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?

Each gets 36 nights of sleep every 45 nights.

Step-by-step 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.

step-by-step explanation: