AMC 10 · 2025 · #8

Grade 6 logic

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

logical-deductionif-then-reasoning casework ↑ Prerequisites: logical-deduction

📏 Long solution 💡 2 insights

Problem

Four statements are written down, each one either true or false, and each one just says how many of the four are true or false ('at least one true', 'at least two true', 'at least two false', 'at least one false'). Using only the statements themselves, decide how many of the four end up false.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Guess and Check

Each statement only talks about how many statements are true or false, so the whole puzzle depends on one number: how many are true. Call it T. There are only five possible values, T = 0, 1, 2, 3, 4, so we can simply try each one and check whether it is self-consistent. The catch is that the statements describe themselves, so the value of T we guess must produce exactly T true statements. Testing all five values and throwing out the ones that clash is faster and safer than trying to reason out the 'right-looking' split directly.

1STEP 1

Name the count of true statements

Let T be the number of true statements; then the number of false ones is 4 - T, and T is 0, 1, 2, 3, or 4.

T = (number true), (number false) = 4 - T, T ∈ {0,1,2,3,4}

2STEP 2

Turn each statement into a rule about T

Rewrite each as a test on T: S1 is T at least 1, S2 is T at least 2, S3 is T at most 2, S4 is T at most 3.

S1: T ≥ 1, S2: T ≥ 2, S3: 4 - T ≥ 2 → T ≤ 2, S4: 4 - T ≥ 1 → T ≤ 3

3STEP 3

Set the self-consistency rule

A statement is true exactly when its test holds, so the number of tests that hold must equal T itself.

(number of rules that hold) = T

4STEP 4

Try every value of T

Tally the tests that hold at each T: T = 0 gives 2, T = 1 gives 3, T = 2 gives 4, T = 3 gives 3, T = 4 gives 2.

T=0 :2, T=1 :3, T=2 :4, T=3 :3, T=4 :2 (rules that hold)

5STEP 5

Eliminate the mismatches

Only 0, 1, 2 and 4 clash with their own tally, so T = 3 survives: S1, S2, S4 true and S3 false.

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

6STEP 6

Count the false statements

With T = 3 true, the count of false statements is 4 - 3 = 1, namely S3 alone, so the answer is (B).

4 - T = 4 - 3 = 1

Answer

Check the surviving case directly: with S1, S2, S4 true and S3 false, exactly one statement is false. S1 ('at least one true') holds since three are true; S2 ('at least two true') holds; S4 ('at least one false') holds since one is false; and S3 ('at least two false') is correctly false since only one is false. Everything lines up, and T = 3 is the only value out of five that survives, so the count of false statements is pinned down as 1, matching choice (B).

💡Key takeaway

When statements talk about themselves, guess how many are true, count how many rules that makes come out true, and keep only the guess that matches its own count.

Name the count of true statements
Turn each statement into a rule about T
Set the self-consistency rule
Try every value of T
Eliminate the mismatches
Count the false statements