43,252,003,274,489,856,000 Rubik's cube combos? Are you sure?

The Toostie Roll owl demonstrated it only takes three licks to get to the center of a Tootsie Pop. Math claims there are 43,252,003,274,489,856,000 combinations to the Rubik’s cube. This is wrong. Stay with me. There is a difference between mathematics and mechanics. And a cartoon owl eating candy.

Before the internet, most everyone aged between 5 to 142 found it was quicker and simpler to disassemble and reassemble the 20 removable pieces of the original Rubik’s cube into the solved position. This is because the removal and replacement of the stickers meant that at least one would fall off, become lost, and you were left with a black spot on your cube. Or 54 black spots. That kid with the sticker-less cube always claimed the fastest solve time - zero seconds.

Disassemble a 3x3x3 original cube and you will verify that there are only 20 removable pieces (12 center edge and 8 corner pieces). Each one has fixed orientation in order to make a solvable cube.

For center edge pieces, there can only be a maximum of 2 orientations at 12 positions for the 12 pieces. Once a location and orientation is chosen for any edge piece, there is a reduction in degrees of freedom. That one chosen location and orientation (there are two possibilities) no longer participates in the math, which eliminates factorials as an option to estimate the number of combinations. Factorials assume no limit to degrees of freedom.

This means, when starting out, there are two choices for reassembling the first generation edge piece. Once a piece, any piece, is put in a location with orientation, each successive choice is binary in option and the available options double in number every generation. This is the rabbit population doubling every generation problem. With 12 edge pieces, the binary orientation means every real, solvable cube has 2^12, or 4,096 combinations of edge pieces per edge location. This gives total of 49,152 real combinations of the 12 locations.

For each corner piece, there can be a maximum of 3 orientations at 8 positions for the 8 pieces. Starting at the beginning, or all eight pieces in your hand as you reassemble the 3x3x3 cube, the first generation has 3 choice branches at a given location. Each succeeding generation triples in choices until the last piece since there is a loss in degrees of freedom once a corner piece is in a position.

This means 3^8, or 6,561 non-sticker removing, real combinations for corner pieces for each corner. This gives a total of 52,488 combinations for the 8 locations.

These possibilities mean that for a real world mechanical Rubik’s cube, there are a maximum of 49,152 * 52,488 = 2,579,890,176 combinations. That is still a huge amount for a human mind to comprehend. Plenty to keep you busy for the rest of your life. Though, those Purdue undergrads (https://engineering.purdue.edu/ECE/News/2025/purdue-ece-students-shatter-guinness-world-record-for-fastest-puzzle-cube-solving-robot) could conceivably solve every one of these combinations within 82 years (assuming 0.103 seconds per solve and 0.103 seconds for the next unique scramble, no removing stickers, no Murphy).

While it is true a 3x3x3 cube with no restrictions on degrees of freedom for the individual stickers can have a ridiculous number of combinations, a mechanical system has limited degrees of freedom. But even in this calculation, you could have an unsolvable cube where one edge piece is twisted.

The Rubik's cube is a dependent system. Since the system must be able to be returned to its solved state, one cannot have a corner piece in its correct location, but twisted one or two colors wrong. The same goes for an edge piece. Or a twisted edge and twisted corner where the rest of the pieces are in their solved state.

So what is the maximum number of real, solvable combinations of the Rubik's cube?

Todd Wickard

23 August 2025