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Imagine testing a pattern one thousand times in a row, and it works every single time. Does that prove it will work on the next try?
In math, an educated guess based on patterns is called a conjecture. No matter how many examples support it, a conjecture is never fully proven just by collecting evidence.
So how do you prove a conjecture is completely false?

The Power of One Counterexample

To destroy a false claim, you only need one single counterexample — an example that fits the given condition (hypothesis) but fails the conclusion.
In 1640, Pierre de Fermat conjectured that the formula 22n+12^{2^n} + 1 always yields prime numbers, which held true for n=0,1,2,3,n=0, 1, 2, 3, and 44. Nearly a century later in 1732, Leonhard Euler showed that for n=5n=5, the formula equals 4,294,967,2974,294,967,297, which factors into 641×6,700,417641 \times 6,700,417 — a single counterexample that overturned Fermat's conjecture.
How do we find these game-changing counterexamples when working with geometric shapes?

Disproving Shape Conjectures

Let's analyze this statement: If a quadrilateral has four congruent sides, then it must be a square.
To test it, we must keep the 'if' part true by drawing a shape with four equal sides, and then check whether it is forced to be a square.
A rhombus has four congruent sides, fulfilling the hypothesis, but its angles are not 90∘90^\circ, disproving the conjecture.