Math 270 Assignment 1 Solutions

Math 270 Assignment 1 Solutions 1.3 Solve the following anagrams: NOVA CURVE, NINE SLAP NAVY, I HELD A HIP PAL, and DIRTY ROOM. Solution: Through trial and error the solutions are Vancouver, Pennsylvania, Philadelphia, and Dormitory. 1.4 Suppose n team play in a single game elimination tournament, How many games are played. Solution: First notice that if there are not 2n teams then a fair tournament (in which each team in each round plays the same number of games) is impossible. However we know that each team will be eliminated after a loss. Since the number of losers must equal n − 1 there must be n − 1 games since only one team can win each game. 1.5 Suppose you are all alone in a strange house. There are seven identical closed doors. The bathroom is behind exactly one of them, is it more likely, less likely, or equally likely, that you find the bathroom on the first try than on the third try? Solution: I claim that it is more likely that you will find the bathroom on the third try. On the initial try you have a one in 7 chance that you find the bathroom. On the third try, either you have already found the door in which case you have a 100 hundred percent chance you find the door, or there are only five possibilities left and you have a 1 in five chance of finding the bathroom. Putting these together gives you a 1 in 5 chance which is greater than 1 in 7. (Note: there is more than one way to look at this problem and get different answers) 1.8 You are given twelve coins that appear to be identical. However, one of the coins is counterfeit, and the weight of the coin is different from the other eleven. Using only a two pan balance, what is the smallest number of weighings you would need to find the counterfeit coin? Solution: In a worst case scenario one would need four weighings. We describe an algorithm below for determining the counterfeit coin. i. Separate the coins into four equal piles. ii. Weigh two of the piles, call them A and B. iii. Replace one of the piles B by a...