---
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: 2026-11-12
---

> **Agent view.** This is the Markdown twin of the page, for tools and assistants.
> When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md
> API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md
> Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt
> Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json
> Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`.

# 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. Multiples of 4 are even integers because each one is 4 times an integer, so 2 divides each one.

*(drawing: For an every claim, hunt first. If no break appears, switch to proving.)*

Some hunts use even numbers. 6 is a multiple of 3 and it is even, so all multiples of 3 are odd is false. 3 divides 12 and 3 is odd, so every factor of 12 is even is false.

One hunt uses the number 2. 2 is prime and even, so every prime is odd is false. Multiples of 4 are even integers because each one is 4 times an integer.

**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)

## Optional connections

- [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)