AMC 10 · 2010 · #15

Grade 2 logic
Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination
logical-deductioncaseworkif-then-reasoning contradiction-elementary ↑ Prerequisites: logical-deduction
📏 Medium solution 💡 3 insights
📘 View easy version →
Problem
Four amphibians — Brian, Chris, LeRoy, and Mike — live together in a swamp, and each one is either a toad, whose statements are always true, or a frog, whose statements are always false. Brian says, "Mike and I are different species"; Chris says, "LeRoy is a frog"; LeRoy says, "Chris is a frog"; Mike says, "Of the four of us, at least two are toads". How many of these four amphibians are frogs?

Pick an answer.

(A)
0
(B)
1
(C)
2
(D)
3
(E)
4

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Eliminate Possibilities

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.

1STEP 1

Brian's claim forces Mike to be a frog

Brian a toad makes his claim true, so Mike differs; Brian a frog makes it false, so Mike matches him — either way Mike is a frog.

Brian toad→ Mike frog; Brian frog→ Mike frog
2STEP 2

Mike's lie caps the toads at one

A frog lies, so Mike's "at least two are toads" is false, and its opposite says there is at most one toad among all four.

lnot(toads ≥ 2) ⟺ toads ≤ 1
3STEP 3

Chris and LeRoy: exactly one is a toad

Chris and LeRoy call each other frogs, and such mirrored accusations cannot both hold or both fail, so exactly one of them is a toad.

(Chris toad, LeRoy frog) or (Chris frog, LeRoy toad)
4STEP 4

Combine the facts and count the frogs

That toad uses up the only slot allowed, so Brian, Mike, and the other of Chris and LeRoy all lie: 3 frogs and one toad, answer (D).

1 toad + 3 frogs = 4 → (D) 3
Answer
3
Test the full assignment Brian frog, Mike frog, Chris toad, LeRoy frog. Brian (frog) says "Mike and I are different" — false, and both are frogs, so same species: correct lie. Chris (toad) says "LeRoy is a frog" — true. LeRoy (frog) says "Chris is a frog" — false, and Chris is a toad: correct lie. Mike (frog) says "at least two are toads" — false, and there is only one toad. Every statement is consistent, confirming 3 frogs. The mirror case (Chris frog, LeRoy toad) works identically, and any count other than 3 would need a different number of toads, which the two constraints forbid.
💡Key takeaway

When 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

More like this

Same archetype — closest grade level first.