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Provably optimal · 11 moves maximum

2x2 Rubik Solver

This 2×2 solver returns the shortest solution that exists, not just a short one. A 2×2 has 3,674,160 positions once you fix one corner — few enough to store the exact distance from every one of them to solved — so the answer here is provably minimal, and never longer than eleven moves. Only U, R and F are ever used, so you never have to re-grip.

Optimal solution

8 moves — the minimum possible

Only U, R and F are ever used, so you never have to put the puzzle down and re-grip. The back-down-left corner stays exactly where it is and everything is solved around it.

Next: Right face anticlockwise

Pick a colour, then click the squares

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A 2x2 in numbers

Positions

3,674,160

7! × 3⁶, with one corner fixed

God's number

11

Half-turn metric

Table size

3.6 MB

One byte per position

Search time

under 1 ms

It is a table lookup, not a search

The whole puzzle fits in memory, so the solver does not search at all. It builds the table once by breadth-first search from the solved state, then walks downhill: from your position it picks any move that lands on a position one step closer, and repeats. That is why the answer is instant and why it cannot be beaten.

For comparison, the same approach on a 3×3 would need a table of 43 quintillion bytes. At current prices that is roughly forty billion dollars of storage.

How far is a random 2x2 from solved?

Half-turn metric, one corner fixed. Half of all 2x2 positions are exactly nine moves from solved; the 11-move cases are vanishingly rare.
Moves from solvedPositionsShare
010.00003%
190.0002%
2540.001%
33210.009%
41,8470.05%
59,9920.27%
650,1361.4%
7227,5366.2%
8870,07223.7%
91,887,74851.4%
10623,80017.0%
112,6440.07%

Note the shape: 2,644 positions out of 3.67 million need the full eleven moves. If your solve came back at nine, that is the single most common answer there is.

Solving a 2x2 by hand

The beginner’s route is to treat it as a 3×3 with no edges and no centres. Build one full face with the layer below it correct, then orient the opposite face with R U R' U R U2 R' — the same Sune from a 3×3 — and finish by permuting the last layer with R U R' U' R' F R2 U' R' U' R U R' F', the T-perm. That is two algorithms you very likely already know.

The catch is that the first layer is harder on a 2×2 than on a 3×3, because there are no centres to tell you what belongs where. You have to work out the colour scheme from the corners themselves. Every algorithm on the algorithms page that touches only corners works unchanged on a 2×2.

Why the answer never needs L, D or B

Turning the back-down-left corner is the same as turning the other three layers the other way and then rotating the whole puzzle. Since a 2×2 has no centres, that rotation is invisible — so half the moves are redundant. Fixing that corner removes the redundancy and leaves a puzzle small enough to solve exactly.

Common questions

What is the maximum number of moves to solve a 2x2?
Eleven, in half-turn metric, where a 180° turn counts as one move. Fourteen if you count quarter turns only. Both numbers are exact, because all 3,674,160 positions of a 2x2 have been enumerated — this solver holds that entire table in memory, which is why its answers are optimal rather than merely short.
Why does this solver only use U, R and F?
A 2x2 has no centres, so nothing is fixed. Pick one corner — this solver picks the back-down-left one — and treat it as the frame of reference. Every other position can then be reached with just U, R and F, so you never have to put the puzzle down and re-grip. It also cuts the search space by a factor of 24.
How many positions does a 2x2 have?
3,674,160 once you fix one corner, or 88,179,840 if you count the 24 orientations of the whole puzzle as different. Compare that with the 3x3's 43 quintillion. The gap is why a 2x2 can be solved perfectly and a 3x3 cannot, at least not in a browser tab.
What is the Ortega method?
A three-step 2x2 method: solve one face without caring about the layer below, orient the opposite face, then permute both layers at once. It averages about 20 moves against the beginner method's 30, needs 12 algorithms, and is what most sub-5-second cubers use. This solver ignores methods entirely and searches for the shortest sequence directly.
My 2x2 has a corner twisted and will not solve. Is it broken?
Almost certainly it was popped and put back wrong. On a real 2x2 the eight corner twists always add up to a multiple of three; if yours does not, the solver says so and tells you to turn one corner in its socket. That is a physical fix, not a puzzle one.

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