2x2 Pocket Cube Solver & God's Number
Scramble a 2x2 cube and get a provably shortest solution — plus why God's number of 11 describes only 0.072% of the puzzle.
God’s number is 11, and it describes 0.072% of the cube
A 2×2 has eight corners and nothing else. Fix the one corner that the U, R and F turns never touch — that removes whole-cube rotations — and the other seven carry the lot: 7! arrangements × 36 twists = 3,674,160 positions. Small enough to walk in full, so here is every one of them, sorted by how far it is from solved:
| Moves from solved | Positions | Share | |
|---|---|---|---|
| 0 | 1 | 0.000% | |
| 1 | 9 | 0.000% | |
| 2 | 54 | 0.001% | |
| 3 | 321 | 0.009% | |
| 4 | 1,847 | 0.050% | |
| 5 | 9,992 | 0.272% | |
| 6 | 50,136 | 1.365% | |
| 7 | 227,536 | 6.193% | |
| 8 | 870,072 | 23.681% | |
| 9 | 1,887,748 | 51.379% | |
| 10 | 623,800 | 16.978% | |
| 11 | 2,644 | 0.072% |
The eleven is real — there are positions no eleven-move sequence can beat. But there are only 2,644 of them, and the shape of that table is the interesting part. Most puzzles pile up at their maximum, because each extra move multiplies the reachable set. This one peaks one short of the end and then collapses: 51.4% of all positions sit at exactly 9 moves, the count falls threefold at 10, and then by a factor of 236.
That is saturation. Each level multiplies by roughly 3.8 while there is somewhere new to go, and then the cube simply runs out of states. The average position is 8.76 moves from solved, and scrambling at random lands you on an eleven-move position about once in 1,390.
And the number changes if you count differently
Everything above counts a 180° turn as one move. Count it as two — which is just as reasonable, and is what a turn metric based on quarter turns does — and the same cube gives a different answer:
| 180° is one move | 180° is two moves | |
|---|---|---|
| God’s number | 11 | 14 |
| Positions needing it | 2,644 | 276 |
| Share of the cube | 0.072% | 0.0075% |
| Average solution | 8.76 moves | 10.67 moves |
| Most common distance | 9 moves | 11 moves |
| Total positions | 3,674,160 | 3,674,160 |
Same cube, same 3,674,160 positions, and the answer is either 11 or 14. So “God’s number for the 2×2” is not a well-defined quantity until you say how you are counting — a caveat that gets dropped surprisingly often.
How the solver here works
It meets in the middle. Every position within five moves of solved is worked out once and kept — that is 12,224 of them, which costs nothing. Then it searches outward from your scramble a level at a time until it hits one, which can never take more than six levels because eleven is the maximum. It keeps going until no shorter answer could exist, so what you get back is genuinely the shortest sequence, not merely a working one.
How to use
- Press scramble, or turn the faces yourself with the move buttons.
- The net shows all six faces unfolded, so nothing is hidden round the back.
- The status line tells you how far from solved you are, and that figure is exact.
- Press solve for the shortest sequence that finishes it.
Frequently asked questions
How many positions does a 2x2 cube have?
Exactly 3,674,160. Fixing one corner removes whole-cube rotations, leaving seven corners that can be arranged in 5,040 ways and twisted in 729, and 5,040 times 729 is the answer. That is small enough to search in full, which is why every figure on this page was computed rather than looked up.
What is God's number for the 2x2?
Eleven, if a 180 degree turn counts as one move. Fourteen if it counts as two. Both are correct, which is why the metric has to be stated — the phrase on its own does not pin down a number.
How likely is a scramble to actually need 11 moves?
About one in 1,390. Only 2,644 of the 3,674,160 positions need the full eleven, which is 0.072 per cent of the cube. You could scramble one every day for years without meeting one.
How far is a typical 2x2 scramble from solved?
Nine moves, more often than any other answer — 51.4 per cent of all positions are exactly nine moves out. The average across the whole cube is 8.76 moves.
Why does the distribution fall at the end instead of rising?
Saturation. Each extra move multiplies the reachable set by roughly 3.8 while there are new positions to reach, so the counts climb steeply — and then the cube runs out. After the peak at nine the count drops threefold, then by a factor of 236. Most search spaces pile up at their maximum; this one does the opposite.
Is the solution it gives really the shortest?
Yes. It works out every position within five moves of solved in advance, then searches outward from your scramble until it meets that set, and carries on until no shorter answer could exist. It is checked against an exhaustive walk of all 3,674,160 positions, so every answer matches the proven optimum.
Why is one corner never turned?
Because turning only U, R and F reaches every position anyway. The back-left-bottom corner stays put and acts as a fixed frame of reference, which is what stops the same cube being counted twenty-four times over in different orientations.
Does this send anything anywhere?
No. The scramble and the search both run entirely in your browser.
🔒 This tool runs entirely in your browser. Nothing you enter is uploaded, logged, or stored.