Tomiko

"Four times versus squared" did it in about ninety seconds. He tried it from a thousand and then immediately tried it from ten to check whether small numbers behaved differently, which I was pleased about.

The argument that works at that age is about what happens when you double the input, because it needs no calculus and it is visibly decisive.

Take the quadratic first. If you double x, then x squared becomes four times what it was. Always four, whatever x you started from. A quadratic grows by a fixed factor when you double the input.

Now the exponential. If you double x, then 2 to the power x becomes 2 to the power 2x, which is the old value squared. Not four times bigger — squared.

That is the whole argument. One of them multiplies by four each doubling; the other one squares. Squaring a number bigger than four beats multiplying by four, and it wins by more every time you double again.

Ask him to try it: start at a thousand. Four times gives four thousand. Squaring gives a million. Double again: sixteen thousand versus a million million. He will not need a third round.

You have asked a genuinely good question and it is worth knowing that you can also build the table yourself with nothing but geometry, which is how it was done for centuries before any of the above existed.

Start from the angles you can construct exactly — 30, 45, 60 — using triangles you can draw. Then use the half-angle identity to get 15, then 7.5, and keep halving. Use the addition identity to combine known angles into their sums. Between those two moves you can reach a dense set of angles, and interpolate between them for the rest.

That is essentially how the first trigonometric tables were built, by hand, over years. Doing five or six steps of it yourself is an afternoon well spent and it makes the calculator feel less like magic.

A gentler one for anyone who wants an example they can verify themselves: among the regular polygons, the square is the only one whose perimeter and area are numerically equal for a whole number side length — side 4 gives perimeter 16 and area 16.

That one is engineered enough that it may not meet your non-trivial bar, but it is the kind that works well with students, because they can find it by hand and then be told that the grown-up versions of the same phenomenon go all the way up to the structure of four-dimensional space.

One where the disappointment was about how rather than whether: the four colour theorem.

It is true — four colours suffice for any map. But the proof required checking an enormous number of cases by computer, and for a long time no human-readable proof existed. Many mathematicians found that unsatisfying in a way that had nothing to do with doubting the result. They wanted to know why it was true, and a verified exhaustive case check does not tell you.

That reaction has softened as computer-assisted proof has become normal, but it opened a question that is still live: what is a proof for? If it is to establish truth, the case check is fine. If it is to produce understanding, it is not.