Conjecture, test and justify — Mathematics, 11–13
Spot a pattern, state a conjecture, and learn that testing cases is not the same as validating — with counterexamples and other ways to justify.
A conjecture is an informed hunch
A conjecture is a statement you believe because you saw a pattern. 2 + 3 = 5, 7 + 8 = 15, 20 + 21 = 41: the sum of two consecutive numbers seems always odd. Classifying helps you see patterns: in 12, 18, 24, 30, 36 and 45, five are multiples of 6 and 45 is the odd one out. Then comes the important part: you test the conjecture on cases, and later you validate it. Those are two different things.
Testing is not enough: four ways to justify
Testing tries cases; validating shows the statement holds for every case, or fails in one. There are different ways to justify. Counterexample: "all primes are odd" is false, because 2 is prime and even. Exhaustion: check every possible case (the multiples of 6 up to 20 are 6, 12 and 18, all even). Generic example: an argument that does not depend on the chosen number. Logical coherence: even + odd is odd. Each suits a different kind of statement.
A case: the sum of two consecutive numbers
Conjecture: the sum of two consecutive numbers is odd. Tests: 2 + 3 = 5, 7 + 8 = 15, 20 + 21 = 41, 99 + 100 = 199, all odd — but that only tests. To validate, use a generic example: 14 + 15 = 14 + 14 + 1 = 2 × 14 + 1, twice a number plus 1. Twice a number is even, and plus 1 is odd. Nothing here depends on 14: with a letter, n + (n + 1) = 2 × n + 1. So it is true for every pair of consecutive numbers.
"It passed three tests, so it's true"
Rui noticed that 3, 13 and 23 are prime (a prime has only two divisors: 1 and itself). His conjecture: every number ending in 3 is prime. He tried one more, 33. But 33 = 3 × 11, so it has more divisors. A single counterexample brings a conjecture down, however many tests it has passed. How to avoid the trap: go looking for the awkward case, and ask "why would it always be so?".
Where you see it: "always" and "never"
Whenever you hear "all", "always" or "never", you are facing a conjecture. "All multiples of 5 end in 5"? Look at 5, 10, 15, 20: the 10 is a counterexample, so it is false. To confirm a statement you need a reason that covers every case. People who write games and apps do the same: they try strange values (0, 1, the biggest possible) precisely to hunt for the counterexample.
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