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twistspace

the order of a cube algorithm, from its cycle structure

based on 1981, Singmaster, Notes on Rubik's Magic Cube

move0 of 4

sexy move: the commutator of R and U, order 6; the most repeated four moves in the puzzle

Solved: every sticker is home.

manual turnsappended to the algorithm, not applied to the cube

net

cycle decompositionhover to highlight on the cube

Order is the least common multiple of the cycle lengths: 6 = 2 . 3. A cycle of length L returns after L applications; the permutation returns when all of them do simultaneously.

displacement under repetitionstickers displaced after k repetitions

Dashed line: k = 6, the order. Displacement is zero there and nonzero at every smaller k above zero, which is what distinguishes the order from a multiple of it.

The permutation

A cube algorithm is a sequence of face turns. Composed, it is a bijection of the stickers onto themselves: an element of a permutation group. The order, the cycle structure and the parity are properties of that element and are computable from it without reference to the puzzle.

Two algorithms written differently can be the same element. Two differing by one move usually are not related.

Order

Every permutation of a finite set decomposes uniquely into disjoint cycles. A sticker in a cycle of length $L$ returns to its position after exactly $L$ applications and not fewer. The cube is solved when all cycles return simultaneously, at the least common multiple of the lengths.

\mathrm{ord}(A) = \mathrm{lcm}(L_1, L_2, \ldots, L_m)

The group is finite and every element invertible, so no algorithm can be repeated indefinitely without returning: $A^i = A^j$ with $i < j$ forces $A^{j-i}$ to be the identity. R U R' U' has order 6; R U has order 105, its cycles being 15, 7, 7 and 3.

The maximum element order on a three by three is 1260.

The size of the group

The reachable positions are counted by placements and orientations, less what the mechanism forbids: eight corners in any order, seven freely twistable and the eighth determined; twelve edges in any order, eleven freely flippable and the twelfth determined; and a halving, since corner and edge permutation parities are constrained to agree.

|G| = \frac{8! \cdot 3^{7} \cdot 12! \cdot 2^{11}}{2} = 43{,}252{,}003{,}274{,}489{,}856{,}000

The product is derived rather than stored, and kept factored as 2^27 . 3^14 . 5^3 . 7^2 . 11. Lagrange's theorem requires the order of every element to divide 43,252,003,274,489,856,000; with both held as prime factorisations, that test is a comparison of exponents.

Solved against looking solved

Both are reported, because they differ. A whole-cube rotation moves all fifty-four stickers and so has order four, while leaving the cube solved throughout. On an even cube the four centre stickers of a face are interchangeable in the hand and distinct in the model, which separates the two counts again for an unrelated reason.

The group-theoretic order is an upper bound on the number of repetitions after which the cube appears solved.

Limits

Only the order is computed, never a solution. The two are not comparably hard: the order follows from the cycle lengths in linear time, whereas establishing that twenty face turns always suffice took three decades and substantial computing time. Nothing here is a claim about distance from solved. The camera mode names what changed between two readings by trying every sequence of at most six turns, which settles positions within six turns of each other and cannot touch a scrambled cube.

The group order and the Lagrange check apply to the three by three only; larger cubes have different groups and the counting argument above does not describe them. Centres are unmarked, so a cube with a marked centre has more distinguishable states than this accounts for, by a factor of $4^6/2$ . Notation is not fully standardised, so the parser writes back its reading of the input. Each of these is listed, with what would falsify it, in the assumptions panel.

Reading a cube

The camera tab puts a physical three by three on screen from six guided captures. Colour is compared rather than measured: the six centres cannot move relative to each other, so whatever they look like under the current light are taken to be the six face colours, and the remaining facelets are assigned relative to those under the constraint that exactly nine belong to each. Nothing leaves the device and no colour constant is stored.

A colouring becomes a position through pieces rather than stickers. A cubie showing three particular colours can only be the one corner that has them, and each of its stickers is then pinned by its own colour. That is exact where a piece is named by its colours, which is why the reading is claimed for the three by three and refused above it.

