Induction Ida

Everything else falls on its own.

The crowd cheers. The town puts up bunting. The bakery sells out of festival biscuits. Somebody hugs a stranger. The dominoes are swept up. Plans are made for next year. The whole town goes home.

This is the festival that made Ida the mathematician she became.

Her family — the Latticeford family, six generations of festival-domino-makers — had run the Cascade preparation for as long as anybody could remember. Ida grew up watching her mother set out long curves of dominoes in the town square. She grew up helping. By the age of seven she could lay a hundred dominoes by herself without knocking any of them over while she worked. (This is harder than it sounds. There is a particular knack to setting down the next domino without bumping the last one. You have to not be in a hurry.)

Ida was, at twelve, a careful and capable festival-domino-setter, but she was not yet a mathematician. She did not become a mathematician until the year the chain was nine hundred and fifty pieces long, the year her mother let her push the first domino.

It was a great honour. It was also, for Ida, a great responsibility. She had spent the whole afternoon laying her share of the chain. She had checked it three times. She was, at twelve, the youngest first-pusher in seventeen years.

She bent down at sunset. She pushed the first domino.

It fell into the second. The second fell into the third. The third fell into the fourth.

Ida watched the chain unfold.

What she noticed, in those thirty seconds — and she noticed it with the kind of clarity that twelve-year-olds occasionally get and adults sometimes forget — was that she had only pushed one domino. The other nine hundred and forty-nine had fallen on their own. She had not touched them. She had not been near them. The whole chain — all of them — had toppled because the first one had toppled and because each one was close enough to the next one to knock it down in turn.

She thought, then and there: That is everything I need to know.

She walked home that night humming. (Her sister Sten, who was nine at the time and is now known as Strong-Induction Sten, did not understand why her older sister was humming, but Sten was also nine and had eaten three festival biscuits and did not care.)

Ida wrote down what she had figured out, in her own twelve-year-old handwriting, in a notebook her grandmother had given her. The notebook page said:

"To knock down all the dominoes, you only need to do two things:

1. Knock down the first one.

2. Make sure each domino is close enough to the next one to knock it down too.

That's it. The rest happens by itself."

She did not know yet, that night, that this principle had a name. She did not know that the principle was hundreds of years old. She did not know that mathematicians called it induction and used it to prove things about every natural number. She did not know that, in eight years, she would arrive at the ProofQuest academy and introduce herself by saying "I am the dominoes person," and that the academy master would look up from his notes and say, "Oh good. We have been waiting for you."