Question

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?

• A. \(10\)
• B. \(9\)
• C. \(8\)
• D. \(7\)
• E. \(6\)

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
There are 6 individuals (3 English, 3 French), each holding 1 unique secret.
They must share all secrets via phone calls where participants mutually exchange all knowledge they currently have.
The communication is heavily restricted by a language barrier: only one specific Englishman speaks French.
We need to find the absolute minimum number of calls required for everyone to learn all 6 secrets.
Step 2: Key Formula or Approach:
This is a variation of the classic "Gossip Problem".
Normally, for $N$ people with full connectivity, the minimum number of calls is $2N - 4$ (which would be 8 calls here).
However, the graph here is a bipartite-like structure with a severe bottleneck: the bilingual Englishman must act as the sole bridge for all cross-language data transfers.
We must trace the optimal routing of secrets to this central hub and back out.
Step 3: Detailed Explanation:

Let the bilingual Englishman be denoted as E1.

Let the other two Englishmen be E2 and E3. They only speak English and can only talk to each other or E1.

Let the three Frenchmen be F1, F2, and F3. They only speak French and can only talk to each other or E1.

Because only E1 can cross the language barrier, all 3 French secrets must reach E1, and all 3 English secrets must reach E1.

First, we consolidate the secrets within the groups before crossing the bridge.

Call 1: E2 calls E3. Both now know 2 English secrets.

Call 2: E3 calls E1. E1 now knows all 3 English secrets.

Call 3: F2 calls F3. Both now know 2 French secrets.

Call 4: F3 calls F1. F1 now knows all 3 French secrets.

Call 5: E1 calls F1. This is the crucial bridge call. E1 shares the 3 English secrets, and F1 shares the 3 French secrets.

After Call 5, both E1 and F1 know all 6 secrets.

Now, the complete information must be disseminated back to the remaining 4 people.

Call 6: E1 calls E3. E3 now knows all 6 secrets.

Call 7: E3 calls E2. E2 now knows all 6 secrets.

Call 8: F1 calls F3. F3 now knows all 6 secrets.

Call 9: F3 calls F2. F2 now knows all 6 secrets.

Everyone now knows all 6 secrets, requiring exactly 9 phone calls.

The theoretical 8-call minimum requires a fully connected 4-person cycle at the core, which is structurally impossible here due to the language barrier preventing any English-French links other than through E1.

Step 4: Final Answer:
The minimum number of phone calls needed is 9.