AMC 10 · 2025 · #4

Grade 6 logic

Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination

logical-deductionif-then-reasoningcasework systematic-enumeration ↑ Prerequisites: logical-deduction

📏 Medium solution 💡 2 insights

Problem

Agnes writes four statements, each claiming how many of the four are true or false: "at least one is true," "at least two are true," "at least two are false," and "at least one is false." Every statement is either true or false. Find how many of the four end up false.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Introduce a Variable

The statements refer to each other, which feels circular until you name one number. Tool #4 (Introduce a Variable) sets T as the number of true statements, and every claim collapses into a simple size check on T. Tool #13 (Convert to Algebra) rewrites each sentence as an inequality. Then the trick: whatever T you pick must make exactly T of those inequalities true. Tool #6 (Guess and Check) tries all five possible values of T, and Tool #3 (Eliminate Possibilities) throws out every value that contradicts itself, leaving the one count that fits.

1STEP 1

Name the number of true statements

Name the count: let T be the number of true statements, so 4 - T are false, with T one of 0, 1, 2, 3, 4.

T = number of true statements, 4 - T = number of false statements

2STEP 2

Turn each sentence into an inequality

Rewrite each claim on T: S₁ is T ≥ 1, S₂ is T ≥ 2, S₃ is T ≤ 2, S₄ is T ≤ 3; each is true exactly when its inequality holds.

S₁: T ≥ 1, S₂: T ≥ 2, S₃: T ≤ 2, S₄: T ≤ 3

3STEP 3

Set the self-check rule

Key link: the number of these four inequalities that hold must equal T itself, or that value of T is impossible.

#{inequalities that hold at this T} = T

4STEP 4

Test all five possible counts

Testing each: T=0→2, T=1→3, T=2→4, T=4→2 all clash, only T = 3 gives three true, so 4 - 3 = 1 false, choice (B).

T=0→2, T=1→3, T=2→4, T=3→3 ✓, T=4→2 → 4-T = 1

Answer

Plug T = 3 back into the original sentences: three statements true and one false. "At least one true" — true. "At least two true" — true. "At least two false" — false, since only one is false. "At least one false" — true. That is three true and one false, exactly matching T = 3, so the answer 1 false statement is self-consistent.

💡Key takeaway

When sentences describe how many of themselves are true, name that count and keep only the count that predicts itself.

Name the number of true statements
Turn each sentence into an inequality
Set the self-check rule
Test all five possible counts

Name the number of true statements

Turn each sentence into an inequality

Set the self-check rule

Test all five possible counts

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