1260.21 – Handshakes; Fly on the Wall

At the start of a political meeting, everybody shakes hands with everybody else exactly once. A fly on the wall (possibly a spy) somehow manages to observe 36 handshakes total. How many people were at the meeting?

Solution

If there are $n$ people at the meeting, each person shakes hands $(n-1)$ times. The tempting total number of handshakes is $n(n-1)$ , but if you think about it you see that each handshake is counted twice. So the formula for the total number of handshakes is $\dfrac{n(n-1)}{2}$ . Set this equal to 36 and you'll arrive at 9 people.

(Questions: why was the fly counting handshakes? Could any fly be quick enough to get an accurate count? What if the fly saw only 34 handshakes? Would it still be possible to come up with 9 people? Why not just count the people instead of the handshakes in the first place?)