Many people think of maths as following rules to reach one right answer. But the people who use maths most, from engineers to puzzle setters, spend much of their time doing something creative: spotting patterns, trying different approaches and asking "what if?". This guide shows what creative maths looks like in practice, with activities you can try at home or in class. Each one comes with the reasoning behind it.
Patterns are the heart of mathematics, and finding one is a creative act. Try adding up odd numbers, one at a time:
The answers are 4, 9, 16 and 25: the square numbers. Why? Draw it. Start with one dot. Add 3 dots around it to make a 2 by 2 square, then 5 more to make 3 by 3. Each odd number is exactly the L-shaped border that grows the square by one. A picture explains what the sums only show.
The times tables are full of patterns like this, and spotting them makes them easier to learn. Our 9 times table trick and skip counting guides are good places to look.
A creative problem has more than one route to the answer. Here is a classic. The young Carl Friedrich Gauss is said to have been asked to add all the numbers from 1 to 100. Instead of adding them in order, he paired them: 1 with 100, 2 with 99, 3 with 98, and so on. Each pair adds up to 101, and there are 50 pairs, so the total is 50 × 101 = 5050.
Try asking children to find two different ways to solve a problem, then compare. For 25 × 16, one person might split 16 into 10 and 6, giving 250 + 150 = 400. Another might notice that 25 × 4 = 100, and 16 is 4 lots of 4, so the answer is 4 lots of 100. Both are right, and talking about them builds understanding. For a collection of these alternative routes, see mental multiplication tricks.
Give yourself four numbers and a target, and use any operations to reach it. With 1, 2, 3 and 4, can you make 24?
There are many answers. Now try making every number from 1 to 10 with the same four digits. It practises arithmetic, brackets and the order of operations, and it rewards trying ideas that might not work.
Why do bathroom tiles come in squares, triangles and hexagons, but never regular pentagons? Look at the corners. For shapes to fit together with no gaps, the angles that meet at each point must add up to 360 degrees.
Children can test this with paper cut-outs before they know any angle facts, then explain what they find. Patterns made of tiles are called tessellations, and they appear in art from Islamic geometric design to the prints of M. C. Escher.
Start with 1 and 1, and make each new number by adding the two before it: 1, 1, 2, 3, 5, 8, 13, 21, 34. Now divide each number by the one before: 8 ÷ 5 = 1.6, 13 ÷ 8 = 1.625, and 21 ÷ 13 is about 1.615. The answers settle towards about 1.618, a number called the golden ratio.
Fibonacci numbers do turn up in nature, for example in the spiral counts of many sunflower heads and pine cones, though not everywhere people claim. Checking a claim like that against real examples is itself good mathematical thinking.
How many times does your heart beat in a day? No one knows exactly, but you can reason your way to a good estimate. At rest, a heart beats roughly 70 times a minute. There are 60 minutes in an hour and 24 hours in a day, so about 70 × 60 × 24 = 100,800 beats, or roughly 100,000. Questions like this, called Fermi problems, need no special knowledge, just sensible guesses and careful multiplication.
Games where you plan ahead build the same reasoning as maths problems: thinking several steps ahead, considering what the other player might do, and changing plans when something doesn't work. Try Quoridor, where you race to the other side while placing walls to block your opponent, or challenge a friend in Player vs Player.
Ask open questions with more than one answer, like "how many ways can you make 24?", and talk about different methods rather than only checking answers. Puzzles, patterns and strategy games all help.
Yes. Spotting a pattern or a shortcut can turn a long calculation into a short one, and being comfortable trying different approaches helps when a question looks unfamiliar.
Adding up odd numbers is a good start: the answers are always square numbers, and drawing dots shows why. It takes five minutes and needs only paper and a pencil.