Kociemba Two-Phase Solver

The Kociemba two-phase solver is a fast, complete Rubik’s Cube method based on decomposing the full cube group into a sequence of two searches. It is widely used as a practical baseline because it solves essentially any scramble extremely quickly, though without optimality guarantees.^[1][2]

Core idea

The algorithm splits solving into two phases:

Phase 1: Reach a restricted subgroup

Phase 1 searches for a move sequence that transforms the cube into a restricted subset of states (a subgroup) that satisfies a set of structural constraints (commonly described via orientation and slice constraints). This dramatically reduces the effective complexity of the remaining problem.^[1:1]

Phase 2: Solve within that subgroup

From the restricted subgroup, Phase 2 completes the cube using a reduced move set that preserves the Phase-1 constraints. This final step is typically fast because the remaining search space is much smaller.^[1:2]

Properties (practical perspective)

• Complete: given sufficient search bounds and standard tables, it finds a solution for any valid cube state.^[1:3][2:1]
• Very fast: typical runtimes are far below learned planning and far below optimal heuristic search on full scrambles.^[2:2]
• Not optimal: solution lengths are usually longer than provably optimal solutions and can be longer than learned planners on some instances.^[2:3]

Position in the solver landscape

Kociemba is best understood as a “structured, engineering-heavy” solver family:

• Optimal-search contrast: IDA* + Pattern Databases target provable shortest solutions but become computationally expensive at high scramble difficulty.^[3]
• Learned-search contrast: DeepCube’s MCTS uses a learned value/policy model to guide search under budgets, often producing short (but not guaranteed optimal) solutions.^[2:4]

References