Definition: A counterexample is a single specific case that satisfies the hypothesis of a statement but fails its conclusion, which is enough to prove the statement false.
Why does one case matter?
A general claim says something is true every time. To show it is false, you only need one time it fails. To show it is true, however, examples are never enough; you need a proof.
How do you find one?
Read the statement as if p, then q. Look for a case where p is true and q is false. Try small numbers, special shapes, and extreme situations such as zero, negative values, or a very thin triangle.
Example. Claim: if two rectangles have the same perimeter, then they have the same area. A 2 by 10 rectangle and a 6 by 6 rectangle both have perimeter 24, but their areas are 20 and 36. This single pair disproves the claim. Another claim: if n is a whole number, then n2 + n + 41 is prime. It works for many values, but n = 41 gives 412 + 41 + 41 = 41 · 43, which is not prime.
What mistakes occur?
Offering a case where the hypothesis is false, which proves nothing. Another is thinking that many working examples count as proof. The counterexample must fit the exact wording of the statement.