The case that breaks it
A claim is a statement that says something is true. Here is one. Multiply a number by the next number, then add 11. The answer is always prime. A prime has exactly two factors, 1 and itself. The claim was checked nine times and held each time.
- 1 × 2 + 11 = 13. Prime.
- Keep going: 2×3+11, 3×4+11, up to 9×10+11 = 101. All prime.
- 10 × 11 + 11 = 121.
- 121 = 11 × 11. Not prime. The claim breaks.
One case is enough
Nine good cases in a row can look like proof. But they are not proof: nine tries are not every number.
121 has three factors, so 121 is not prime. A case that shows a claim false is a counterexample.
The rule
A claim that says always true is false if one case fails. Find none? Then all you know is that you have not found one yet.
The Collatz path gives a claim like that. Start with any number. Halve it if it is even. If it is odd, triple it and add 1. Repeat on each answer. Every start tried reaches 1. Nobody has a proof for every start.
The two jobs are not the same size. Showing a claim false takes one case. Proving a claim takes a reason that covers every number. Trying every number is impossible, because the numbers go on without end.
Pick a claim. Try cases, or type your own. Stop when a case fails.
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