Mathematics: Counterexample hunting

What you'll be able to do: Find the cases your working definition doesn't cover, which is where mathematical definitions get their precision, and where exams get their questions.

Why mathematics rewards this more than any other subject

In most subjects a counterexample weakens a generalisation. In mathematics a counterexample destroys it, immediately and completely, and everybody agrees.

That makes this the subject where the method is sharpest. There's no negotiation about whether the case counts. One function that's continuous everywhere and differentiable nowhere ends the argument that continuity implies smoothness, permanently.

It also explains something students find baffling: why definitions are worded so carefully. Every clause in a mathematical definition is scar tissue from a counterexample. The definition of continuity looks fussy because the intuitive version ("you can draw it without lifting your pen") was killed by specific, constructible functions. Knowing which ones makes the fussiness legible instead of arbitrary.

How to run it

1. State your working rule confidently and broadly. Hedging pre-empts the method. "A continuous function is one you can draw without lifting the pen."
2. Ask for cases where it gives the wrong answer: with no explanation.
3. Work out yourself what's wrong with your formulation before reading anything.
4. Notice whether you narrowed the rule or replaced it. Both are legitimate; confusing them is how you end up believing you've refined something you've actually abandoned.
5. Ask which counterexamples are practically important and which are pathological. Both exist in mathematics and they deserve different amounts of your attention.

Here's a rule I'd state confidently: [rule]. Give me three cases where it gives the wrong answer. Don't explain them, just the cases. I'll work out what's wrong with my version.

The counterexamples worth meeting

Each of these kills a specific intuitive rule that students genuinely hold:

"Continuous means you can draw it without lifting the pen." The Weierstrass function, continuous everywhere, differentiable nowhere. Also a function continuous at exactly one point. The pen-lifting picture isn't a bad intuition; it's a bad definition, and separating those is the lesson.

"Multiplication makes things bigger." Any fraction. This one is held below conscious level by enormous numbers of students and quietly damages everything involving scaling.

"If the derivative is zero it's a maximum or minimum." y = x³ at the origin. Kills the reflex and forces the second-derivative condition to mean something.

"If a sequence's terms go to zero the series converges." The harmonic series. The single most productive counterexample in analysis.

"A function is a formula." A piecewise definition, or an arbitrary set-theoretic mapping. This one reorganises what students think the word means.

"Squaring both sides is safe." Any equation where it introduces a spurious root. Students meet this as a rule to remember; as a counterexample it becomes a reason.

"If AB = 0 then A = 0 or B = 0." True for numbers, false for matrices. Produces the idea of zero divisors.

"Correlation near zero means no relationship." A perfect parabola. Kills a misreading that survives into professional life.

"An average tells you what's typical." Any bimodal distribution. Or a single billionaire in a sample of ten.

The exam-preparation use

I'm about to be tested on [topic]. What are the edge cases an examiner is most likely to use to find out whether I understand the rule or just the examples?

This is unusually effective in mathematics because examiners deliberately construct boundary cases, that's how they discriminate between a student who memorised worked examples and one who understands the conditions. Asking for the boundary cases in advance is asking for the examiner's strategy.

Undergraduate and beyond

At university the method changes character: you construct the counterexamples yourself, and that becomes a substantial part of the work.

"Does this hold without the compactness assumption?": try to build a counterexample; if you fail in an informative way, you've usually found the proof. Attempting a counterexample and failing is one of the standard routes to a proof, and it's rarely taught as a technique.

Pitfalls

1. Hedging the rule first. A rule stated with "generally" can't be broken and can't teach you anything.
2. Collecting counterexamples as trivia. The point is the reshaped rule. If you finish with three odd functions and the same intuition, it failed.
3. Silent rule-switching. Check whether you narrowed or replaced.
4. Treating pathological and practical cases alike. The Weierstrass function and a fraction are both counterexamples and they matter to you differently.
5. The tell: you can recite three counterexamples and still state the original over-broad rule when asked casually. The boundary never attached.

Try this today

Write down a rule from your current topic as confidently and broadly as you can state it, no hedging.

Ask for three cases where it gives the wrong answer, with no explanation. Work out what's wrong with your version before reading anything.