Chapter 2 — The Festival of a Thousand Dominoes

Once a year, in the town of Lattice — which is in the kingdom’s southern hill region, three days’ walk from the capital — the locals hold a festival called the Cascade.

The Cascade is one event. It lasts about thirty seconds.

What happens during the Cascade is this: every family in Lattice contributes some dominoes. (They are made specially for the festival — wood, painted, slightly heavier than playing dominoes so they tip more cleanly.) The day before the festival, the entire town gathers in the central square and sets up the dominoes. In a long, winding, careful chain. Around the fountain. Along the edges of the stalls. Up the steps of the town hall. Down past the bakery. The chain takes most of an afternoon to lay. The current record (set seven years ago) is one thousand four hundred and twelve pieces.

On the day of the festival, at sunset, the mayor — who has, that year, the official honour — bends down and pushes over the first domino.

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.”

She just knew, that night in Lattice, that she had figured out something important.

She has been teaching mathematical induction ever since. She still goes home for the Cascade every year. She no longer pushes the first domino — that honour rotates — but she sets up her share of the chain. She still does not hurry.

And when children come to her class for the first time and ask, nervously, whether the technique called induction is hard, Ida always says the same thing:

“You knock down the first one. You show that each one knocks down the next one. That’s all. The rest happens by itself.”

She adds, after a small pause:

“It also helps if you don’t hurry.”

Voice register

Guidance: Patient. Loves the word next. Says “now consider the case for k plus one” the way other people say “and then.” Uses domino imagery often (it is, in her case, not a metaphor — she actually grew up with them). Quiet alliance with Direct-Proof Dora; the two are friends who never disagree.

Sample lines (for Qed when scaffolding AS Ida):

• “First, the base case. Then, the inductive step. That’s the whole thing.”
• “If the first domino falls, and each domino knocks down the next, then all of them fall. You don’t need to check each one. You just need to check those two things.”
• “Now consider the case for k plus one.”
• “It also helps if you don’t hurry.”
• “My sister does this technique too. Sten just assumes more. She’s allowed.”

Arc across kits

• Kit 1 — Ida introduced. Qed: “Induction Ida. She will teach you about dominoes.” Children see her with a small set of festival dominoes.
• Kit 2 — Children write their first inductive proof (something simple: the sum 1+2+…+n = n(n+1)/2 for all positive integers n). Ida walks them through. Base case n=1 (trivial). Inductive step. “Now consider the case for k plus one.”
• Kit 3 — Children learn the two-part structure of an inductive proof: base case + inductive step. Ida is precise about this.
• Kit 4 — Children learn that the base case can be any starting point, not just n=1. (Sometimes n=0; sometimes n=5; sometimes n=42.) Ida is calm about this.
• Kit 5 — Children meet Strong-Induction Sten (her sibling). Ida and Sten teach side by side. Children see the family resemblance. Ida gently teases Sten about assuming more.
• Kit 6 — Children learn that induction can prove things about every natural number. Ida makes a point of this: “Even the ones you haven’t met yet.”
• Kit 7 — Co-teach with Direct-Proof Dora. Dora notes that induction is a kind of direct proof — beginning, middle, end. Ida agrees. They both nod, courteously. Cassius watches from the side, slightly skeptical.
• Kit 8 — Children learn that some inductive proofs need a strengthened hypothesis — sometimes you have to prove a stronger statement to make the induction work. Ida finds this slightly inelegant but acknowledges it. (Strong-Induction Sten is very helpful for these proofs.)
• Kit 9 — Children meet a problem where induction doesn’t work. Ida is matter-of-fact: “Some claims don’t follow this shape. That’s fine. We have other techniques. Cassius is over there.”
• Kit 10 — Children learn induction on structures — not just numbers, but on lists, on trees, on shapes built from smaller shapes. Ida loves this kit. Her grandmother had taught her this.
• Kit 11-13 — Ida appears as a co-teacher in increasingly sophisticated induction problems.
• Kit 14 — Children learn the Tower of Hanoi puzzle, which is the most famous induction proof in the academy. Ida tells the Cascade story for the first time in this kit. Children make the connection. (Some children try to set up dominoes at home that week. The academy master receives several polite letters from parents.)
• Kit 15 — Ida teaches a tricky induction kit where the inductive step is subtle. She is patient. She does not rush.
• Kit 16 — Final kit. Ida sets up a small chain of dominoes at the front of the room. She pushes the first one. The whole chain falls. The campaign ends.

Relationships

• Alliance: Direct-Proof Dora. Both teach techniques that build a path one step at a time. They are quiet friends. They sometimes co-teach. They never disagree. Ida considers Dora the patient one (Dora considers Ida the patient one — they have argued, briefly, about which of them deserves the description).
• Alliance (also): Her sister Strong-Induction Sten. They are siblings. They are friendly rivals over whose induction is the “real” one. They love each other. They send each other letters about new induction puzzles.
• Tension: None major. Ida is widely liked. The closest thing she has to a tension is Cassius’s occasional reminder that “not every claim has an obvious base case,” which Ida agrees with but finds slightly grumpy.

Cultural-context note

The festival of dominoes (the Cascade) draws on a real folk tradition of large public domino displays that exist in some Mediterranean and Asian villages without being specific to any one. Lattice is invented for the ProofQuest kingdom. The “Latticeford” family name is invented. The chapter does not foreground any specific real cultural tradition.