How many combinations does a Rubik's Cube have?
A standard 3×3×3 cube has exactly 43,252,003,274,489,856,000 reachable positions. The count is smaller than a naive calculation suggests because the mechanism constrains it: you cannot twist a single corner in place, flip a single edge, or swap just two pieces, so only one twelfth of the arrangements you could make by pulling the cube apart are reachable by turning it. Every one of those 43 quintillion positions can be solved in 20 moves or fewer — a result proved in 2010 and known as God's number.
How the figure is arrived at
8 corner pieces can be permuted 8! ways and 7 of them independently twisted (3^7, the eighth being determined); 12 edges permuted 12! ways with 11 independently flipped (2^11); and the total is divided by 2 because corner and edge permutation parity must match. So 8! × 3^7 × 12! × 2^11 ÷ 2 = 43,252,003,274,489,856,000. The divisions by 3, 2 and 2 are the mechanism showing up in the arithmetic.
Other numbers and possibilities counts
- How many ways can you shuffle a deck of cards? about 8 × 10^67
- How many possible chess games are there? about 10^120 games; about 4.8 × 10^44 legal positions
- How many combinations are there in the lottery? 292,201,338 for Powerball
- How many digits of pi have been calculated? about 202 trillion
- How many possible passwords are there? 6.6 quadrillion for eight characters — 540 sextillion for twelve
- How many Sudoku puzzles are there? 6,670,903,752,021,072,936,960 completed grids
- How many prime numbers are there under a million? exactly 78,498
- How many possible positions are there in Go? about 2.08 × 10^170
Common questions
- How many combinations does a Rubik's Cube have?
- 43,252,003,274,489,856,000 — about 43 quintillion
- How is that estimated?
- 8 corner pieces can be permuted 8! ways and 7 of them independently twisted (3^7, the eighth being determined); 12 edges permuted 12! ways with 11 independently flipped (2^11); and the total is divided by 2 because corner and edge permutation parity must match. So 8! × 3^7 × 12! × 2^11 ÷ 2 = 43,252,003,274,489,856,000. The divisions by 3, 2 and 2 are the mechanism showing up in the arithmetic.
Related pages
- How many ways can you shuffle a deck of cards? about 8 × 10^67
- How many possible chess games are there? about 10^120 games; about 4.8 × 10^44 legal positions
- How many combinations are there in the lottery? 292,201,338 for Powerball
- How many digits of pi have been calculated? about 202 trillion
- How many possible passwords are there? 6.6 quadrillion for eight characters — 540 sextillion for twelve
- How many Sudoku puzzles are there? 6,670,903,752,021,072,936,960 completed grids
- How many prime numbers are there under a million? exactly 78,498
- How many possible positions are there in Go? about 2.08 × 10^170
- How many people are there in the world? Living things and the natural world
- How many stars are there in the universe? Space and the universe
- How many countries are there in the world? Things people built
- How many bones are in the human body? The human body
- How many grains of sand are there on Earth? Living things and the natural world
- How many ants are there in the world? Living things and the natural world
Sources
- Calculated on this page — At One Place