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Provably optimal · 11 turns maximum

Pyraminx Solver

Enter the colours of your Pyraminx and this solver returns the shortest solution that exists. All 933,120 positions of the puzzle are held in memory as exact distances, so the answer is not merely short — it is minimal, and never more than eleven turns. The four tips are listed separately because they can be twisted at any time.

Optimal solution

7 turns plus 3 tips — 10 total

Body — do these in order

U R' U' R' B' R' L

Tips — any time, any order

u' l' r

Front
Front tip — blueFront edge — yellowFront axial — greenFront edge — blueFront tip — greenFront axial — yellowFront edge — yellowFront axial — greenFront tip — blue
Left
Left tip — greenLeft edge — greenLeft axial — redLeft edge — greenLeft tip — blueLeft axial — blueLeft edge — blueLeft axial — greenLeft tip — red
Right
Right tip — redRight edge — yellowRight axial — blueRight edge — redRight tip — yellowRight axial — blueRight edge — blueRight axial — yellowRight tip — yellow
Bottom
Bottom tip — redBottom edge — greenBottom axial — redBottom edge — redBottom tip — greenBottom axial — yellowBottom edge — redBottom axial — redBottom tip — yellow

Hold the Pyraminx with one vertex up and one face towards you. That top vertex is U, and the three others are L, R and B going clockwise from the bottom left.

A Pyraminx in numbers

Positions

933,120

Excluding the four tips

With tips

75,582,720

× 3⁴

God's number

11

Plus up to four tip twists

Pieces

14

4 tips, 4 axials, 6 edges

The 933,120 comes apart cleanly: 34 = 81 orientations for the four axial pieces, 360 even permutations of the six edges — a turn three-cycles them, and a three-cycle is always even — and 32 flip patterns, because the number of flipped edges is always even. 81 × 360 × 32 = 933,120.

Every one of those was visited by breadth-first search when you loaded this page, which takes under a second. After that, solving is not a search at all: the solver reads the distance from your position and walks downhill one move at a time.

Notation

Hold the Pyraminx with one vertex up and one face towards you. The top vertex is U; L, R and B go clockwise from the bottom left.
WrittenTurnsWhich pieces move
UThe top vertex wedge, clockwiseTip, axial and three edges
U'The same, anticlockwiseTip, axial and three edges
L R BThe other three verticesSame shape, different corner
u l r bThe tip aloneOne piece, nothing else

Solving one by hand

The usual beginner route is three stages. First twist the four tips to match their axial pieces — free, and always possible. Second, orient the four axials so each shows the right colour on each face; this is guessable and takes a few turns. Third, solve the six edges, which is the only part that needs thought.

For the edges, most people learn two three-cycles and a flip algorithm, or use the L4E method — solve two edges, then handle the last four as a single recognised case. A competent Pyraminx solve takes about 25 turns against this solver’s eleven, because a human method never risks disturbing what it has already fixed.

Top competitors average under three seconds. That is not because the puzzle is easy — it is because eleven turns is genuinely all it takes, and the fifteen-second inspection is long enough to plan almost the entire solve.

How the geometry is derived

The move tables here are not transcribed from a diagram. Each turn is applied as a real 120° rotation about the axis through a vertex, to the actual 3D positions of the pieces and the actual face normals — which is what produces both the piece permutation and the edge flips without a hand-written cycle list to get wrong. The same approach builds the 4×4 and 2×2 models.

Common questions

What is the maximum number of moves to solve a Pyraminx?
Eleven turns for the body, plus up to four tip twists that can be done at any time. That figure is exact: all 933,120 positions of a Pyraminx excluding tips have been enumerated, and this solver holds the complete distance table, so it returns the true minimum rather than a short answer.
Why are the tips solved separately?
Because they are independent of everything else. A tip is attached to nothing but itself — twisting it changes no other piece — so it contributes a free factor of 3 to the state count and can be fixed at any point in the solve. Serious Pyraminx solvers do the tips first, during inspection, so the timed part is only the body.
How many positions does a Pyraminx have?
933,120 ignoring the tips, or 75,582,720 counting them. The breakdown: 3^4 = 81 for the four axial pieces, 360 for the even permutations of the six edges, and 32 for their flips. Multiply and you get 933,120 — small enough to breadth-first search in under a second.
What is the difference between an uppercase and lowercase move?
Uppercase U, L, R and B turn the two-layer wedge at that vertex: the tip, the axial piece under it, and the three edges around it. Lowercase u, l, r and b turn only the tip. That is the WCA convention and it is what the solver's output uses.
Why can a Pyraminx be solved optimally when a Rubik's cube cannot?
Size. A Pyraminx has 933,120 positions and a 3x3 has 43,252,003,274,489,856,000 — forty-six trillion times as many. One fits in a megabyte-sized table; the other would need more storage than exists. That is the whole difference between a solver that looks up the answer and one that has to search for it.

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