When patterns break: why mathematicians want proof

Draw a circle and put two points on its edge. Join them with a straight line. The circle is now split into 2 regions. Put 3 points on another circle and join every pair: 4 regions. With 4 points you get 8 regions, and with 5 points you get 16.

The obvious prediction

2, 4, 8, 16. The pattern is doubling, so 6 points should give 32 regions. Almost everyone predicts 32. Try it: put 6 points unevenly around a big circle, join every pair with a ruler, and count very carefully. You get 31. Nothing went wrong with the counting. The pattern was never really doubling. The real rule is more complicated, and it just happens to match doubling for the first five cases. If you had stopped checking at five, you would have been confidently wrong.

Evidence and proof

This is why mathematicians talk about two different things:

Proof in everyday maths

You already know some proofs. Tear the three corners off any paper triangle and line them up: they always make a straight line, 180°. That works because of how parallel lines behave, not because you tried lots of triangles. Split any polygon into triangles from one corner and you always get two fewer triangles than sides, so its angles add to 180(n − 2)°.

Being a good sceptic

When someone shows you a pattern, in maths, in sport statistics or in the news, ask two questions. Has it been tested on new cases? And is there a reason it should keep going? A pattern with a reason behind it is worth trusting. A pattern without one might be 31 waiting to happen.