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The relay of messages

Master playsLevel 4 · Advanced · ●●●●○

Four spies A, B, C and D each have a different secret. In each phone call, the two participants tell each other everything they know up to that moment.

Question 1: What is the minimum number of calls for all four to know the four secrets? Question 2: For $n$ spies (with $n\ge4$), what is the overall minimum?

Hints

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  1. Optimal strategy: A calls B $\Rightarrow$ A and B know $\{a,b\}$.
  2. With 3 calls it is not possible for the four to accumulate complete information.
  3. Response (4 spies): minimum 4 calls.

Solution

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Response (4 spies): minimum 4 calls.
Optimal strategy:

  1. A calls B $\Rightarrow$ A and B know $\{a,b\}$.
  2. C calls D $\Rightarrow$ C and D know $\{c,d\}$.
  3. A calls C $\Rightarrow$ A and C know $\{a,b,c,d\}$.
  4. B calls D $\Rightarrow$ B and D also know everything.

With 3 calls it is not possible for the four to accumulate complete information.
Generalization: for $n\ge4$, the minimum is:

$$ 2n-4. $$


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