Puzzle #52: How Many People Partying?

Thanks to Anirudh Baddepudi for contributing this puzzle. I also do not know the answer yet and hence looking forward to enjoying this one.

At a party, seven people know seven others, and the remaining people each know 5 other people who are at the party. Knowing is mutual, so if person A knows person B, then person B knows person A as well. From this information, what is the smallest number of people that could have been at the party?

Happy partying!

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4 Responses to Puzzle #52: How Many People Partying?

  1. Vikas Desai says:

    #52: 12 people.

  2. challapali says:

    The smallest number of people that could have been at the party is 10.

    Say there ‘x’ other people who knew 5 others each.
    Now, consider a Graph(G) with (7+x) vertices each vertex representing a person and an edge between two vertices (A, B) represents A knows B.
    Total vertices = (7*7 + x*5)/2
    Which implies x must be odd.

    To find minimum x. Lets consider a complete graph with 7+x vertices.
    Total vertices = (7+x)(6+x)/2

    Number of vertices to be taken out from the current to get G = (x^2 + 8x – 7)/2.

    When x = 1, this number will be 1, which is not possible.
    So any x > 1 and which is odd satisfies the condition. So minimum x is 3.

    Minimum number of persons is 10.

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