AMC 10 · 2016 · #5
Grade 6 number-theoryPick an answer.
The conjecture is a claim about every even number greater than 2, so breaking it is not a matter of finding a pattern — it is a matter of finding one number where the claim fails. Switching focus from "all cases work" to "one case fails" produces an exact template with two required halves, and each of the five descriptions can then be held against that template. Concrete test numbers finish the job: a description that is already satisfied by some number alive today cannot be a counterexample, because the problem says no counterexample has ever been found.
Write the claim as if-then
The claim is one if-then statement.
Naming the number turns a sentence into a rule with one clear condition and one clear promise.
6.EE.A.2Introduce A VariableSee what breaks a for-every claim
A counterexample needs both halves.
A claim about every case is only as strong as its weakest case, so one number that meets the condition and misses the promise sinks the whole thing.
A claim about every case is only as strong as its weakest case, so one bad number sinks it.
▸ Why?
A single case that meets the condition and misses the promise refutes the whole statement.
▸ Why?
Cases that never meet the condition say nothing at all, so only the qualifying ones can testify.
Odd numbers cannot testify
A number of the wrong kind cannot testify.
You cannot break a promise that was never made about your number.
2.OA.C.3Eliminate PossibilitiesA bad split is not a failure
A single bad split is not a failure.
Finding one bad way to split a number does not erase the good way that also exists.
4.OA.B.4Eliminate PossibilitiesAn example is not a counterexample
A working example is not a counterexample.
Cases that obey a rule can pile up forever without ever breaking it.
4.OA.B.4Guess And CheckKeep the one that fails
Only one candidate breaks it, choice (E).
A counterexample is the claim's opposite packed into one number: condition met, promise broken.
6.EE.B.5Change Focus Count The ComplementTo break a claim that says "every", you need one case that fits the claim's condition and still fails its promise — nothing weaker counts.
- Write the claim as if-then
- See what breaks a for-every claim
- Odd numbers cannot testify
- A bad split is not a failure
- An example is not a counterexample
- Keep the one that fails