Question

There are ten meerkats in a colony. Every night, two meerkats stand guard while the others sleep. During a certain period of nights, every meerkat stands guard with every other meerkat exactly once. During that period, how many nights of sleep does each meerkat get?

Answer 1

Each gets 36 nights of sleep every 45 nights.

Explanation:

This is called the handshake problem, which is the same problem as n guests (including host) attend a party. At the end, everyone shakes hands with everyone else. The number of handshakes is each person shakes hands with (n-1) other persons to give n(n-1) handshakes. However, this counts each handshake twice (A with B and B with A), so the actual number of handshakes is n(n-1)/2

For n=10, then number of "teams" of two meerkats is 10(10-1)/2 = 45.

So each meerkat gets to be on duty 9 nights out of 45, or gets 36 nights of sleep every 45 nights.