Sanity-Check the Conclusion With One Concrete Number Before Writing the Proof

You have twenty minutes before the test, and a problem set that reads "prove: for every positive integer n, n^2 + n + 1 is divisible by 3." You roll up your sleeves, write "Assume n is a positive integer," and start unfolding definitions. Twenty minutes later you are stuck, sweating, and wondering what trick you are missing.

There is no trick. The statement is wrong. For n = 3, the expression is 3^2 + 3 + 1 = 13, which is not divisible by 3. You just burned twenty minutes trying to prove something false.

This is the most avoidable kind of disaster in proof-writing. The fix takes twenty seconds: before you write a single line of proof, plug in a small concrete number and check. If the claim fails for n = 3, the claim is broken and you move on. If it holds for three or four values, the claim is worth attempting.

The rule in one sentence

Never start writing a proof until you have verified the claim for at least one concrete case.

This is not a universal verification — checking three values does not prove the statement. But it is a cheap filter that catches the wrong claims before you commit time to them.

A worked example: which claims pass the filter

Here are four claims. Only three of them are true. Without writing any proof, figure out which.

For every integer n, n^2 + n is even.
For every positive integer n, n^2 + n + 1 is divisible by 3.
For every integer n \ge 2, n^3 - n is divisible by 6.
For every integer n, 3n^2 + 3n is divisible by 6.

Plug in n = 1, 2, 3:

Claim	n=1	n=2	n=3	Verdict
1. n^2 + n even	2 ✓	6 ✓	12 ✓	worth attempting
2. n^2+n+1 div by 3	3 ✓	7 ✗	—	broken
3. n^3 - n div by 6	—	6 ✓	24 ✓	worth attempting
4. 3n^2 + 3n div by 6	6 ✓	18 ✓	36 ✓	worth attempting

Claim 2 dies at n = 2. Seven seconds of arithmetic saved you from twenty minutes of futile algebra.

Why the filter works: a universal claim ("for every n, …") is falsified by a single counterexample. You do not need to check every n to debunk — one failure is enough. So even a quick check of n = 1, 2, 3, 4 is surprisingly powerful: it cannot confirm the claim, but it can kill a false one cheaply.

Where bad claims come from

You might think that "who would give me a false claim?" The answer: everyone does, eventually.

Typos in textbooks. "Prove n^2 + n + 41 is prime for all n" — this is a famous near-miss; it fails at n = 40. A book that dropped "for n = 1, 2, \dots, 39" from the problem statement would send you chasing an impossible proof.
Typos in your own transcription. You copied the problem wrong from the board. Happens constantly.
Claims that are almost true but need a condition you forgot. "n^2 - n is divisible by 2" is true for every integer n, but "n^2 - n is divisible by 4" fails at n = 2 (giving 2). If you drop the important "divisible by 2" and replace with "divisible by 4," you are proving something false.
Your own exploratory claims. When you are solving a problem yourself, you sometimes conjecture a stronger claim than what is actually true. The sanity check is how you find out.

The visualisation: five numbers, five lamps

Drag the slider below to watch a universal claim tested against n = 1, 2, 3, 4, 5. Each lamp lights green if the claim holds for that n and red if it fails. One red lamp means the universal claim is dead.

Four claims tested against the first five integers. Two claims come up all green — they pass the sanity filter and are worth trying to prove. The other two have red lamps — those claims are false, and writing a proof for them is wasted time. The filter takes seconds. The alternative is a half-page of algebra that was never going to work.

What counts as a "good" concrete test

Not every test is equally informative. Some guidelines:

Use small, honest values. n = 1, 2, 3 usually exposes the structure. n = 0 is worth trying if the claim allows it (many claims quietly fail at 0). If the claim is about integers in general, try a negative value too — n = -1 or n = -2. A claim that holds for positive integers but fails at -1 is a claim with a missing condition.
Pick values that could expose corner cases. If the claim involves divisibility by 3, check a value that is not a multiple of 3. If it involves primes, check both a prime and a composite.
Avoid values that trivially satisfy the claim. n = 0 makes everything divisible by everything; n = 1 makes a lot of things equal to 1. A "good" test hits a non-trivial case.
Two or three values is enough. More than that is diminishing returns; if three values all work, you have good evidence the claim is probably true and you should start writing the proof.

When the sanity check passes but the proof is still hard

Passing the sanity check is not a guarantee. It just rules out the stupidly false cases. If n = 1, 2, 3, 4 all work but the proof still eludes you, the claim is probably true and the difficulty is genuine — go find your technique (direct proof, induction, contradiction, contrapositive). The sanity check did its job: it told you the theorem is real, so your effort is not being wasted.

The sanity check also occasionally helps you find the proof. Watch the pattern of values as n grows. For "n^3 - n is divisible by 6," plug in n = 2, 3, 4, 5: you get 6, 24, 60, 120. Look at the factorisations: 6 = 2 \cdot 3, 24 = 2^3 \cdot 3, 60 = 2^2 \cdot 3 \cdot 5, 120 = 2^3 \cdot 3 \cdot 5. A factor of 6 = 2 \cdot 3 is present every time. Why? Because n^3 - n = n(n-1)(n+1) is the product of three consecutive integers — which always contains a multiple of 2 and a multiple of 3. The concrete values pointed directly to the factoring that finishes the proof.

The habit to internalise

Before starting a proof, do three things:

Read the claim.
Plug in n = 1, 2, 3 (or relevant small cases).
Either a counterexample appears — move on — or all cases pass — start the proof.

Thirty seconds, every time. It is the cheapest insurance in mathematics: a tiny, reliable investment that protects you from a large, likely disaster.

When not to sanity-check

Two kinds of claim are not easy to sanity-check by plugging in:

Existence claims — "there exist x, y such that …" — because you have to find the witness, which is itself the hard part. A sanity check can still help: try to find an example by guessing. If you cannot find one after a few tries, maybe the claim is false, or maybe the example is unusual.
Claims about structures — "every group with prime order is cyclic," "every vector space has a basis" — because "plug in a small case" requires choosing a structure, which may not be easy at first. For these, the sanity check shifts to: "have I checked the claim for the three smallest / simplest examples of this kind of structure?" The spirit is the same: try before you prove.

The short summary

Before writing any proof, plug in n = 1, 2, 3 (or other small values) and verify the claim by arithmetic.
If any test fails, the universal claim is false — stop, move on.
If all tests pass, the claim is probably true — proceed to write the proof.
This check costs seconds and saves minutes of wasted algebra on broken statements.
Patterns in the values you compute often suggest the structure of the proof (factoring, divisibility, symmetry).

The technique is boring on purpose: boring steps are the ones you can remember to take under pressure. Sanity checks save you from the single most demoralising experience in proof-writing — working hard on something that was never going to be true.