The following problem appeared in an undergraduate problems competition (The Joe Konhauser Problemfest, Macalester College April 17, 1993). The game that is described here is a simplification of the Mass Millions game. Currently, Mass. Megabucks is exactly like this game, but you only pick six numbers from 1, 2, 3, ..., 42.
How to Win at the Lottery
A $1 ticket in the Massachusetts Lottery consists of 6 different numbers chosen from 1, 2, 3, ..., 48. On lottery day, 6 numbers from this set are chosen at random (without repetition); a winning ticket is one that has at least 5 of these 6 numbers (order is irrelevant). Show that if one buys all tickets for which the sum of the entries is divisible by 47, then one is guaranteed of having a winner.