Four spies—A, B, C and D—each have a different secret. Every time two spies talk on the phone, they both tell each other everything they know at that moment. What is the minimum number of calls necessary for the four of them to know the four secrets?
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The relay of messages
Four secrets start separately and each call mixes everything its two participants know. The key is not to pass messages one by one, but to have two pairs accumulate information before crossing them.
Hints
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- A call does not transmit just a secret: it transmits everything accumulated by both spies.
- Start by forming two pairs: this way each pair brings together two secrets.
- Then cross the pairs so that complete information reaches everyone.
Solution
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Answer: 4 calls are needed. A sequence that achieves this is: 1. A calls B. A and B know A and B's secrets.
- C calls D. C and D know C and D's secrets.
- A calls C. A and C exchange what they know, so they both know the four secrets.
- B calls D. B and D exchange what they know, so they both know the four secrets. In the end, the four spies know the four secrets. 3 calls cannot be enough. With only 3 calls between 4 people, at least one person participates in a single call. That person can only learn what his interlocutor knew at that moment; It cannot receive information that arrives later through other calls. Therefore, not everyone can finish the four secrets. So the minimum is 4.
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