Rubik's Cube & puzzles
The maths hidden in a Rubik's Cube: ideas for children and parents
Counting pieces and stickers, factorials, symmetry, 3D shapes, patterns and algorithms: how to use a Rubik's Cube to talk about maths at home, with activities for different ages.
A Rubik's Cube is a puzzle, but it is also a box full of school maths. You do not need to solve it to use it: even a scrambled cube works. Here are ideas you can try with a child, with the maths they connect to.
Counting and multiplication
How many stickers are there? Each of the six faces has 9 squares, so 6 × 9 = 54 stickers. You can also count by piece: 8 corners with 3 stickers, 12 edges with 2 and 6 centres with 1. That is 24 + 24 + 6 = 54 again. Getting the same answer two ways is a nice check.
How many little cubes? A 3×3×3 cube is 3 × 3 × 3 = 27 little cubes. One is hidden in the middle, so 26 can be seen: 8 corners, 12 edges and 6 centres.
Shape and space
- Faces, edges and corners. A cube has 6 faces, 12 edges and 8 corners. Count them on the real thing.
- Nets. Cut open a cardboard cube and flatten it: this is a net. Try our cube nets game to see which flat shapes fold into a cube.
- Surface area and volume. If each small cube is 1 cm across, the whole cube has a surface of 54 cm² and a volume of 27 cm³.
Symmetry
A cube is very symmetric. There are 24 different ways to turn a cube so that it fits back into the same cube-shaped hole: 6 choices for the face on top times 4 turns of that face. Colours break the symmetry: on a solved cube only one of those 24 holds puts every colour back where it was. Ask: what changes when you add colours, and what changes when you scramble it?
Big numbers and factorials
The cube has 43,252,003,274,489,856,000 positions. That number comes from multiplying: the 8 corners can be arranged in 8! = 40,320 ways, and each of the first 7 can be twisted 3 ways (3⁷ = 2,187); the 12 edges can be arranged in 12! = 479,001,600 ways and each of the first 11 flipped 2 ways (2¹¹ = 2,048). A rule about swaps cuts the total in half. Read more on the maths of the Rubik's Cube.
For a child, a simple version is: if you want to arrange 4 books on a shelf, there are 4 × 3 × 2 × 1 = 24 ways. That is 4 factorial, or 4!.
Patterns and algorithms
An algorithm is a list of steps that does the same thing every time. The four moves R U R' U' repeated six times bring a cube back to where it started. Ask a child to predict why, and then try it on the virtual cube. Repeating patterns and cycles is the same thinking as multiples and lowest common multiples.
Statistics
Time a few solves with the cube timer. It shows the best time, the mean, and the average of five (which drops the fastest and slowest). Ask: why does the average ignore the best and worst? When is a mean misleading?
Activities by age
| Age | Try |
|---|---|
| 5 to 7 | Count the faces and stickers. Sort the colours. Make the checkerboard pattern. |
| 8 to 10 | Work out 6 × 9 and 3 × 3 × 3. Fold a net. Learn the first step of the beginner method. |
| 11 to 13 | Explore factorials and powers. Time solves and work out averages. |
| 14 and over | Look at permutations, parity and group ideas on the maths page. |
Common questions
Does solving a Rubik's Cube improve maths?
Not directly: it does not teach the school curriculum. But it practises patterns, planning and persistence, and the cube is an excellent starting point for talking about maths.
Is it too hard for a child?
Not if you start in small steps. Many children enjoy the 2×2 or the first steps of the 3×3 before attempting a full solve.
Do you offer classes?
Yes: our Rubik's Cube and speedcubing classes and maths classes use puzzles like this to build confidence.
This guide is general information. For exams, follow your school, teacher and official exam board, and use your own judgement about what suits your child.