How cubes are solved (LBL, CFOP, two-phase, reduction)

Methods for solving a Rubik's Cube fall into two broad families: the ones people memorise, and the ones computers search. They optimise for different things. Human methods keep the amount to memorise manageable and value moves that flow naturally under the fingers. Computer methods ignore memorability entirely and chase short solutions found quickly. Below are four representative approaches, arranged by what each is trying to achieve.

LBL — building one layer at a time

LBL (Layer By Layer) is the most straightforward method: finish the bottom layer, then the middle, then the top. The order runs cross, first-layer corners, second-layer edges, last-layer orientation, last-layer permutation. Only about ten algorithms are needed, but solutions often run past 100 moves. Because each stage must leave the finished layers intact, the method constantly takes the long way round — breaking a layer and restoring it. That very idea of "break it and put it back" is the most basic tool in cubing.

CFOP — memorise more to move less

CFOP (Cross, F2L, OLL, PLL) is the dominant speedsolving method. It differs from LBL in two ways. First, it stops treating first-layer corners and second-layer edges separately and inserts them as pairs (F2L). Second, it reorganises the last layer into two clean stages: orient everything (OLL), then permute it (PLL). Solutions drop to roughly 50–60 moves, at the cost of learning 57 OLL cases and 21 PLL cases. Memorisation and move count trade off against each other — that principle runs through every human method.

Two-phase — splitting the search in half

Computers most often use the two-phase method (the Kociemba family). A 3x3x3 has about 43 quintillion positions, so searching head-on for the shortest solution is hard. The trick is a subgroup. Positions reachable using only G1 = ⟨U, D, R2, L2, F2, B2⟩ form a far smaller world than the whole cube. The method first searches for moves that drop the current position into that smaller world, then searches within it to finish. Splitting one large search into two small ones yields a short-enough solution in practical time. The price of the split is that the result is near-optimal, not necessarily the true minimum.

Reduction — making a big cube behave like a 3x3x3

A 4x4x4 or 5x5x5 has parts a 3x3x3 does not: several centre pieces per face, and edges made of two or three separate pieces. Reduction turns this to its advantage. First gather the same-coloured centre pieces onto each face, then pair up the loose edge pieces. Once that is done the big cube reads as though each face were a single block, structurally identical to a 3x3x3, and any 3x3x3 method finishes the job. One thing appears along the way that a 3x3x3 never produces: parity, which looks like a single pair sitting swapped. Nothing is broken — it is a property inherent to even-sized cubes.

How cubepic solves

cubepic picks a method per size. The 3x3x3 uses the two-phase method. The 2x2x2 rides on the same two-phase search with only the corners embedded in it. The 4x4x4 and 5x5x5 use reduction, handing off to the two-phase solver once the cube has been reduced. Every size returns a near-optimal solution, with no optimality guarantee. A few famous preset positions (such as the superflip) show a solution proven shortest in HTM (half-turn metric). Searching further to shorten the solution is planned as a paid tier. The target need not be the solved state either: any legal position can be registered. cubepic then solves current-to-solved and target-to-solved, and appends the second one reversed and inverted.

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