Problem 17 · 1987 AJHSME
Hard
Logic & Word Problems
negate-both-clues
Abby, Bret, Carl, and Dana are seated in a row of four seats numbered #1 to #4. Joe looks at them and says:
"Bret is next to Carl." "Abby is between Bret and Carl."
However each one of Joe's statements is false. Bret is actually sitting in seat #3. Who is sitting in seat #2?
Show answer
Answer: D — Dana.
Show hints
Hint 1 of 3
The clues are FALSE — that's the gift. Each false 'X is here' tells you X is NOT there, which is often more powerful than a true clue.
Still stuck? Show hint 2 →
Hint 2 of 3
Start from the fact you're handed: Bret is in #3. 'Bret is next to Carl' is false, so Carl can't be in either seat touching #3.
Still stuck? Show hint 3 →
Hint 3 of 3
Once Carl is forced, the second false clue ('Abby between Bret and Carl') eliminates exactly one seat for Abby — the seat strictly between Carl and Bret.
Show solution
Approach: negate each false clue, then place seats one at a time
- Bret sits in #3. Clue 1 ('Bret next to Carl') is false, so Carl is NOT in #2 or #4 — the only seat left for Carl is #1.
- With Carl in #1 and Bret in #3, the seat strictly between them is #2. Clue 2 ('Abby between Bret and Carl') is false, so Abby is NOT in #2. The remaining seats for Abby and Dana are #2 and #4, so Abby takes #4 and Dana takes #2.
- Seat #2 holds Dana.
- Why this transfers: a false 'A is next to B' statement is a constraint in disguise — flip it to 'A is NOT next to B' and use elimination just like a true clue.
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