AMC 10 · 2010 · #12
Grade 2 logicPick an answer.
Tool #3 (Eliminate Possibilities): each amphibian has only two possible species, so I test a species, check whether it makes the statements consistent, and throw out any choice that forces a contradiction. Tool #16 (Count the Complement): Mike lies about "at least two are toads," so the useful fact is the opposite quantity — "at most one toad" — which caps the whole count. Tool #7 (Identify Subproblems): I settle Mike first, then the Chris-LeRoy pair, then combine those partial results into the final frog count.
Brian's claim forces Mike to be a frog
One claim pins a speaker down either way.
When both possible species for Brian lead to the same conclusion, that conclusion is locked in.
When both possible species for the speaker lead to the same conclusion, that conclusion is locked in.
▸ Why?
The speaker is one species or the other and never both, so the two cases cover everything.
▸ Why?
When every case that remains forces the same answer, no alternative survives to be considered.
Mike's lie caps the toads at one
That liar's false claim caps the truth-tellers at one.
Flipping a false "at least" statement turns it into a hard ceiling on the count.
1.OA.A.1Change Focus Count The ComplementChris and LeRoy: exactly one is a toad
The mutual accusation gives exactly one truth-teller.
Two opposite accusations can't both be true or both be false, so they split into one truth-teller and one liar.
1.OA.D.7Eliminate PossibilitiesCombine the facts and count the frogs
So 3 of the four are liars, choice (D).
One guaranteed toad plus a ceiling of one toad means every remaining amphibian must be a frog.
2.OA.A.1Identify SubproblemsWhen both guesses for one speaker lead to the same result, that result is certain — then a liar's "at least" claim flips into a firm ceiling that pins down the rest.
- Brian's claim forces Mike to be a frog
- Mike's lie caps the toads at one
- Chris and LeRoy: exactly one is a toad
- Combine the facts and count the frogs