Proof by Counter-Example: GCSE Maths Guide

What is a conjecture and when is it false?

A conjecture is a statement claimed to be true for all values within some domain. It is false if even one case fails.

For example: "All square numbers are odd."

• 1² = 1 (odd ✓), 3² = 9 (odd ✓), 5² = 25 (odd ✓) — these support the claim.
• But 2² = 4 (even) — this disproves the conjecture.

The fact that infinitely many examples support the claim is irrelevant once a single counter-example is found.

How do you find a counter-example efficiently?

Rather than trying random numbers, use these strategies:

1. Try small numbers first: 0, 1, 2, 3, −1 often catch failures that larger numbers miss.
2. Try edge cases: negative numbers, fractions, zero. Many conjectures fail for n = 0 or for fractions.
3. Read the conjecture carefully: identify what the statement says should happen, then look for a situation where it cannot happen.

Worked examples of counter-examples

Example 1: "n² > n for all positive integers n."

Try n = 1: 1² = 1. Is 1 > 1? No. Counter-example: n = 1, since 1² = 1 and 1 is not greater than 1. ✗

(Note: for n ≥ 2 the statement is true, and for 0 < n < 1 it is also false. The question specifies positive integers, so n = 1 suffices.)

Example 2: "The sum of two prime numbers is always even."

Most pairs work: 3 + 5 = 8, 7 + 11 = 18. But try 2 + 3 = 5, which is odd. Counter-example: 2 and 3, since both are prime and their sum is 5, an odd number. ✗

(The number 2 is the only even prime — pairing it with any other prime gives an odd sum.)

Example 3: "If n is a positive integer, then 2n + 1 is always prime."

• n = 1: 2(1) + 1 = 3 ✓ prime
• n = 2: 2(2) + 1 = 5 ✓ prime
• n = 3: 2(3) + 1 = 7 ✓ prime
• n = 4: 2(4) + 1 = 9 = 3 × 3. Not prime. Counter-example: n = 4, giving 2n + 1 = 9, which is not prime. ✗

Example 4: "For all real numbers x, √(x²) = x."

Try x = −3: √((−3)²) = √9 = 3, not −3. Counter-example: x = −3, since √(x²) = 3 ≠ −3 = x. ✗

(In fact, √(x²) = |x|, the absolute value.)

What must a valid counter-example include?

An exam answer for a counter-example question must show:

1. The specific value(s) used: state the counter-example clearly (e.g. "n = 4").
2. The calculation: show the working to confirm the values satisfy the conditions.
3. Why it contradicts the claim: state which part of the conjecture fails.

One line is often enough:

"Counter-example: n = 4. Then 2n + 1 = 9 = 3 × 3, which is not prime. So the conjecture is false."

Do not write "it doesn't always work" without giving specific values — that scores no marks.

How is this different from algebraic proof?

Counter-example questions are often phrased as: "Prove that [statement] is not always true," or "Show that [statement] is false," or "Disprove [conjecture]." All three ask for the same thing: one specific case that fails.

Frequently asked questions

Does a counter-example prove a statement is always false?

No. A counter-example proves the statement is not always true — it fails for at least one value. The statement may still be true for many other values. For example, n² > n fails at n = 1 but is true for all integers n ≥ 2.

How many counter-examples do I need?

Exactly one is sufficient and necessary. More than one is not wrong, but wastes exam time. If the question asks you to "find a counter-example," stop once you have presented one clearly.

What if I cannot find a counter-example?

If you cannot find one, the conjecture may actually be true — in which case a counter-example does not exist and the question requires an algebraic proof instead. In a GCSE exam, the question wording will tell you which approach is required: "prove" → algebraic; "disprove" or "show it is not always true" → counter-example.

Can I use a calculator to help find a counter-example?

Yes — especially for conjectures involving large expressions. Try several values quickly on a calculator, looking for one that breaks the pattern. Once found, write the calculation clearly in your answer booklet. Stating "I used a calculator and found n = 4 works" is not a valid answer — you must show the numerical check.

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