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Minimum phone calls for secret exchange

CAT 2005 Slot 1 · Quantitative Ability · Hard · Logical Reasoning

This is a hard Quantitative Ability question from the CAT 2005 Slot 1 paper. It tests Logical Reasoning. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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Three Englishmen and three Frenchmen work for the same company. Each of them knows a secret not known to others. They need to exchange these secrets over person-to-person phone calls so that eventually each person knows all six secrets. None of the Frenchmen knows English, and only one Englishman knows French. What is the minimum number of phone calls needed for the above purpose?

  1. A.

    5

  2. B.

    10

  3. C.

    9

  4. D.

    15

Answer

C

Explanation

Let Frenchmen be F1,F2,F3F_1, F_2, F_3 and Englishmen be E1,E2,E3E_1, E_2, E_3, where E1E_1 knows French. Round 1 (Gathering secrets to E1E_1):

  1. F1F_1 calls F2F_2 (2 secrets)
  2. F2F_2 calls F3F_3 (3 secrets)
  3. F3F_3 calls E1E_1 (E1E_1 gets all 3 French secrets)
  4. E2E_2 calls E3E_3 (2 secrets)
  5. E3E_3 calls E1E_1 (E1E_1 gets all 6 secrets) Calls used =5= 5.

Round 2 (Distributing secrets from E1E_1): 6. E1E_1 calls E3E_3 (E3E_3 gets all 6) 7. E3E_3 calls E2E_2 (E2E_2 gets all 6) 8. E1E_1 calls F3F_3 (F3F_3 gets all 6) 9. F3F_3 calls F2F_2, F2F_2 calls F1F_1 / etc. Minimum total calls required =9= 9.

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