Count True and False Statements Precisely
准确统计真话与假话
Logic words such as exactly, at least, and one false are mathematical conditions. Translate the truth quota first, then test statements without confusing false with not yet known.
Translate the truth rule before solving
The same three statements behave very differently under “exactly one true” and “exactly one false.” Write the quota in numbers first.
Translate each rule
Separate false from undecided — and spot complements
A statement is false only when the facts contradict it. Before the value is known, the statement may simply be undecided.
Check the language
Audit a puzzle with exactly one false claim per speaker
Each student makes three claims. Every speaker has exactly one false claim, so every speaker must have two true claims.
Complete apple-count puzzle
Liu Hong: “I have 22 apples.” “I have two fewer than Chen Ming.” “I have one more than Li Xiaoming.”
Chen Ming: “My count is greater than at least one of the other two counts.” “Li Xiaoming’s count differs from mine by 3.” “Li Xiaoming has 25 apples.”
Li Xiaoming: “I have fewer apples than Liu Hong.” “Liu Hong has 23 apples.” “Chen Ming has 3 more apples than Liu Hong.”
Each person has exactly one false statement. Find all three apple counts.
Test a candidate triple
The board evaluates all nine claims.
Study the worked apple example (reveals the counts)
This example leaves your entries unchanged. Liu, Chen and Li have 23, 25 and 22 apples respectively. Liu’s claims are false, true, true; Chen’s are true, true, false; Li’s are true, true, false. Each speaker has exactly one false claim. Mission 4 explains why the other branch fails.
Submit the valid triple
Follow an efficient proof chain through the apple puzzle
When several claims are comparisons, begin with a claim containing a definite number. Reveal one justified step at a time.
Assume Liu Hong’s first claim, “I have 22 apples,” is false.
Liu’s other two claims are true: Chen has 2 more than Liu, and Li has 1 fewer than Liu.
Chen is not the smallest, and Chen differs from Li by 3. Those two are true, so “Li has 25” is false.
Li is below Liu, while “Chen has 3 more than Liu” is false. Therefore “Liu has 23” must be true.
Liu = 23, Chen = 25, Li = 22. Each speaker has 2 true claims and 1 false claim.
Check the proof links
Use complementary claims to solve the apple-box puzzle
A count may be zero. The first two estimates already use up the one allowed true statement.
Complete original practice
- “There are at least 20 apples.”
- “There are fewer than 20 apples.”
- “There is at least one apple.”
Exactly one estimate is correct. How many apples are in the box?
Explain with objective choices
Check a self-referential truth count for consistency
Five people each make one claim. Here “lying” means that this particular quoted claim is false, not that the person always lies. They make different claims about how many of these five claims are false. The correct claim must create exactly the liar count it announces.
Complete practice problem
- “All five of us are lying.”
- “Exactly one of us is lying.”
- “Exactly two of us are lying.”
- “Exactly three of us are lying.”
- “Exactly four of us are lying.”
Compare the assumed count with the number of false claims it creates.
Find the consistent count
Split compound replies into separate clauses
Here, each person says two clauses: exactly one clause is true and one is false. Count the clauses separately.
Complete bad-deed puzzle
Exactly one of A, B, and C did a bad deed.
A: “I did not do it, and B did not do it.”
B: “I did not do it, and C did not do it.”
C: “I did not do it, and I do not know who did it.”
Each person says one true clause and one false clause.
Record the valid branch
Transfer exact truth counts to three quick mysteries
Test every possible case and count the true statements; do not choose by tone or personality.
Good deed
Exactly one of A, B, C, and D did the good deed, and exactly one of their four quoted claims is true.
- A: “B did it.”
- B: “D did it.”
- C: “I did not do it.”
- D: “B’s claim that D did it is false.”
Who can drive?
Exactly one of A, B, C can drive, and exactly one statement is true.
- A: “I can drive.”
- B: “I cannot drive.”
- C: “A cannot drive.”
Who is the reporter?
A, B, C have different jobs; exactly one is the reporter and exactly one statement is true.
- A: “I am the reporter.”
- B: “I am not the reporter.”
- C: “A’s claim that A is the reporter is false.”
Check all three
The choices here and beside the puzzles stay in sync. You can enter or change each answer in either place.
Fresh workshop
Try all six fresh checks. Use a hint after your first attempt.
A box holds a nonnegative whole number of beads. Three claims are made: “At least 12”; “Fewer than 12”; “At least one.” Exactly one claim is true. Separately, a three-statement speaker has exactly one false statement.
One hint
The first two claims cover every possible count without overlap.
Worked reasoning — open after trying
One of the first two is always true. The third must be false, so the count is zero. At zero the claims are F,T,F. Exactly one false among three means two true.
Fresh exit challenge
Complete Missions 2 and 4, this workshop, and all five exit checks to earn your certificate. Reflection and extra exploration are optional.
One of three boxes A, B, C contains a token. The claims are: “The token is in A”; “The token is not in A”; “The token is not in B.” Exactly one claim is true.
One hint
The complement pair already supplies one truth.
Worked reasoning — open after trying
The third claim must be false, so the token is in B. The claims are F,T,F. At C they would be F,T,T.
Lesson checkpoints completed
Precise Truth-Count Detective
You completed the checkpoints on exact truth counts, undecided claims and clause-by-clause audits. Revisit any steps for which you needed solution help.
Teaching notes
Every original puzzle used here is restated in English.The page adds interactive truth audits and objective checks.