1. Prove that if n is any integer, then 3n3 + n+ 5 is odd. (Suggestion: do one case for n odd and one case for n even.)
2. A cold-footed centipede has a drawer filled with many, many socks. And yes, that centipede does have 100 feet. If the centipede only owns green and brown socks, how many must it pull from the drawer in the dark of the morning to be assured that it has a matching set for all of its feet? What if the centipede also owns polka-dotted socks? What if the centipede's drawer has many, many socks of k different colors?