How many moves does the hardest position need?

The answers

Two things stand out. First, the hardest positions are rare: only 2,644 of the 3,674,160 positions of a 2×2 need eleven moves. Second, the typical position is close to the worst. Half of all 2×2 positions need exactly nine moves, and the average is almost nine. A random scramble is very unlikely to have a short solution.

The 3×3 Rubik's Cube is not in this table. Its 43 quintillion positions are far too many to list on one computer, and God's number for it, 20 with half turns counted as one move, was proved in 2010 using large-scale computing and symmetry arguments (see cube20.org). This site's 3×3 solver finds short solutions but does not promise the shortest.

How it is computed: layer by layer from solved

The method is a breadth-first search, and it is easier than it sounds. Start with the solved puzzle: that is the only position at distance 0. Apply every possible move to it; each new position you reach is at distance 1. Apply every move to each distance-1 position; anything not seen before is at distance 2. Continue until a round produces no new positions.

A position is recorded the first time it is reached and never again. Because the rounds go outward one move at a time, the first time is always by a shortest route. When the search stops, every reachable position has been found exactly once with its true distance, and the last non-empty round is God's number. Nothing is estimated or sampled.

The cost is memory: the program must remember every position it has seen. The 2×2 census holds 3,674,160 positions at once. That fits easily in a laptop, which is why these puzzles can be settled completely while the 3×3 cannot.

The same search also checks the model. If a move were defined wrongly, the search would reach positions the real puzzle cannot, and the total would come out too large. Each total on this page matches the count derived independently from the puzzle's mechanism (below) and the figures published by Jaap Scherphuis.

What counts as a move changes the answer

God's number is not a property of the puzzle alone. It depends on which moves you allow and what each one costs.

Half turns. If a half turn of a 2×2 face counts as one move, God's number is 11. If only quarter turns count, so a half turn costs two, it rises to 14, and the average rises from 8.76 to 10.67. Both are correct answers to different questions. Puzzle literature calls these the half-turn (or face-turn) and quarter-turn metrics, and a solver should say which one it uses. This site's 2×2 solver counts half turns as one move.

Turning the whole puzzle. Holding a 2×2 differently does not solve anything, so positions that differ only by how the cube is held are treated as the same. The search does this by never moving the bottom-back-left corner: it only turns the top, right and front layers, which can still reach every arrangement of the other seven corners. Without that rule you would count each position 24 times, once per way of holding the cube, and get 88,179,840.

Pieces you cannot see. The centre pieces of a Pyramorphix carry one triangular sticker each, so turning one in place changes nothing visible. The engine tracks them anyway, so it finds three solved states that look identical: the whole puzzle rotated about the held centre. Each visible position appears three times in the raw search, which reaches 408,240 states. Dividing by three gives the 136,080 positions you can actually tell apart.

Trivial pieces. The four Pyraminx tips can each be twisted on their own, so each tip can be put right with at most one move at any time. The table leaves them out, as is customary, and multiplies by 3⁴ = 81 at the end if you want the full count: 75,582,720 positions. The exact God's number with tips included is not computed here.

Where the totals come from

The search totals can be predicted from how the pieces move, which is a good check that the model is right.

2×2. One corner is held still, leaving seven to arrange in 7! = 5,040 ways. Each can be twisted three ways, but legal turns always keep the sum of all twists a multiple of three. Six corners can therefore be twisted freely and the seventh is forced: 3⁶ = 729. 5,040 × 729 = 3,674,160. A single twisted corner is the classic sign that a cube has been taken apart and rebuilt wrongly.

Pyraminx, tips aside. The six edges can be arranged in only half of their 720 orders, because every layer turn rearranges edges in an even permutation: 360. Each edge can sit two ways round, but flips come in pairs, so five edges choose freely and the sixth is forced: 2⁵ = 32. The four centre pieces each turn three ways independently: 3⁴ = 81. 360 × 32 × 81 = 933,120.

These conservation rules are exactly what the solvers check when they reject an entry. A colouring that breaks one, like a single flipped edge, cannot be reached by any sequence of turns. Our article on positions with correct colour counts that still cannot be solved shows examples on the 3×3.

Full distance tables

Each row gives how many positions need exactly that many moves at minimum, and that number's share of all positions.

2×2 cube: half turn counts as one move

Moves: U, R, F and their inverses and half turns (9 moves); the bottom-back-left corner is held still.

2×2 cube: only quarter turns count as one move

Moves: U, R, F and their inverses (6 moves); a half turn costs two.

Pyraminx (tips ignored): 120° layer turn counts as one move

Moves: The four main layers, each clockwise or counter-clockwise (8 moves).

Skewb: 120° corner twist counts as one move

Moves: Four corner axes, each clockwise or counter-clockwise (8 moves).

Pyramorphix: half turn counts as one move

Moves: U, R, F and their inverses and half turns (9 moves); the bottom-back-left centre is held still.

The early rows show how quickly the count grows. From solved, a 2×2 has 9 positions one move away but only 54 two moves away, not 81, because some pairs of moves cancel or combine into one. Repeating the same face, as in R then R′, is never part of a shortest solution.

Sources and how to check this page

All counts were computed by this site's own puzzle engines on 1 November 2026 and agree exactly with the tables Jaap Scherphuis has published for the 2×2, Pyraminx, Skewb and Pyramorphix. These are not new results; they are an independent reproduction. Its value here is that the solvers you use are the same programs that produced these tables. That is why the 2×2, Skewb and Pyramorphix solvers can promise a shortest solution. The Pyraminx solver promises the shortest route for the main layers and then adds any tip twists separately.

The search is rerun as an automated test, which fails if any count changes. To see the method behind the solvers and their limits, read how this site verifies solutions. To learn a method you can use by hand, try the 2×2 beginner method.