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!
Alok Goyal's Puzzles Page 
18?
#52: 12 people.
#52: Think I was wrong, 14 would be the right answer. Can anyone confirm?
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.