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Since its introduction to the wider public in 1980, the Rubik’s Cube has fascinated mathematicians worldwide. The deceptively complex puzzle holds many unanswered questions. There are constantly new finger tricks, new algorithms being developed, and new challenges coming up. One fascinating question that took 30 years for mathematicians to calculate was God’s Number.
God’s Number is the highest number of moves needed to solve any of the 43,252,003,274,489,856,000 possible combinations of the cube. The number was calculated by Google Supercomputers and a team of mathematicians in 2010. The term was coined because in order to actually solve the cube in only 20 moves you would need to possess a mind with a thousand times the capacity of the human brain, that could test millions of combinations within seconds- an almost godlike ability.
While this number may appear impossibly low, in many cases it is even lower! Only a fraction (0.0000011328955%) of these- 490,000,000 necessitate all 20 moves to be solved. 1.5 quintillion combinations can be solved with 19 moves. On average, God’s Number is closer to 19 than 20. However, since some configurations require 20 moves, the greatest common denominator is 20.
The most well-known scramble that takes 20 moves to solve is the super-flip position. To get to the scramble, follow the following moves R L U2 F U’ D F2 R2 B2 L U2 F’ B’ U R2 D F2 U R2 U from any orientation on a solved cube. You can instantly recognize this position- all of the corners are solved in their place, and all edges are flipped in their place. This was the first position that needed 20 moves to solve, raising the lowest number of moves to 20.
History of God’s Number
In 1981 Morwen Thistlewaite began pursuing God’s Number when he proved, by inventing a complex algorithm that you can solve any of the 43 quintillion different scrambles in a maximum of 52 moves. As more efficient methods were created, this number was lowered. In the ’90s this number went down to 29, plateauing in the 2000’s. In the 2000’s research started again using Google supercomputers and the number again was lowered to 20.
In order to prove this theory, not all of the combinations were actually tested individually. Patterns were identified that allowed mathematicians to reduce the permutations tested. To illustrate, if you were to execute the super-flip algorithm on a Rubik’s Cube then flip it, you would have theoretically created another of the 43 quintillion combinations without increasing the number of moves needed to solve it. This is because 43 quintillion is the number of positions, not unique patterns. Rotating the puzzle adds new positions, therefore there are 24 ways to position the cube for all scrambles. This means that out of the 43 quintillion possible positions you need to test only 4%, or 1,802,166,800,000,000,000. This number is further brought down by factoring in mirrors and other pattern interrelations.
Each twisty puzzle has its own unique God’s Number, which can also be calculated. For example the 2x2x2 has 3,674,160 different positions. The 2x2x2 God’s Number has been calculated to be 11 or 14 moves depending on whether you use a half-turn metric or a quarter-turn metric.
Although fascinated by the Rubik’s Cube, mathematicians have not yet discovered God’s Number for the 4x4x4 and up. Looking at the exponential development of computing powers today, we do expect this number to pop up soon though! There is always more to discover about the cube- read more about the opposite of “God’s Number”, or the Devil’s number.