3x3 Speedcubing: The Ultimate Limits Forecast
History · Methods · Hardware · Math · Biomech · Top Cubers · Training · Statistics — toward a single & average forecast.
God's Number Evolution (1981-2010)
God's number is the Cayley graph diameter of the cube group under the chosen metric. Thistlethwaite (1981) gave the first upper bound, 52 HTM. 29 years later Rokicki et al. (2010) used ~35 CPU-years of Google compute to push HTM to exactly 20, with at least one position (superflip composite) requiring exactly 20.
| Year | HTM bound | By | Method |
|---|---|---|---|
| 1981 | 52 | Morwen Thistlethwaite | four-stage group reduction |
| 1990 | 42 | Hans Kloosterman | improved cosets |
| 1995 | 29 | Michael Reid | pruning tables |
| 2007 | 26 | Kunkle & Cooperman | parallel coset search |
| 2008 | 22 | Tomas Rokicki | symmetry reduction |
| 2010 | 20 | Rokicki / Kociemba / Davidson / Dethridge | ~35 CPU-years on donated Google compute (proven optimal) |
- HTM = 20 (Rokicki et al. 2010, SIAM J. Discrete Math. 27(2))
- QTM = 26 (Rokicki & Davidson 2014; superflip × 4-spot)
- STM unsettled (16 ≤ ? ≤ 20);superflip is 16-STM-solvable
- NxN Θ(N² / log N) (Demaine et al. 2011, arXiv:1106.5736)
- NxN optimal solving NP-complete (Demaine, Eisenstat, Rudoy 2017)
Superflip (8 corners correct, 12 edges flipped in place) is the classic 20-HTM-hard position; sits in the center of the cube group. Solvable in just 16 STM — source of the 16-STM lower-bound.
Two-phase (Kociemba 1992). Phase 1 reduces to subgroup H = ⟨U, D, R², L², F², B²⟩; phase 2 within H. Phase 1 coset count = 2,217,093,120. Cube Explorer + Tronto's nissy are modern optimal-solver standards.