## 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. Patson says:

18?

2. Vikas Desai says:

#52: 12 people.

• Vikas Desai says:

#52: Think I was wrong, 14 would be the right answer. Can anyone confirm?

3. 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.