How do you convince a thirteen-year-old that an exponential will always overtake a quadratic eventually?

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.

If he likes concrete pictures, the rice-on-a-chessboard story is the same fact in narrative form and it lands with almost everyone.

One grain on the first square, doubling each square. Halfway across the board it is still an amount you could carry. By the end it is more rice than has ever been grown. Nothing about the doubling changed; the numbers simply had room to run.

Then ask him to imagine a competing story where the number of grains is the square of the square number. It starts far ahead — 64 squared is 4096, while doubling from 1 has only reached about 9 billion by square 34, wait, that already lost. Which is exactly the point, and the fact that it is hard to even construct a fair-looking race is itself the lesson.

A second angle that handles the "but what about a really big quadratic" objection head-on, which is the actual question he asked.

The coefficient — the 10 in 10 x squared, or a million if he likes — only shifts when the crossover happens. It never changes whether it happens.

Here is why, in a form he can check. Suppose the quadratic is a x squared for some enormous a. The exponential is 2 to the x. Now compare the two at x and at x plus 1:

• The quadratic goes up by a factor of (x+1)² / x², which gets closer and closer to 1 as x grows. By x = 100 it is about 1.02.
• The exponential goes up by a factor of exactly 2, forever.

So past some point the quadratic is growing by about two percent per step while the exponential is growing by a hundred percent per step. Whatever head start a huge coefficient bought, it gets eaten, because one side's growth rate is dying towards nothing and the other's is constant.

The slogan version: a big coefficient buys a delay, never a victory.

Worth mentioning where this stops being a homework curiosity, because it motivates the whole thing: this is exactly why algorithm speed is discussed in these terms.

An algorithm whose work grows like x squared is slow but usable at large sizes. One whose work grows like 2 to the x is fine on small inputs and impossible not much later — not slow, impossible, as in longer than the age of the universe for inputs that fit on a page.

The crossover he is being asked about is the reason that distinction is the first thing anyone learns about algorithms.