Problem

Source: 2003 Estonia National Olympiad Final Round grade 12 p5

Tags: combinatorics, game



On a lottery ticket a player has to mark $6$ numbers from $36$. Then $6$ numbers from these $36$ are drawn randomly and the ticket wins if none of the numbers that came out is marked on the ticket. Prove that a) it is possible to mark the numbers on $9$ tickets so that one of these tickets always wins, b) it is not possible to mark the numbers on $8$ tickets so that one of tickets always wins.