AMC 10 · 2025 · #8
Grade 6 logicAgnes writes the following four statements on a blank piece of paper.
∙ At least one of these statements is true.
∙ At least two of these statements are true.
∙ At least two of these statements are false.
∙ At least one of these statements is false.
Each statement is either true or false. How many false statements did Agnes write on the paper?
Pick an answer.
AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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.
Givens: There are exactly four statements, and each is either true or false.; S1 says 'at least one of the four statements is true'.; S2 says 'at least two of the four statements are true'.; S3 says 'at least two of the four statements are false'.; S4 says 'at least one of the four statements is false'.
Unknowns: How many of the four statements are false.
Understand
Restated: 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.
Givens: There are exactly four statements, and each is either true or false.; S1 says 'at least one of the four statements is true'.; S2 says 'at least two of the four statements are true'.; S3 says 'at least two of the four statements are false'.; S4 says 'at least one of the four statements is false'.
Plan
Primary tool: #6 Guess and Check
Secondary: #4 Introduce a Variable, #16 Change Focus / Count the Complement, #3 Eliminate Possibilities
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.
Execute — Answer: B
6.EE.A.2 Step 1 Name the count of true statements
- Let T be the number of statements that are true.
- Then the number of false statements is 4 - T, because the four statements split into exactly true and false.
- T can only be 0, 1, 2, 3, or 4.
💡 Every statement only cares about counts, so a single number T carries all the information we need.
6.EE.B.8 Step 2 Turn each statement into a rule about T
- Each statement is a claim about a count, so rewrite it as a condition on T.
- 'At least one true' means T is 1 or more; 'at least two true' means T is 2 or more.
- For the two statements about false ones, use the complement: 'at least two false' means 4 - T is 2 or more, which is T at most 2; 'at least one false' means 4 - T is 1 or more, which is T at most 3.
💡 Counting the false ones is just 4 minus the true ones, so every statement becomes a simple size test on T.
6.EE.B.5 Step 3 Set the self-consistency rule
- A statement is true exactly when its rule holds.
- So the number of rules that hold is the same as the number of true statements, which is T.
- That gives the key requirement: for the guess to be consistent, the count of rules that come out true must equal T itself.
💡 The statements describe themselves, so a good value of T has to predict its own head count.
6.EE.B.5 Step 4 Try every value of T
- Go through T = 0, 1, 2, 3, 4 and count how many of the four rules S1..S4 hold for each.
- T = 0: only S3 and S4 hold, so 2 rules.
- T = 1: S1, S3, S4 hold, so 3 rules.
- T = 2: all four hold, so 4 rules.
- T = 3: S1, S2, S4 hold but S3 fails, so 3 rules.
- T = 4: only S1 and S2 hold, so 2 rules.
💡 Plug in each guess and simply tally which of the four size tests pass.
6.EE.B.5 Step 5 Eliminate the mismatches
- Keep only the value of T where the count of rules that hold equals T.
- T = 0 gives 2 (not 0), T = 1 gives 3 (not 1), T = 2 gives 4 (not 2), and T = 4 gives 2 (not 4), so all four are impossible.
- T = 3 gives 3, which matches.
- So T = 3 is the only consistent case: S1, S2, S4 are true and S3 is false.
💡 Only the guess that predicts its own head count can survive; the rest contradict themselves.
4.OA.A.3 Step 6 Count the false statements
- With T = 3 true statements, the number of false statements is 4 - 3 = 1.
- So exactly one statement (S3) is false, and the answer is (B).
💡 Once you know three are true, the leftover count of false ones is just four minus three.
6.EE.A.2 Let T be the number of statements that are true. Then the number of false statem 6.EE.B.8 Each statement is a claim about a count, so rewrite it as a condition on T. 'At 6.EE.B.5 A statement is true exactly when its rule holds. So the number of rules that hol 6.EE.B.5 Go through T = 0, 1, 2, 3, 4 and count how many of the four rules S1..S4 hold fo 6.EE.B.5 Keep only the value of T where the count of rules that hold equals T. T = 0 give 4.OA.A.3 With T = 3 true statements, the number of false statements is 4 - 3 = 1. So exac Review
Reasonableness: 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).
Alternative: Instead of testing all five values, reason from the ends. S1 and S4 can be checked fast: if all four were true, S4 ('at least one false') would be false, a contradiction; if all four were false, S1 would still have to be false yet 'at least one false' is clearly true, another contradiction. So T is strictly between 0 and 4, and a short check of the middle values again isolates T = 3, giving 1 false statement.
CCSS standards used (min grade 6)
6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Letting T stand for the number of true statements and writing the number of false statements as 4 - T.)6.EE.B.8Write an inequality of the form x > c or x < c and graph on a number line (Rewriting each 'at least' statement as an inequality on T, such as T >= 2 or T <= 2.)6.EE.B.5Understand solving an equation or inequality as a process of finding values (Testing each candidate value of T to see whether the number of rules it satisfies equals T, keeping only the value that makes the self-consistency requirement true.)4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Computing the number of false statements as 4 - 3 = 1 once T is known.)
⭐ 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.
⭐ 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.
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