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Can You Solve The Spies Sharing Secrets Puzzle?

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Channel: MindYourDecisions
Categories: Math  
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After a successful mission, each of N spies obtains a unique secret. By protocol, the spies share secrets by a series of phone calls between 2 spies at a time. During a call, a spy can share all the secrets the knows, including secrets learned from previous calls. What is the minimum number of phone calls so that all spies learn all secrets? Solve if the phone call is 1-way (only the caller can transmit information) and if the phone call is 2 way (both spies can learn each other's information).

Video with proofs of minimum number of calls:

Blog post (with links to proofs and many references):

This puzzle is also known as the "gossip problem," and MathWorld has an entry on that:

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