Chapter 3 — Sten Inherits Everything

Strong-Induction Sten is, as the chapter title suggests, the kind of person who inherits things.

He inherited his nose from his father. He inherited his height from his mother. He inherited his love of festival biscuits from his grandfather. He inherited his belief that you should use everything you already have from a series of older cousins who pointed out, repeatedly, throughout his childhood, that there was a perfectly good library at the end of the street.

This is a chapter about why Sten became the kind of mathematician he is.

Sten was, as the previous chapter mentioned, three years younger than his sister Ida. He grew up in the same town (Lattice), in the same family (Latticeford), in the same domino-festival tradition. He helped set up the Cascade every year. He pushed the first domino at fifteen, which was a great honour, and he too watched the whole chain fall, and he too noticed that he had only pushed one piece.

He thought: Yes. I see why Ida likes this.

But Sten also noticed, that day and other days, that his sister’s domino technique had a particular shape. She only used the previous one. She knocked down domino k, and domino k knocked down domino k+1, and the rest happened by itself.

Sten thought, watching her work: That’s fine. But what about the dominoes she already knocked down?

This will sound like a small thought. It was not small.

What Sten realised — and he realised it slowly, over the course of several years — was that when you are proving things about, say, the natural numbers, by the time you are trying to prove the case for n=10, you have already proved n=1, n=2, n=3, …, n=9. They are already established. You are allowed to use them. All of them.

Ida’s technique used the previous one. Sten thought: Why not use all of them?

He brought this up at the dinner table when he was sixteen.

Ida (who was nineteen and home from her first year at the academy) said: “You can. It’s allowed. It’s just a different version of the technique. It’s called strong induction.”

Sten said: “How is that different?”

Ida said: “In ordinary induction, you only assume the case for k. In strong induction, you assume the cases for everything up to k. It’s still valid. Some proofs need it. Most don’t.”

Sten said: “Why would anyone NOT use the strong version?”

Ida said: “Because it isn’t always necessary.”

Sten said: “It isn’t always necessary, but it’s never wrong. So I’d just use it.”

Ida said: “You’re allowed to. Most mathematicians don’t, because it feels heavier.”

Sten said: “It only feels heavier.”

Their mother (who was excellent at deflecting domino arguments at the dinner table) suggested they pass the bread.

Sten went on to study mathematics, like his sister. He arrived at the ProofQuest academy three years after Ida did. He introduced himself by saying, “Hello. I am the dominoes-but-also-everything-already-fallen person.” The academy master said, “Oh. We have your sister. Are you also the dominoes person?” Sten said, “Yes. But I assume more.” The academy master, who had been at the academy for a long time, immediately understood and hired him.

Sten teaches strong induction. He is the cast member who, when proving something about case k+1, gets to use every previously-proven case. This is occasionally exactly what a proof requires. There are theorems — about the structure of prime numbers, about the depth of certain trees, about the way certain recursive algorithms terminate — that cannot be proved by ordinary induction at all. They need strong induction. Sten teaches all of them.

He is, in person, mildly relaxed. He says “obviously” a lot, which is sometimes annoying but is usually accurate. He believes — and has said, more than once, in front of his sister — that strong induction is just induction, but with more friends in the room.

Ida finds this analogy slightly grumpy.

She finds it more grumpy because it is correct.

Sten and Ida are very close. They write letters when they are at different academies. (Sten now teaches at the southern branch; Ida at the central.) Their letters are warm and full of next case discussions. They send each other interesting recursion problems. They argue about whose technique is more elegant (Ida says ordinary; Sten says strong; neither has changed her or his mind). Their mother, who is now seventy-two and still runs the family domino business in Lattice, considers this argument the only thing her children ever fight about.

At the next Cascade festival, both of them will be home. They will set up their share of the chain — Ida’s careful, Sten’s slightly faster — and they will help the youngest cousin push the first domino, and they will watch the chain fall, and afterward they will go to the bakery and eat festival biscuits and continue arguing about which kind of induction is the real one.

Their mother has, by now, learned to bring earplugs.

Voice register

Guidance: Casual, slightly dry. Uses “obviously” often (it is usually accurate). Says “assume we’ve already shown all the cases below” freely. Has the rhythm of someone who likes to use everything in the room. Sibling-bond with Ida; mild rivalry over whose induction is the real one.

Sample lines (for Qed when scaffolding AS Sten):

• “Assume we’ve shown the claim for every value below k. Now prove it for k. Obviously.”
• “Strong induction is just induction, but with more friends in the room.”
• “My sister only uses the previous one. I use all of them. We’re both right.”
• “Some claims need strong induction. The structure-of-recursion proofs, for instance. Ordinary induction won’t reach them.”
• “Obviously this is allowed. I said all the cases below. Use them all.”

Arc across kits

• Kit 1-4 — Not present. Sten enters after Ida is established.
• Kit 5 — Sten introduced as Ida’s sibling. Qed: “Strong-Induction Sten. He assumes more.” Children meet him. Sten and Ida bicker briefly. Children laugh.
• Kit 6 — Children learn that strong induction assumes more in the inductive step but is still valid. Sten is matter-of-fact.
• Kit 7 — Children meet a recursion problem that ordinary induction cannot prove. Sten demonstrates the strong-induction approach. Ida nods. (Ida always nods at strong-induction proofs when they are necessary. She just prefers ordinary when it is sufficient.)
• Kit 8 — Children learn the well-ordering principle — every set of natural numbers has a smallest element — which is equivalent to strong induction. Sten teaches this. He is unusually thoughtful for this kit.
• Kit 9 — Co-teach with Construction Cole. Cole’s proof by construction + Sten’s strong induction combine for several existence theorems. They are mutually amused.
• Kit 10 — Children learn the prime factorisation theorem — that every integer greater than 1 has a unique prime factorisation. The proof uses strong induction. Sten is in his element.
• Kit 11 — Children learn induction on structures — recursively defined objects. Sten teaches this alongside Ida. Their styles complement. (Ida proves the simple cases; Sten proves the ones where the induction needs to look back several steps.)
• Kit 12 — Children meet a tricky recursion that needs to look back more than one step. Sten handles it gracefully. He says, casually: “Obviously you need to assume more than just k-1. Use everything.”
• Kit 13 — Children learn that strong induction can prove things about non-numeric structures (trees, lists, sets). Sten teaches this with mild relish.
• Kit 14 — A complicated proof requires both ordinary and strong induction in different parts. Ida and Sten teach it together. Children see the family teamwork.
• Kit 15 — Sten teaches a kit where the strong-induction hypothesis is the only thing that makes the proof work. He is, briefly, vindicated. He does not gloat.
• Kit 16 — Final kit. Sten arrives at the academy on horseback. He has just walked back from the Cascade. He says: “I’ve been pushing dominoes.” Children solve the final puzzle. Campaign ends.

Relationships

• Alliance: His sister Induction Ida. They are siblings. They write letters. They co-teach when needed. They argue about which kind of induction is more elegant — they always will.
• Alliance (also): Construction Cole. Cole and Sten teach a particularly rich combination — strong induction + proof by construction — for recursive-existence theorems. They are friendly without being especially close.
• Tension: Mild rivalry with his sister. The argument over whose induction is real will not be resolved. Both characters are content with this.

Cultural-context note

The “inheriting things from older relatives” framing is intentionally generic — the kind of folk-tale opening that exists in many cultural traditions without being specific to any. Sten shares Ida’s Latticeford family background. The bread-passing dinner scene is borrowed in spirit from Beverly Cleary’s family-life-at-the-dinner-table register without being a direct reference. No specific cultural register is foregrounded.