The single habit that separates students who get unstuck on proofs from students who sit staring at the page is almost embarrassingly simple: keep two pieces of paper. One is scratch, where you run the logic in whichever direction makes progress — usually backwards from the conclusion. The other is the fair copy, where the final proof has to run forwards from the hypothesis. The two columns do different jobs, and confusing them is what makes proofs feel impossible.
This article is not about any particular proof. It is about the workflow that turns a blank page into a finished argument — a routine you can apply to every direct proof in your homework, every week, without thinking about which technique to use.
The two columns
Imagine your desk split down the middle. On the left, scratch paper. On the right, the fair copy.
- Scratch paper (left column). Messy. Bidirectional. Full of crossed-out lines, question marks, arrows going both ways, half-finished algebra. Nobody reads it but you. Its only job is to find the route from P to Q.
- Fair copy (right column). Clean. One-directional. Every step justified. The only reader is the grader — or your future self. Its only job is to communicate the route, running forwards.
The scratch paper is where you think. The fair copy is where you write. These are different activities, and collapsing them into one leads to two symmetric failures: you either try to write a clean proof before you understand it (and get stuck on line three), or you turn in a scratch sheet (and the grader cannot follow the argument).
The recipe
Here is the routine, for any direct proof "if P then Q."
On scratch paper:
- Write P at the top. Write Q at the bottom. Leave space between them.
- Unpack P using its definitions — write whatever that unlocks just below P.
- Unpack Q using its definitions — write what the conclusion will need to look like, just above Q.
- Now work the gap. Try forwards from P's unpacking and backwards from Q's unpacking. Write arrows pointing both ways. Mark any step that is not reversible with a warning sign (see Working Backwards).
- When the two halves meet, you have a route. Box the sequence of intermediate statements.
On the fair copy:
- Rewrite the boxed sequence, running from P at the top to Q at the bottom. Every step gets a justification (by definition, by algebra, by a previously proved theorem). Use because for assertions, if only for genuine hypothetical scopes (see Because vs If).
- End with a \blacksquare or QED so the reader knows the argument is done.
The handoff from step 5 to step 6 is the whole trick. On the scratch sheet the argument bends and circles; on the fair copy it runs in one direction.
A picture of the handoff
A tiny worked example
Claim. If n is an integer, then n(n+1) is even.
Scratch. I want n(n+1) even, i.e. n(n+1) = 2k for some integer k. Case split: n even or n odd. If n even, n = 2m, then n(n+1) = 2m(n+1) — done. If n odd, then n+1 is even, n+1 = 2m, then n(n+1) = n \cdot 2m = 2nm — done. Route found.
Fair copy. If n is even, write n = 2m for some integer m. Then n(n+1) = 2m(n+1), which is 2 times an integer, so n(n+1) is even. If n is odd, then n+1 is even (consecutive integers alternate parity), so n+1 = 2m for some integer m. Then n(n+1) = n \cdot 2m = 2(nm), which is 2 times an integer, so n(n+1) is even. In either case, n(n+1) is even. \blacksquare
The scratch sheet was one line. The fair copy is three times longer — but every sentence in the fair copy does a job the scratch line does not: it names the case, cites the definition, and makes the conclusion explicit. The scratch sheet found the route; the fair copy communicates it.
What this habit buys you
Three things.
1. You stop getting stuck. A blank page is intimidating when your only allowed move is "write the first line of the final proof." A scratch sheet lets you write anything — guesses, rearrangements, backward chains. Once the route is visible, writing the final version is mechanical.
2. You catch your own errors. The two directions give you two chances to check each step. If the scratch chain claims A \Leftrightarrow B but the forward version only shows A \Rightarrow B, you catch it during the handoff. Mistakes that would slip past a single-pass proof get caught by the flip.
3. You stop believing textbook proofs came out of nowhere. Every textbook proof was a scratch sheet once. Seeing your own scratch-sheet-plus-fair-copy workflow demystifies the published version. You are not missing some genius instinct; the textbook author just did not show you the left column.
A diagnostic checklist
Before handing in a proof, run these three checks:
- Does the fair copy mention P? If not, you proved the conclusion without using the hypothesis, which is usually a sign the hypothesis is load-bearing and you forgot to cite it.
- Does every step in the fair copy have a one-word justification? (by definition, by algebra, by substitution, by the previous line.) If a step has no justification, it came from the scratch sheet and you forgot to translate it.
- Can you read the fair copy aloud in one direction from top to bottom without ever referring to a later line? If no, a step is quietly backwards and needs rewriting.
Three yeses means the two-column handoff worked. You have a proof that was discovered by the most effective method available, and presented in the form the reader needs.
The short summary
- Scratch paper: explore freely, backwards from Q if needed, bidirectional.
- Fair copy: run forwards from P to Q, every step justified.
- The handoff between the two is where proofs stop being mysterious.
- Keep the columns physically separate — two sheets of paper, or two halves of one sheet. The separation is the point.
If you build this habit, the question "where do I start?" disappears. You start on the scratch paper, always, and the fair copy writes itself once the route is found.
Related: Mathematical Proof — Direct Proof · Scratch Work vs Final Proof · Working Backwards: Cheating or Not? · Because vs If Inside a Proof