A hard question on logic puzzles for P4–P5, with a step-by-step solution, diagrams and the answer. The Hua Xia Cup's P4 qualifying round and finals have true-or-false statements and match-scoring questions. A favourite thinking workout, too.
The question
At a class party, 5 pupils each draw a card saying True or False.
Anyone with a True card only tells the truth; anyone with a False card only lies.
At least one of them has a True card.
Each says one thing.
Number 1:
"Exactly 3 of us have False cards."
Number 2:
"Exactly 1 of us have False cards."
Number 3:
"Exactly 4 of us have True cards."
Number 4:
"Exactly 2 of us have True cards."
Number 5:
"Exactly 1 of us have False cards."
How many of them have a True card?
Show the hints (3)
💡Hint 1: Everyone with a True card tells the truth. Could they disagree about how many True cards there are?
💡Hint 2: Turn every "so many False cards" into "so many True cards" (there are 5 pupils).
💡Hint 3: If v pupils have True cards, then exactly v pupils must say "v have True cards". Try each number.
Worked solution
Have a go first, then read the solution step by step.
1.Everyone with a True card states the same fact, so they all give the same count of True cards: the real one. Anyone with a False card is lying, so never says that count.
2.With 5 pupils, each statement as a count of True cards: number 1 says 2, number 2 says 4, number 3 says 4, number 4 says 2, number 5 says 4.
3.If v pupils hold True cards, exactly v must say "v": 2 say 2 (fits); 3 say 4 (no).
4.A count nobody says can't be right either: there is at least one True card, and its holder states the real count.
5.So 2 pupils hold True cards.
⚠️Common mistake: trusting a number because many people say it. The number of people saying it must equal the number itself.
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