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Generalising Urgent Meetings

First see the following problem ::

There are M people at a conference. Initially nobody at the conference knows the name of anyone else. The conference holds several N - person meetings in succession, in which each person at the meeting learns (or relearns) the name of the other people present in that meeting. What is the minimum number of meetings needed until every person knows everyone elses name?

I found that if M = 2N and N is divisible by 2 , then answer is 6 ; if M = 3N and N is divisible by 3 , then answer is 12..

In other words if M is an integral multiple (let the integer be G) of N, and N is divisible by that same integer (G), then the answer can be found out easily (unless the integer is large i.e. of 2 digits in which case the cases will become numerous).

Is there any easy way of computing if G > 10 ?

Can I generalise it for (say) M = 4N + 5 or M = 5N - 4 ?

Can I generalise it for any M , N ?

Note by Santanu Banerjee
4 years ago

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The actual problem was ::

There are 32 people at a conference. Initially nobody at the conference knows the name of anyone else. The conference holds several 16-person meetings in succession, in which each person at the meeting learns (or relearns) the name of the other fteen people. What is the minimum number of meetings needed until every person knows everyone elses name? Answer: 6 .

AoPS

Santanu Banerjee - 4 years ago

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