---
title: "Finding a Counterexample"
description: "Disprove a claim by producing one case where it fails, and see why a single case is enough to sink a general statement."
canonical: https://lightmysky.com/learn/mathematics/finding-a-counterexample-mt_EoLyw6eP87
source: https://lightmysky.com/learn/mathematics/finding-a-counterexample-mt_EoLyw6eP87.md
retrieved: 2026-09-12
---

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# Finding a Counterexample

Disprove a claim by producing one case where it fails, and see why a single case is enough to sink a general statement.

Subject: Mathematics · Area: Mathematical Thinking · Ages 13 to 14
Page: https://lightmysky.com/learn/mathematics/finding-a-counterexample-mt_EoLyw6eP87

## Ready when they can

- Produce a counterexample to a plausible claim such as every odd number being prime
- Explain the asymmetry: one case disproves, no number of cases proves
- Decide, for a given claim, whether to hunt for a counterexample or attempt an argument

## Lesson: One case that breaks the claim

Disprove a claim by producing one case where it fails. The claim that every odd number is prime dies at 9, because 9 equals 3 times 3. Note that 1 is a special case: neither prime nor composite.

See the asymmetry: one case disproves, while no number of cases proves. Dan tests 3, 5, and 7, finds all prime, and declares every odd number prime. He has not proved it. The untested 9 breaks his claim.

**Example.** Hunt where claims look weakest. Against every number ending in 3 is prime, try 33: it equals 3 times 11. Against all multiples of 3 are odd, try 6. Against every factor of 12 is even, try 3. Against every prime is odd, try 2, the only even prime.

**Tip.** For a claim with the word every, hunt for a counterexample before attempting an argument. If a solid search finds no break, then switch to proving. Counterexample first, argument second.

**Recap.** One failing case sinks every, while passing cases never prove it.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [From Examples to a General Argument](https://lightmysky.com/learn/mathematics/from-examples-to-a-general-argument-mt_vm7cigeRtg)
- [Prime numbers](https://lightmysky.com/learn/mathematics/prime-numbers-mt_y1XCVsIelg)

## Opens up

- [Proof by Contradiction and Counterexample](https://lightmysky.com/learn/mathematics/proof-by-contradiction-and-counterexample-mt_pbFTv4v6k8)