How to write up a proof so it gets full marks

There is a particular kind of disappointment that only happens in proof contests. You solved it. You know you solved it. The paper comes back with most of the marks missing, and the comment is not that you were wrong.

Writing up is a separate skill from solving, and almost nobody is taught it. This is that skill: not what makes an argument correct, but what puts the correctness where a marker can find it.

What the marker is actually doing

A marker reads your page once, in the order you wrote it, and cannot ask you a question. They are not reconstructing your reasoning; they are looking for specific things and ticking them off. A mark scheme splits a problem into a small number of named achievements, and each one is either visibly present on your page or it is not there at all.

Two consequences follow, and they are the whole article.

• Credit attaches to claims, not to effort. A page covered in true, relevant work that never states a finished claim scores far worse than three sentences that prove one clean thing.
• A missing step and a wrong step are not the same size. A gap costs you the achievement it sat under. A false assertion can cost the ones you had already earned, because a marker who catches you asserting something untrue stops believing the sentences around it.

Write in dependency order, not discovery order

Solving is a search: you tried the wrong substitution, noticed something, backed up. Writing is not a report of that search. A write-up that opens with first I tried hides the achievements inside a narrative, and the marker has to dig for what you proved.

Write the finished argument in the order a reader needs it. Objects get named before they are used. Every lemma is stated in full, proved, and then applied. If you found the key idea last, it still goes near the front.

The hypothesis you never used

Here is a test a marker can run in seconds, and you can run it first. Take every condition the problem gave you and find the sentence where you used it. A condition you never used is either unnecessary, which is rare, or your proof is broken.

Take the classic: if $p$ is a prime greater than 3, then $24$ divides $p^2-1$. The bound matters, and it is not decoration. At $p=2$ you get 3, at $p=3$ you get 8, and neither is a multiple of 24. So any write-up that reaches the conclusion without anywhere using that $p$ is odd and that $p$ is not a multiple of 3 is provably not a proof, whatever else is on the page.

Name the use explicitly. Not "since $p$ is prime", but "since $p>3$ is prime, $p$ is odd, so $p-1$ and $p+1$ are consecutive even numbers". The marker is looking for that sentence.

Induction: three lines carry the marks

Induction is where the most marks are lost to write-up rather than to mathematics, because the ritual is so familiar that people stop reading their own version of it. Three lines carry everything.

1. The statement, with its index. Write $P(n)$ out as a sentence. Not "the formula" — the actual claim, for a general $n$.
2. The base case at the right index. Not automatically $n=1$. The right base is wherever your step starts working.
3. The step, saying where the hypothesis went in. A step that never visibly uses $P(n)$ is not an induction, it is an unrelated calculation with the word assume in front of it.

Line 2 is the quiet one. Consider proving $2^n > n^2$. The inductive step needs $(n+1)^2 \le 2n^2$, and that is false at $n=1$ and $n=2$ and true from $n=3$ onwards. Meanwhile the statement itself is true at $n=0$ and $n=1$, then false at $n=2$, $n=3$ and $n=4$, and true again from $n=5$. Base at $n=1$ and you have proved nothing; base at $n=5$ and the argument closes. The write-up difference is one line, and so is the mark difference.

Proving a degree-$n$ polynomial has at most $n$ real roots is a good one to write out in full for practice, because the induction is on degree rather than on a counter, and the step has a branch in it that is easy to leave implicit.

The sentence that says "and that is all of them"

Whenever your argument splits into cases, the marker needs one sentence establishing that the cases cover everything. Whenever you claim a list is complete, the marker needs the sentence that rules out anything else. These are not pedantry; they are usually a whole achievement on the scheme, and they are usually one line.

Take the product of any four consecutive integers being divisible by 24. "Obviously there are enough factors of 2 and 3" gets nothing. What earns the marks is saying which: among four consecutive integers exactly two are even and exactly one of the four is a multiple of 4, which gives a factor of 8, and at least one of the four is a multiple of 3. Same idea, same length, and now every claim is checkable by the person reading it.

When the algebra is done but the proof is not

For every integer $n>1$, $n^4+4$ is composite is the cleanest trap of this kind. The identity $n^4+4=(n^2-2n+2)(n^2+2n+2)$ is the idea, and a page that stops there feels finished.

It is not finished, and the problem tells you so. At $n=1$ the first factor is exactly 1, and $1^4+4=5$ is prime. A factorisation only proves compositeness once both factors are shown to exceed 1, which is also precisely where the condition $n>1$ gets used. One sentence, and it is the sentence the scheme is looking for.

When you have not solved it

Most partial credit is lost by people who had something worth marks and wrote it in a form that could not be credited.

• Detach and state. A lemma you proved is worth marks even if you never used it. Give it a number, state it as a standalone claim, prove it, and say what you hoped it would give you.
• Do not assert a plausible lemma you have not tested. "If $a$ divides $bc$ then $a$ divides $b$ or $a$ divides $c$" sounds like a fact and is false: 4 divides $2 \times 2$ and divides neither factor. It becomes true when $a$ is prime. Thirty seconds on small numbers is the difference between a clean partial and a page the marker now distrusts.
• Label verification as verification. Checking $n^2+n+41$ for $n$ up to 39 gives a prime every time, and at $n=40$ it is $41^2$. A table of confirmed cases is evidence; presented as a proof it is a false claim, and it is scored as one.

Four phrases that cost you

• "Clearly" and "it is easy to see" mark the exact place a marker looks hardest. If the step really is easy, the one line that proves it is cheaper than the suspicion.
• "Similarly" is fine only after you have said what the similarity is. Name the swap that turns the written case into the unwritten one.
• **"WLOG"** needs its licence stated. Say which symmetry of the problem lets you reorder, in the same sentence.
• "So we are done" without restating what was proved. The final line should be the problem's own statement, quantifier and all.

A ninety-second read-back

Read only the first sentence of every paragraph, in order, and ignore the rest. That is roughly what a tired marker retains, and if those sentences alone do not spell out a complete argument, your structure is hiding your work rather than showing it.

Then run three greps by eye: every hypothesis appears in a sentence that uses it, every case split is followed by a covering sentence, and the last line answers the question that was asked.

Lemma teaches the named techniques alongside proof problems that carry hints and full worked solutions, so you can compare your write-up against a complete one line by line rather than guessing what was missing. The problem archive holds the ones above with difficulty and expected time, and the daily problem posts a fresh one every day with no account needed. Writing up two short proofs a week, properly, moves scores faster than almost anything else you could do with the same hour.