How Big Is a Rubik's Cube? Absurdly

There is a particular kind of vertigo that comes from doing mathematics correctly and arriving somewhere your intuition refuses to follow. Richard Elwes, a mathematician at the University of Leeds, induces this feeling efficiently in a recent Numberphile video—using, of all things, a toy.

The standard 3x3 Rubik's Cube has 43 quintillion possible states. That figure is familiar enough that it has almost lost its strangeness, which is itself a small cognitive failure worth noting. Forty-three quintillion is 4.3 × 10¹⁹. Written out: 43,000,000,000,000,000,000. Elwes does not dwell on the 3x3. He picks up a 6x6 cube and, methodically, builds the argument from the ground up—starting with a 2x2, constructing the machinery of permutations piece by piece, then letting the numbers run.

The numbers run very far.

The Machinery

The elegance of Elwes's approach is that it never asks you to simply accept a large figure. Every number is earned.

For a 2x2 cube, the counting logic is transparent: eight small cubical pieces, eight positions. The first piece can go anywhere—eight choices. The second, seven. And so on, until you've multiplied 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1, which mathematicians write as 8! (eight factorial) and which equals 40,320. But position alone doesn't determine state. Each corner piece can sit in its slot in one of three rotational orientations. Seven of those orientations are free choices; the eighth is determined once the others are fixed—a constraint baked into the mechanics of the puzzle. So the count multiplies by 3⁷.

There's one more correction. A cube sitting on a table can be physically rotated 24 different ways without any actual scrambling occurring—six faces to put on top, four ways to orient from there. Those 24 orientations are all the "same" state for counting purposes. Divide by 24.

The result: roughly 3.6 million distinct states for a cube that fits in a child's palm.

This is the scaffolding Elwes calls C—the corner arrangement count—and it recurs unchanged in every larger cube, because corners are always corners. What changes as the cube grows is what happens to the edges and centers.

The Corridors

Here is where the combinatorics becomes genuinely interesting rather than merely mechanical.

A 4x4 cube introduces two new categories of piece beyond corners: edges (24 of them, two along each of the cube's 12 edges) and centers (also 24, four per face). The edge count is simply 24!—a number Elwes notes is "around 6 × 10²³, so that's 620 sextillion." The centers require dividing out the redundancy of identical pieces on each face: four indistinguishable centers per face, six faces, so divide by 4! six times over, yielding roughly 3 × 10¹⁵, or 3 quadrillion.

Multiply corners × edges × centers, divide by 24 for whole-cube rotations: the 4x4 clocks in at approximately 7 × 10⁴⁵. The 3x3's 43 quintillion now looks quaint—4.3 × 10¹⁹ versus 7 × 10⁴⁵ is not a doubling or a tripling. It is a leap of 26 orders of magnitude.

The 6x6 is where the piece-type structure becomes notably richer. Edges on a 6x6 come in two flavors: "central" edges that run along the middle of each cube edge, and "wing" edges flanking them. As Elwes explains it, pieces are constrained to their "brand of corridor"—a central edge can reach any other central edge position, and a wing edge can reach any wing edge position, but the two populations never mix. That gives two independent edge arrangements, each equivalent to the 4x4 edge count E. The centers on a 6x6 subdivide into four distinct orbit types; each type contributes another factor of K, the center arrangement count from the 4x4.

Work through the multiplication. The 6x6 comes out at approximately 1.6 × 10¹¹⁶. There are estimated to be around 10⁸⁰ atoms in the observable universe. The 6x6 surpassed that benchmark by 36 orders of magnitude. Elwes notes this with the particular flatness of someone who has made peace with the fact that language is not going to help: "There's more possible scrambles of this puzzle than there are atoms in the universe."

The General Formula and Where It Goes

The machinery Elwes builds is scalable. For any even-sided cube—call it a 2n cube, where n is half the side length—the number of possible states follows a general formula: C × E^(n−1) × K^((n−1)²) ÷ 24.

The exponent on K is (n−1)². That squared term is the structural reason the numbers grow so ferociously. This isn't exponential growth in the ordinary sense. The exponent itself grows as a quadratic function of the cube's size. Double the cube's side length and the exponent on K roughly quadruples.

For a 10x10 cube—which Elwes mentions as a "recent and rash purchase" he hasn't yet dared to fully scramble—n equals 5. Plugging in: C × E⁴ × K¹⁶ ÷ 24. The result is approximately 10³⁴⁹.

To put 10³⁴⁹ in context, Elwes offers an analogy that is either clarifying or dizzying depending on your tolerance: "If you took every atom in the universe and replaced it with a whole universe, and then repeated that four times, and then counted the number of atoms in that whole thing, this has still got more scrambles than that."

That cascade of nested universes gets you to roughly 10^(80 × 4) = 10³²⁰. Still not there. Still 10²⁹ short.

What's Actually Being Counted

It is worth pausing on what these numbers represent, because "possible states" is doing conceptual work that's easy to slide past.

These are not random samples from a probability distribution. They are not outcomes of a process. They are a precise, exhaustive count of distinct configurations—every arrangement of every piece in every valid orientation, with duplicates carefully removed. The counting methodology matters here, and Elwes is deliberate about it: he identifies the overcounting problems (center pieces that look identical when swapped, whole-cube rotations that represent no real change) and corrects for each one explicitly.

This is combinatorics as taxonomy—cataloguing the possible before any particular sequence of moves takes you there. Whether a given scramble is reachable from the solved state through legal moves is a separate question (for the standard 3x3, exactly half of all naively-countable arrangements are unreachable, which is why the true count divides by 2—a constraint Elwes handles for the smaller cubes via the factorial machinery). The number of states a puzzle can occupy tells you something fundamental about its structure without telling you anything about strategy.

The video does not address odd-sided cubes—the 3x3, 5x5, and so on—noting only that the calculation "gets a bit more technical and mathematical." The odd-cube calculation involves additional parity constraints that require separate treatment; Elwes covers this in an extended version on Numberphile 2. That the even-cube formula is already this involved should calibrate expectations for the odd-cube case.

The 10x10 sits on Elwes's desk, unscrambled, containing 10³⁴⁹ possible states. Most of those states no human will ever visit. The question of whether "number of states" constitutes a meaningful property of a physical object—or whether it's a feature of our mathematical description of that object—is the kind of question combinatorics raises and then quietly declines to answer.

Priya Sharma is a science and health correspondent for Buzzrag.