The Collatz conjecture: a question a child can ask and nobody can settle

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What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

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What to assume they know

Kaprekar's 6174: a process that always ends in the same place: a procedure applied over and over, and the idea that it might always end in the same place
Even and odd numbers, and halving an even number
Multiplying by 3 and adding 1
Every number sequence is a rule, not a list: a sequence is a rule, not a list — and Table 1 of Chapter 1, where the powers of 2 are printed

What they should be able to do

Work out the rule behind a set of printed sequences by comparing consecutive terms
Apply the rule to generate a sequence from any starting whole number
Explain why an odd step is always followed by an even step
Show that a sequence can rise before it falls, and give an example from the page
Explain why reaching a power of 2 settles the rest of a sequence
Explain why two sequences that meet can never separate again
State the conjecture Collatz made, and the year he made it
Explain what it means for a problem to be unsolved, and why running more cases does not settle it
Prove the conjecture for the one family where it can be proved: the powers of 2
Generate and check a sequence from a given start, such as 100

Where it usually goes wrong

"Unsolved means nobody has tried." It means many people have tried and failed. The book says so directly.
"Unsolved means we do not know whether it is true, so it is probably false." It has been checked on enormous ranges of starting numbers without a single failure. What is missing is an argument, not evidence.
"If it works for the first thousand numbers, it works." There is no last whole number, so no amount of checking finishes the job. This is the sentence the whole topic exists to deliver.
"The sequence only goes down." Sequence (c) more than triples in one step. Every odd number is thrown upward. That is exactly why the conjecture is hard.
"Two odd numbers in a row would be a problem." They never occur. Tripling an odd number gives an odd number, and adding one makes it even, so an odd term is always followed by an even one.
"Different starting numbers give completely different sequences." They join up. Once two sequences share a term they are the same from that point, because the next term depends only on the current one. The book's (b) and (d) demonstrate it on the page.
"A conjecture is just a guess." It is a claim precise enough to be tested and to be wrong. Collatz's is stated exactly, which is what allows it to have resisted for nearly ninety years.

Questions to check understanding

Given a start, write out the sequence until it reaches 1
Given a sequence, state the rule that generated it
"Why is the conjecture certainly true for the powers of 2?" — the chapter's own question 8
Check the conjecture for a stated start, such as 100
Explain why an odd term is always followed by an even one
Explain what would be needed to settle the conjecture, and what would refute it
Reasoning items on the difference between checking many cases and proving all — assessed again against Kaprekar's 6174: a process that always ends in the same place
Items asking the student to say, with a reason, whether they believe the conjecture — the book asks this directly and it is marked on the reasoning

Examples worth working on the board

The four printed sequences (§3.10, p.68). Printed before the rule is given, so that finding the rule is the first task:
- a. 12, 6, 3, 10, 5, 16, 8, 4, 2, 1
- b. 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
- c. 21, 64, 32, 16, 8, 4, 2, 1
- d. 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 These are ordinary type and extract cleanly.
The rule (§3.10, p.69). An even number is halved; an odd number is tripled and one is added; then you start again on whatever you got.
What the four sequences are chosen to show. Worth pointing out, because the selection is deliberate:
- (a) starts even and immediately reaches an odd number that sends it back up
- (c) hits a power of 2 on its very first step and then simply falls
- (d) is (b) with two extra terms in front — the two sequences run into one another at 17 and are identical from there on
- all four finish on the same five terms: 16, 8, 4, 2, 1
The claim (§3.10, p.69). Collatz, a German mathematician, conjectured in 1937 that the sequence reaches 1 whatever whole number you start from. The book states that this is still not settled, and ranks it among mathematics' best known open problems.
The powers of 2 (chapter-end Figure it Out, p.73, question 8). Start from a power of 2 and every term is even, so the rule only ever halves, and the sequence walks straight down the doubling table to 1. This is the one case the class can settle completely, and it is the right place to show what a proof looks like next to a pile of checked examples.
Starting from 100 (chapter-end Figure it Out, p.73, question 9). A start that is neither small nor a power of 2, and one that rises before it falls.
The rise. Any odd number is sent above itself — tripling and adding one is always an increase. The book's own sequence (c) rises from 21 to 64, more than threefold in a single step. Show the rise; an explanation that shows only descents makes the conjecture look obvious and drains the section of its point.

Figures to have open

The four printed sequences (§3.10, p.68), each laid out so that the rule can be shown moving between consecutive terms. Standard schematic; the book prints them as plain lists.
A height plot of one sequence — term number along the bottom, value up the side — so that the rises are visible. Not in the textbook, and essential: the rises are the whole difficulty and a list of numbers hides them.
A diagram of two paths merging at 17, for section 7. Not in the textbook.
The powers-of-2 column from Chapter 1, Table 1, referenced by the chapter-end question. Cite the earlier chapter rather than redrawing it fresh.
No photograph, table or dataset from this chapter is required; §3.10 is two pages of running text and four lists.

Where this sits in the book

NCERT Class 6 Mathematics (Ganita Prakash), Chapter 3 "Number Play": §3.10 An Unsolved Mystery — the Collatz Conjecture!, pp.68–69
The four sequences, p.68
The rule, the attribution to Lothar Collatz and the year 1937, the statement that the problem remains open, and the two closing prompts, p.69
The chapter-end Figure it Out block, p.73, questions 8 and 9
Summary, p.73, which names Collatz's conjecture as its example of a problem easy to pose and hard to solve
Backward link: Chapter 1, Table 1, for the powers of 2 (Every number sequence is a rule, not a list)
Backward link: §3.6, pp.62–63 (Kaprekar's 6174: a process that always ends in the same place), the procedure that looks like this one and is not in doubt