An exhaustive search you can interrupt with a toy
Eleven Moves From Anywhere
Turn and scramble a pocket cube while this page exhaustively searches every position in its rotation-quotiented state space. The browser builds the optimal-distance table, draws the histogram it actually found, and lets you hunt a farthest position. The pocket-cube results are recomputed here; the famous result for the larger cube is kept clearly cited, not claimed as a browser proof.
Preparing the exhaustive search.
Interactive fixed-corner pocket cube
The pocket cube is solved.
Optimal distance in the half-turn metric
Building the distance table. You can turn the cube now.
Quarter turns
Half turns, each counts as one HTM move
moves: none
The outlined down-back-left corner is fixed in place and orientation. In this normalized frame, U, R, and F generate every rotation-equivalence class.
The whole universe, one ring at a time
Breadth-first search begins at solved. Its first ring contains every position one move away, the next ring contains every unseen position two moves away, and so on. The first time a state appears is therefore its shortest possible distance. This worker does not sample and does not stop after finding a hard case. It exhausts the graph.
The histogram this browser found
States reached
waiting
HTM diameter
waiting
QTM diameter
waiting
Farthest states
waiting
The worker is filling this chart from its own BFS.
The top of the graph will appear only after it is found.
The chart is not a remembered table painted onto the page. Its bars and labels are created only when the worker returns its newly computed depth counts. The button above then chooses one of the states the same run actually placed in its final ring. From there, Take one optimal move follows the distance table downhill, one guaranteed-shortest step at a time.
The objection is the scale
The fair dismissal
A pocket cube is a small finite graph. Exhausting it in one browser tab does not make the original cube's much larger result fall out of this page.
What the small graph shows
The logic is visible without taking the larger result on faith: define legal moves, cover the state graph, and inspect its last occupied ring. Scaling that demand required different machinery, not a different meaning of “farthest.”
For the 3x3x3 cube, Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge partitioned the state space into cosets, reduced it with symmetry, then used search and prepasses to prove the HTM diameter is 20. Their final computation ran in 2010. Their 2013 paper reports an extrapolated 34.75 CPU-years. Those two figures are cited results. This browser does not recompute either one. The literal algorithm here is a full in-memory BFS; the larger proof is a more elaborate exhaustive coset computation.
The check
The green values below are inserted only from arithmetic or completed exhaustive searches in this tab. If the worker fails, they stay visibly incomplete.
Convention, not measurement: fixing the down-back-left corner quotients out whole-cube rotation. The unquotiented sticker-state count is larger by the number of spatial orientations of a cube. The page derives that factor as choices for the top face times choices for the front face, then checks that the quotient agrees with 7! × 36.
rotation quotient: waiting
Free choices: HTM counts a quarter turn or half turn of one face as one move. QTM counts a half turn as two. The color scheme, flat net, fixed corner, and choice of U/R/F generators are representations, not measured properties. Changing the fixed corner relabels the same quotient graph.
Uncertainty: there is no sampling error in a completed finite enumeration.
There can still be a software error or an interrupted worker. The standalone dependency-free
Node verifier re-implements the same cubie move model, rebuilds the move tables and both
searches from scratch, asserts every histogram entry, and exits nonzero on disagreement:
node research/eleven-moves-from-anywhere/verify-eleven-moves-from-anywhere.mjs.
Device speed is deliberately not turned into a mathematical claim.
Sources and limits
- Pocket-cube count and distance tables. Jerry Bryan, “God's Algorithm for the 2x2x2 Pocket Cube”, Cube-Lovers archive, 4 December 1993. Bryan explains the fixed-corner count, the factor from whole-cube orientations, and publishes both complete histograms. The page and standalone verifier reproduce those results in separate executions.
- Earlier QTM computation. Dan Hoey, “The pocket cube: correction, calculation, and conjectures”, Cube-Lovers archive, 7 August 1984. This is an archival primary report of the exhaustive quarter-turn calculation.
- The larger cube. Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge, “The Diameter of the Rubik's Cube Group Is Twenty”, SIAM Journal on Discrete Mathematics 27(2), 2013, 1082-1105. The paper documents the proof, coset method, verification work, and extrapolated computation cost. It is cited here, not rerun.