THE NUMBER
Cubing a number
Multiply a number by itself three times: 3³ = 3 × 3 × 3 = 27, the blocks in a three-by-three-by-three stack.
Try the cube builder- Multiply three times
- Measure volume
- Count positions
- 20 moves is enough
00NUMBERS IN THREE DIMENSIONS
Three times over.
Cubed, explained.
From a single block to 43 quintillion puzzle positions.
Cubing a number means multiplying it by itself three times. It is how we measure volume, why a litre is a box 10 cm on each side, and the reason a palm-sized puzzle cube can be scrambled into about 43 quintillion different positions. Here is the maths, in plain English.
Plain English, with the sources linked.
Build a cube01Principles
02THE SHORT ANSWER
Small sides, huge totals.
Cubes grow fast. That is true of blocks,
of boxes and of puzzle positions.
THE PUZZLE
43 quintillion positions
The 3×3×3 puzzle cube can reach 43,252,003,274,489,856,000 different positions by turning its faces.
Where that comes from
THE PROOF
God’s number is 20
In 2010 a team showed that every position can be solved in 20 moves or fewer, and some need all 20.
How it was proved03A QUICK CUBE BUILDER
How many blocks
make the cube?
Pick the length of one side, in blocks. The builder shows how many identical blocks you need to make a solid cube that size.
The bar compares each cube with the 1,000 blocks of a 10-sided cube. Notice how small the first few are: cubes grow very quickly.
04TWO RESULTS WORTH KNOWING
Big numbers,
neat answers.
One took decades of computer time.
The other began with a taxi’s number.
20 moves
Moves needed for the hardest position
Tomas Rokicki, Herbert Kociemba, Morley Davidson and John Dethridge split the 43 quintillion positions into 55,882,296 groups and solved them all, using about 35 CPU-years of idle computer time donated by Google. The proof was published in 2013.
THE PUZZLE CUBE · 2010
1³+12³ = 9³+10³
Two cubes, two different ways
1729 is the smallest number that is the sum of two positive cubes in two different ways. It is called the Hardy-Ramanujan number, after a famous conversation between the two mathematicians about a taxi’s number.
CUBE NUMBERS · 1729
Both are about cubes, but in different senses: one counts the arrangements of a physical cube, the other is pure arithmetic with cube numbers.
05TRY IT YOURSELF
Four ways to
think in cubes.
Square it, then once more
To cube in your head, square first and multiply by the number again: 4² = 16, and 16 × 4 = 64.
Double the side, times eight
Double every edge of a box and its volume grows eight times, because 2³ = 8. Triple the edges and it grows 27 times.
Count cubic centimetres as litres
A box 20 cm by 10 cm by 10 cm holds 2,000 cm³. Divide by 1,000 and you have its volume in litres: 2 litres.
Learn the last digits
The cubes of 0 to 9 all end in different digits. 8³ = 512 ends in 2 and 2³ = 8 ends in 8, so a cube’s last digit points back to its root’s.
06CUBE FACTS
Five facts,
cubed.
The first cubes
The cube numbers start 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1,000. Each one is a whole number multiplied by itself three times.
A litre is a cube
The litre is a special name for the cubic decimetre: a cube 0.1 m (10 cm) on each side. A thousand of them fill one cubic metre, because 10³ = 1,000.
43 quintillion positions
Turning the faces of a 3×3×3 puzzle cube can produce 43,252,003,274,489,856,000 different positions, about 4.3 × 10¹⁹.
Twenty moves always suffice
In July 2010 a team proved that God’s number, the most moves the hardest position needs, is 20. As recently as March 2008, the best proven limit was 25.
The superflip
The first position shown to need 20 moves is the superflip: every piece in its home spot, with all the edges flipped. Michael Reid showed this in 1995.
07QUICK QUESTIONS
Cubes, answered.
Short answers to the questions people ask most.
What does it mean to cube a number?
Cubing a number means multiplying it by itself three times. Three cubed, written 3³, is 3 × 3 × 3 = 27. The name comes from geometry: 27 small blocks fit together into a cube three blocks long, wide and tall.
How many positions does a 3×3×3 puzzle cube have?
There are 43,252,003,274,489,856,000 positions that can be reached by turning the faces, about 43 quintillion. That figure counts only positions you can reach by turning, not ones made by taking the cube apart.
What is God’s number?
God’s number is the largest number of moves needed to solve the puzzle cube from its hardest position, if you always use the shortest solution. In July 2010 a team proved it is 20, counting any turn of a face as one move.
Why is 1729 a famous cube number fact?
1729 is the smallest number that can be written as the sum of two positive cubes in two different ways: 1³ + 12³ and 9³ + 10³. It is known as the Hardy-Ramanujan number.
Is there a quick way to add up cube numbers?
Yes. The first n cubes always add up to the square of 1 + 2 + … + n. For example, 1³ + 2³ + … + 10³ = 3,025, which is 55², and 55 is 1 + 2 + … + 10.
08Sources
Where the facts come from
- God’s Number is 20cube20.org (Rokicki, Kociemba, Davidson, Dethridge)
- The Diameter of the Rubik’s Cube Group Is TwentySIAM Journal on Discrete Mathematics, 2013
- Twenty-Five Moves Suffice for Rubik’s Cube (Rokicki)arXiv, 2008
- SI units: volume (the litre and the cubic metre)NIST
- Cubic numberWolfram MathWorld
- A000578: The cubesOEIS Foundation