1: Place a knight on each square of a 7x7 chessboard. Is it possible for each knight to simultaneously make a legal move?
2: Suppose n is an odd integer. The numbers 1, 2, . . . , 2n are written on a blackboard. I pick two arbitrary numbers on the blackboard, say a, b with b ≥ a, erase them, and write down b - a on the blackboard. I repeat this process until only one integer is left. Is the number at the end odd or even?
3: At first, a room is empty. Each minute, either four people enter or one person leaves. After exactly 1 hour, could the room contain 101 people?
4: There is one stone at each vertex of a square. In any move, you can change the number of stones at each corner according to this rule: you may take away any number of stones at one vertex and add twice as many stones to the pile at an adjacent vertex. Is it possible to get 2003, 2004, 2005, 2006 stones at consecutive vertices after a finite number of moves?
5: Consider an 8 × 8 × 8 cube with two diagonally opposite corners removed. For which n is it possible to completely fill this object with 1 × 1 × n boxes?
6: Are you Positive about this Polynomial? Suppose f(x) is an unknown polynomial with unknown degree and non-negative integer coefficients. Your goal is to determine f(x), but you are only allowed to ask questions of the following form: for a specific number k, "what is f(k)?". What is the fewest number of questions needed to determine f(x)?