Examples are not a reason
A rule can pass forty tests and still be false. Try to break this one. For any whole number n, n × n + n + 41 is a prime. A prime has exactly two factors: 1 and itself.
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- n = 0 to 9 → every case is prime.
- n = 10 to 39 → still prime. 40 cases pass.
- n = 40 → 1600 + 40 + 41 = 1681 = 41 × 41.
- One case ends the whole claim.
The claim, and the pile of passes
Take a whole number n. Work out n × n + n + 41. The claim: the answer is always prime.
n = 1 gives 43. n = 2 gives 47. n = 3 gives 53. Every case up to n = 39 gives a prime. That is forty cases, all passing.
Then n = 40. The value is 1600 + 40 + 41 = 1681. And 1681 = 41 × 41. It is not prime. This one case is a counterexample — a case that breaks the claim. The claim is over.
What geometry needs instead
Geometry makes claims about every triangle anyone could draw. No pile of drawings can cover that. A claim about every case needs a proof. A proof starts from things we agree on. Then each step is forced by the one before.
Rule: A pile of passing cases is examples, not a reason. One failing case ends the claim.