Puzzles Hilary Term 2013

1. A 23×23 square is divided into smaller squares of dimensions 1×1, 2×2 and 3×3. What is the minimum possible number of 1×1 squares?
2. In what base of notation is the multiplication 166 x 56=8590 correct?
3. There is a company sitting at a round-table. They all leave the table for a while, and after returning to the table they all find that their neighbours differ from those they had previously. How many people may be sitting around the table?
4. Let n be a positive integer, and let a1,a2,…,an be pairwise different integers. Prove that the polynomial below cannot be expressed as a product of two polynomials of integer coefficients whose degrees are at least one.
   
5. Three members of the FIGHT CLUB: Alan, Bob and Craige, engage in a combat. Alan is very quick, but not particularly strong: 1/3 of his attacks result in a knock-out. Bob attacks after Alan, but he gets it half the time. Finally, there’s Craige – once he’s decided on the target, they’re dead. Alan attacks first, then Bob, then Craige, then Alan again… until there’s only one left. – Who should Alan target first (he can only attack one of his opponents, or neither)? – (*) How does the answer look like given some different knock-out probabilities for Alan, Bob and Craige (but keeping the order)?
6. This week in flight club: A mysterious voice ordered John to fill out the whole (euclidean) 3D space with these STRANGE SOLIDS (see picture below). Is that even possible? (optional) And what is this poster in the back?

Describe what the following pictures represent: Picture 1:

Picture 2:

7. A thousand wires hang on a very high tower, so high that you cannot see what tip belongs with what bottom. This is something you are interested in knowing. You have a battery and a light bulb which will light up if two wires connect it to the battery with appropriate polarity (i.e. the battery and bulb each have two contact points, and one of each is + and the other side is -). Wires may be tied together to form longer wires, and you can see the bulb light up even if you are on the opposite side of the tower. Since the tower is so high, you want to minimize the number of times you have to climb up and/or down the staircase, regardless of how much you have to do while you are at the top or bottom. What is the minimum number of traversals required?