Exact Completion Multiplicities from Five Observed Faces of a Rubik's Cube

Exact Completion Multiplicities from Five Observed Faces of a Rubik's Cube How much information about a Rubik's Cube remains unknown when five of its six faces are completely observed? Although five observed faces may uniquely determine a cube configuration, they do not always do so. This paper provides a complete mathematical classification of the number of legally reachable configurations compatible with any given observation of five complete faces of a standard 3×3×3 Rubik's Cube. The main result establishes that exactly four completion multiplicities are possible: [N_5\in{1,2,3,12}.] No other positive number of legal completions can occur. Furthermore, the paper derives the exact probability distribution of these multiplicities when a complete legally reachable cube configuration is chosen uniformly at random: 1 completion: 393/440 ≈ 89.3182% 2 completions: 3/55 ≈ 5.4545% 3 completions: 17/330 ≈ 5.1515% 12 completions: 1/1320 ≈ 0.0758% Thus, five complete faces uniquely determine the cube in approximately 89.32% of uniformly sampled legal configurations, despite providing no universal guarantee of uniqueness. The proof combines cubie permutation parity, orientation constraints, combinatorial reduction, and exhaustive finite enumeration. The resulting classification is supported by reproducible Python reference programs, exact integer computations, and explicit examples realizing all four multiplicities. The computational nature of the classification and the scope of the verification procedures are stated explicitly. The accompanying source archive provides the programs required to reproduce the reported enumeration. This work extends the reconstruction results established in: Eigler, D. (2026). How Much of a Rubik's Cube Does a Three-Face Photograph Determine? Zenodo. https://doi.org/10.5281/zenodo.23846140 Whereas the earlier work establishes the limitations of reconstruction from partial face observations and the universal six-face reconstruction guarantee, the present paper determines precisely how much ambiguity remains when exactly five faces are known. The results provide a complete quantitative answer to the five-face completion problem and contribute to the mathematical understanding of partial information, combinatorial reconstruction, and state identification in the Rubik's Cube.