AMC 10 · 2017 · #6

Grade 6 number-theory
Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination
logical-deductioncontrapositiveif-then-reasoning contrapositive ↑ Prerequisites: logical-deduction
📏 Medium solution 💡 2 insights
Problem
A teacher promises: anyone who gets every multiple-choice question right will receive an A. Among five if-then statements about Lewis, decide which one MUST be true as a logical consequence of that promise.

Pick an answer.

(A)
If Lewis did not receive an A, then he got all of the multiple choice questions wrong.
(B)
If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.
(C)
If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A.
(D)
If Lewis received an A, then he got all of the multiple choice questions right.
(E)
If Lewis received an A, then he got at least one of the multiple choice questions right.

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Eliminate Possibilities

The promise is a one-way if-then: "all right → A." Tool #11 (Work Backwards) builds the one statement that must travel with it — the contrapositive, which flips the arrow and negates both sides. Tool #16 (Count the Complement) handles the tricky negation: the opposite of "all right" is "at least one wrong," not "all wrong." With the contrapositive in hand, Tool #3 (Eliminate Possibilities) tests all five choices against it; only the true contrapositive survives, and the converse/inverse traps fall away.

1STEP 1

Write the promise as one-way if-then

One way only: all right leads to an A. It never says an A comes only that way, so "A means all right" reads in too much.

Promise: (all right) → (A)
2STEP 2

Flip and negate: the contrapositive

Every if-then has one guaranteed partner — its contrapositive — made by reversing the arrow and negating both sides: not A → not all right.

(all right) → (A) ≡ (not A) → (not all right)
3STEP 3

Negate "all right" the careful way

"Not all right" means at least one wrong, not all wrong — a single missed question already breaks "all right."

not(all right) = (at least one wrong) ≠ (all wrong)
4STEP 4

Test all five, keep the contrapositive

Test all five: (A) too strong, (C) inverse, (D)/(E) backward — only the true contrapositive survives, so the answer is (B).

(A) ✗, (C) inverse ✗, (D) converse ✗, (E) ✗, (B) contrapositive ✓ → (B)
Answer
If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.
Stress-test (B) with a concrete case: suppose Lewis missed exactly one multiple-choice question. Then he did not get them all right, so the promise no longer guarantees his A. If he indeed did not receive an A, statement (B) says "at least one wrong," which is exactly true here. Meanwhile (A) would falsely demand he got them ALL wrong. The single-miss scenario confirms (B) and kills (A), (C), (D), (E).
💡Key takeaway

An "if you do X you get Y" promise only forces one extra fact — its contrapositive: "no Y means you didn't do X" — and remember the opposite of "all right" is just "at least one wrong," not "all wrong."

  • Write the promise as one-way if-then
  • Flip and negate: the contrapositive
  • Negate "all right" the careful way
  • Test all five, keep the contrapositive

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