Three quantities are conserved by every turn, so a misread facelet usually breaks one. In 360 readings taken under light bad enough that most of them failed, no wrong position was accepted.

Model

v1.1August 2026

a camera can put a real cube on screen. Six guided captures, and the position is recovered exactly, with nothing sent anywhere: the classification, the reconstruction and the checks are arithmetic that runs in the page.
colour is compared rather than measured. The six centres of a three by three cannot move relative to each other, so whatever they look like under the current light are taken to be the six face colours, and the other forty-eight facelets are assigned relative to those under the constraint that exactly nine belong to each. Nothing anywhere holds an idea of what red is. A common change of exposure is cancelled exactly and a colour cast is absorbed, both checked rather than claimed.
the assignment is solved rather than approximated. Nearest colour facelet by facelet has no way to know a face has nine stickers and will happily return ten of one and eight of another, so the count is imposed and the whole thing becomes a minimum cost assignment. Being sure about eight facelets then settles the ninth.
a colouring is turned into a position through pieces, not stickers. A cubie showing white, red and blue can only be the one white-red-blue corner, and each of its stickers is then pinned by its own colour. This is exact on a three by three and impossible on a four by four, where a face has four indistinguishable centres, and the reconstruction says which case it is in rather than guessing.
the piece grouping is derived from the slot convention rather than tabulated, so the two cannot disagree, and it is checked against the turn engine: every layer turn of a three by three and a four by four carries every piece to a piece.
three quantities conserved by every turn are checked before a reading is accepted. They reject one reassembly in twelve, which is the standard count, and in three hundred and sixty readings taken under light bad enough that most of them fail, not one wrong position was accepted.
what changed between two readings is named by exhaustive search from both ends, bounded at six turns. This is a search and not a method: it declines the superflip rather than answering, because that position is twenty turns away. The turns it does name are written onto the algorithm, so the cycle rings and the return length go on describing a cube that is now a real one.
the position on screen no longer has to start solved. A scanned cube goes in as the base that the algorithm is read from, and nowhere else: the order of an algorithm is a property of the permutation and is the same element of the group whatever position it acts on, so the analysis is untouched.

v1.0August 2026

turns are index permutations of the sticker array rather than rotations of a geometric model. The rendering still needs geometry, but nothing the mathematics claims depends on it, and the whole model runs without a browser.
the order of an algorithm is carried as a prime factorisation instead of an integer. That is partly because the order of an element on a seven by seven runs past what a double holds exactly, and partly because 2^2 . 3^2 . 5 . 7 explains 1260 in a way that 1260 does not.
Lagrange's theorem becomes a comparison of exponents rather than a long division, and is claimed only for the three by three, whose group order is the one being derived.
parity is measured rather than assumed. A single quarter turn is an odd permutation of the stickers on a three by three and an even one on a four by four, so what parity tells you about an algorithm depends on the cube, and the playground asks instead of guessing.
the first return is computed twice by different routes: once as the least common multiple of the cycle lengths, and once by turning the cube until it comes back. The calibration checks that they meet at Singmaster's figure.
the difference between a cube that is solved and a cube that looks solved is measured rather than mentioned. A whole-cube rotation has order four and a visual return of one; on an even cube the interchangeable centre stickers separate the two numbers again.
the notation parser reduces every form to one shape. A wide turn, a slice and a whole-cube rotation differ only in which layers they take, so `Rw`, `M` and `x` are the same operation with different depth sets rather than three special cases.
the commutator and conjugate brackets expand before anything else sees them, so [A,B] is analysed as the eight moves it is rather than as a notation.
turning the cube by hand writes onto the algorithm rather than onto the cube. There is no second place for state to live, so the cycle rings, the return length and the orbit describe what is on screen whether it was typed or pressed, and cannot drift apart from it.

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