The Cubing Corner

The Devils Number - History of the Cube Mystery

Devils Number in the world of Rubiks Cube

The Devils Number – History of the Cube Mystery

“God’s number” is a famous concept in the world of cubing. This number, or 20, is defined as the maximum number of moves needed to solve any configuration of a scrambled 3×3 cube. What about the opposite idea, or the “Devil’s Algorithm” and the “Devil’s Number”? Is it possible to find one algorithm that can solve any starting position? If we were to discover this secret, anyone could solve the Rubik’s Cube by learning just one algorithm.

The “Devil’s Algorithm” is an algorithm that someone can use regardless of the starting position on the cube, and through repetitions of the algorithm will get to the solved state somewhere along the sequence of moves. This is based on the idea that there’s an algorithm that goes through all 43,252,003,274,489,856,000 possible positions of the cube and applying this algorithm to any random scramble must go through the solved state at some point.

The “Devil’s Number” is the number of moves in the shortest  “Devil’s Algorithm”.

Unfortunately, this number remains elusive to the cubing world since finding this number is close to impossible. The algorithm would need to be trillions and trillions of moves long, and impossible to calculate using today’s technology. Finding the “Devil’s Algorithm” is much more difficult than finding the perfect solution to any scramble, or “God’s number” due to the enormous number of moves needed to calculate.

History of The Devils Number

Mathematical analysis of this idea has been thus far on the 2×2 since there are substantially fewer positions. In 2011 cuber Bruce posted his theory for the first “Devil’s Algorithm” on the WCA forum by using a Hamiltonian circuit on the 2×2. The algorithm of 800,000 moves can be repeated up to 5 times, and it runs through every single position on the 2×2, returning to its starting position at the end.

The maximum number of moves needed to solve using this algorithm would be approximately 2.9 million, while the minimum number of moves needed to solve would be over 100 thousand. While theoretically interesting this algorithm is certainly impractical to actually perform.

The 3×3 has an impossibly large number of positions, and using a Hamiltonian circuit would produce as the minimum number of movies 30 quadrillion moves. In 2012 Cuber Bruce published his theory for this sequence. This contains a huge amount of repetition and sub-sub algorithms that form a gargantuan beast.

This is probably why research around the number has stalled. The numbers are too massive for even computers to calculate, even for the 2×2. Additionally, it is difficult to even begin coming up with proof.

This theory confirms that although computers will always be infinitely better than humans at calculating, there are questions that continue to elude computers. Good luck to the next mathematician who decides to take on the challenge!

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