How to build a problem-solving notebook that actually works

Most problem-solving notebooks fail the same way. You solve something hard, or read a solution you liked, and copy it out neatly under a topic heading. Months later the notebook is thick, tidy, and you have not opened it since you wrote it. On the next contest it does nothing for you.

That is not a discipline problem. The notebook was built to be read, and you never need to read it. In a contest what you need is for an idea to come to mind while you look at a statement you have never seen. A list of solutions sorted by topic cannot do that, because you do not know the topic yet. Working out the topic is the hard part.

So build the notebook the other way round: index it by what you see in a problem, not by what the solution turned out to be.

One entry, four lines

Every entry has the same shape, and none of them is a full solution.

1. Trigger. The feature of the statement that should have made you reach for the idea. Write it as words you could spot on the page: "prove a tiling is impossible", "a prime greater than 3", "for every integer $n$".
2. Move. The first thing to do, in one line. Not the whole argument: the opening decision.
3. Source. The problem you got it from, so you can go back to it.
4. Why it works. One sentence. If you cannot write it, you have copied a step without understanding it, and the entry will not transfer.

Four lines is deliberate. A copied solution takes twenty minutes and teaches you nothing new the second time. Four lines take two minutes, and they record the one thing that solution had which you did not: the decision about how to start.

Five entries, written out

Here is what a working page looks like. Each one is a real problem from the archive, and each entry stops at the move.

Trigger: prove that some covering or tiling is impossible. Move: colour the board and count each colour. Source: the mutilated chessboard. Two opposite corners are the same colour, so removing them leaves $32$ squares of one colour and $30$ of the other. Every domino covers one of each. Why: a tiling has to cover equally many of each colour, so a count that comes out unequal rules it out. That is the colouring idea in one line.

Trigger: "a prime greater than 3". Move: the condition is there to exclude $2$ and $3$, so use it. Such a prime is $6k \pm 1$, and $p^2 - 1 = (p-1)(p+1)$ is two even numbers either side of $p$, one of them a multiple of $3$. Source: $p^2 - 1$ is divisible by $24$. Why: "greater than 3" tells you which small primes matter, and factoring shows you where they divide.

Trigger: "divisible by a fixed number for every integer $n$". Move: factor the expression, split the divisor into prime powers, and check residues modulo each prime separately. Source: $n^5 - n$ is divisible by $30$. Here $n^5 - n = (n-1)\,n\,(n+1)(n^2+1)$ and $30 = 2 \cdot 3 \cdot 5$. The three consecutive integers handle $2$ and $3$, and a five-row residue table handles $5$. Why: a statement about every integer is a statement about finitely many residues. That is modular arithmetic doing the work.

Trigger: "the number of things with an odd count is even". Move: add up the counts. Each handshake is counted twice, once at each end. Source: the handshake lemma. The sum of everyone's handshakes is twice the number of handshakes, so it is even, and an even total cannot have an odd number of odd terms. Why: double counting turns a parity question about individuals into one about a single total.

Trigger: something is switched on and off once per divisor. Move: ask how many divisors each number has, and when that number is odd. Source: the hundred lockers. Divisors pair up as $d$ and $n/d$, and the pairing fails only when $d = n/d$, so only perfect squares have an odd count and stay open. Why: "toggled once for each divisor" means "open if and only if the number of divisors is odd".

Merge entries, do not stack them

Look at the second and third entries again. Both moves begin the same way: factor first, then check each prime separately. When two sources share a move, do not keep two entries. Put the second source under the first entry.

That merge is the most useful thing the notebook does. One entry with one source is an anecdote. One entry with four sources is a technique you discovered yourself, and its trigger is the most reliable thing in the book. After a few months, the entries with the most sources are the ones to read before a contest. The rest can stay where they are.

Test it from the trigger side

A notebook you only reread is a notebook you only recognise. To check whether an entry has stuck, cover everything except the trigger and say the move out loud. Then uncover it.

1. Once a week, cover the move column and go down the triggers. Say each move before you look.
2. Any entry you could not reproduce gets a sharper trigger. "Number theory" is not a trigger. "A prime greater than 3" is.
3. Any entry you have not needed in two months can be dropped. A short notebook is one you will actually reopen.

This is the same cold test that revising a topic you keep missing relies on, applied to your successes as well as your mistakes. The two belong in different places, though. A mistake log records where you broke. This notebook records the moves that worked, so you can make them again.

What to fill it from

Entries come from problems you have just worked on. They do not come from rereading theory. After each session, choose the one problem whose first move surprised you and write those four lines while it is still fresh. One good entry a day beats a full page of copied solutions.

The problem archive gives each problem's topic, difficulty and expected time, so you can work through one area until its triggers repeat. The technique pages give each idea's name, which helps when two of your entries turn out to describe the same move. And the daily problem posts a new problem every day at three levels with no account needed, which is a cheap, steady supply of first moves to write down.