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

The belief there are 43,252,003,274,489,856,000 combinations to the Rubik’s cube is false. I know. A bold statement. Stay with me. The proof that a 3x3 cube with 9 movable stickers of 6 different colors on each face does have 43,252,003,274,489,856,000 possible combinations. Note, however, there is a difference between mathematics and mechanics.

There are much, much fewer possible Rubik’s cube combinations for a standard 3x3x3 Rubik’s cube because it is a mechanical, not a mathematical system.

Anyone who grew up in the late 70’s and 80’s and aged between 5 to 142 found it was quicker and simpler to disassemble and reassemble the 20 removable pieces of the 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 and you were left with a black spot on your cube. Or 54 spots. That guy always claimed the fastest solve time - zero seconds.

As all of us then discovered in the disassembled cube, the central points of every face are fixed. In mathematical terms, this means there are no degrees of freedom for these points. In playing with the disassembled cube, the truth that the six center points of the cube are immutable axes is clearly apparent to the player.

I referenced https://rubiks-cube-solver.com/ for the standard color positions for holding an official Rubik’s cube. The solver presents a cube with White on top and Yellow on bottom. This website presents the red face to the right, the green face to the left, and, continuing clockwise, orange and blue. When the cube is stripped to the rotation axes, there is only one arrangement no matter how you rotate. White is opposite Yellow, Green is opposite Blue, and Red is opposite Orange. There are no degrees of freedom for the colors to change orientation. This is how an undergraduate team could make a machine that could solve a cube in milliseconds.

If white is used as a point of reference, at 90 degrees from White (W) there is Red (R). Starting from R, viewing from W, going in a clockwise direction, the sequence is always R, G, O, B. Y is always opposite W. In order for there to be quintillions of combinations, these central axes must be able to be one of the six colors, or only two, three, four, or five. The fact the mechanics does not allow this immediately disproves the so-called ‘proof’ of 43,252,003,274,489,856,000 combinations of the Rubik’s cube.

Disassemble a 3x3x3 cube and you will verify that there are only 20 removable pieces and each one has fixed orientation in order to make a soluble cube. Every center edge piece (2 colors) has only two allowable orientations. Every corner has only 3 allowable orientations (3 colors in fixed orientation). There are 12 center edge pieces and 8 corner pieces.

Corner pieces can be rotated into the three different orientations. Center edge pieces are binary and can only be ‘flipped’ at each of the 12 center edge positions. Each piece, due to the rotation of the axes mechanism, can be viewed as being ‘independent’ of the rest of the pieces.

For center edge pieces, there can only be a maximum of 2 orientations at 12 positions for the 12 pieces. This means the math is very straightforward: 2 * 12 * 12 = 288.

For each corner piece, there can be a maximum of 3 orientations at 8 positions for the 8 pieces.

This means: 3 * 8 * 8 = 192.

These possibilities mean that for a real world mechanical Rubik’s cube, there are a maximum of 192 * 288 = 55,296 combinations. That is still a huge amount for a human mind to comprehend. Actually count that many fans in the stands the next time you are at a large stadium.

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. The axes must always stay in fixed orientation, like Stargate’s concept for interplanetary locator identification (for those who’ve forgotten – 6 constellation glyphs for fixed spatial location of the destination. The seventh is for the starting point.). Bottom line is the cube’s axes are not available for combinations which are absurd and insoluble, such as all blue on the six center faces of the axes, all white corners, and yellow only on center edge pieces.

Also, I believe there is a proof for why is only takes a maximum of 20 moves to solve any cube due to the fact there are only 20 effectively movable pieces. In the end, I ask where did Rubik's rings go?

Todd Wickard

23 August 2025