Problem 19 · 2025 Math Kangaroo
Hard
Logic & Word Problems
casework
Three square Martians and three round Jupiterians are sitting at a table as shown. One of the six has the key to the spaceship. Everyone from one planet always tells the truth, and everyone from the other planet always lies. When asked “Does any of your neighbours have the key?” all six answer as shown. Who has the key?
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Answer: B — B
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Hint 1 of 3
The key-holder's two neighbours truly have a neighbour with the key, while everyone non-adjacent to the key truly does not.
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Hint 2 of 3
Whoever holds the key answers about their own neighbours, who do NOT have it—so the holder's truthful answer would be "No."
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Hint 3 of 3
Test each candidate and keep the one that yields exactly three truth-tellers and three liars.
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Approach: assume each candidate holds the key and count consistent types
- Around the table the neighbours are \(A\!-\!B\!-\!C\!-\!D\!-\!E\!-\!F\!-\!A\); the answers are A:Yes, B:Yes, C:No, D:No, E:No, F:Yes.
- Suppose \(B\) has the key: then \(A\) (neighbour) truly says Yes; \(B\) sees no key neighbour yet says Yes (lie); \(C\) says No but a neighbour has it (lie); \(D,E\) truly say No; \(F\) says Yes but neither neighbour has it (lie).
- That is truth-tellers \(A,D,E\) and liars \(B,C,F\)—exactly three each, the only candidate that works—so the key is with B, choice (B).
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