds a dimension: a lozenge tiling can be interpreted as the visible surface of a pile of unit cubes inside a $3\times3\times3$ box. A legal twist corresponds to adding or removing exactly one cube. Transforming the empty configuration into the completely filled one therefore requires at least
33=273^3=27
moves. The second proof removes a dimension: considering only the nine horizontally oriented lozenges turns them into beads sliding along one-dimensional tracks. Each of the $9$ beads must move $3$ positions, while one twist moves only one bead by one position, giving again
9×3=27.9\times3=27.
For the uncolored version, this reasoning can in fact be pushed further to show that the worst-case distance is exactly $27$.
The article also connects the puzzle with the mathematics of lozenge tilings, plane partitions, combinatorics, and configuration spaces. Propp collaborated with puzzle designer Oskar van Deventer and Dmitry Andreev to turn the mathematical idea into an actual mechanical puzzle, called the Propp Twist. Particularly interesting are updates added two days after publication: Tom Rokicki showed that the order-$2$ Flat Cube has God’s number exactly $27$, while two arbitrary states can be as far as $30$ moves apart. More strikingly, Rui Viana subsequently proved that for the order-$3$ Flat Cube, God’s number is at least $81$, and Rokicki found a pair of states separated by 91 moves. Thus the problem is substantially richer than the original $27$-move argument suggests.
Key takeaways
- The Flat Cube converts a geometric tiling problem into a Rubik-like optimization problem.
- A lozenge rotation can be interpreted either as adding/removing a cube or as moving a bead one step.
- Both viewpoints give the beautiful lower-bound calculation $27=3^3=9\times3$.
- The later results show that the full colored order-$3$ puzzle is considerably harder: its God’s number is already known to be at least $81$, with some pairs of states 91 moves apart.
ARTICLE [PDF]