HomeWork I

Due Sep. 23rd

Think about all seven problems, and come up with some ideas about how to solve them. If you want you can write them. Then choose two of the problems and write up a careful solution. Your solution should be convincing for a skeptical classmate.
  1. You have seven glasses on a table, all face down. You are allowed to turn over four glasses at a time, as many times as you want. Is it possible to end up with all seven glasses face up?
  2. You are given a piece of rope and you are told that it takes 60 minutes to burn from one end to the other. However you are also told that it burns "unevenly", in the sense the first part of the rope might burn faster than the last- you simply don't know. Find a way to measure out a time interval of exactly 30 minutes. (No, you aren't allowed to use a watch)
  3. Suppose you have a mug of milk and a mug of tea (equal amounts of each). You take three table spoons of milk and put it in the tea, and stir it well. Then you take three table spoons of the mixed milk and tea and put it back into the milk. Which is greater: the percentage of milk in the tea or the percentage of tea in the milk?
  4. In a certain year, there were four Fridays and four Mondays in January. What day of the week was January 17th that year?
  5. If   SAGE + SUAVE + SAGE = 46933, what word will correspond to the number 46933? (Each number is represented with at most one letter)
  6. We have a 4*5 Chocolate bar and we went to break it down into units (1*1 pieces) of chocolate. To do so, each time we can break a piece of chocolate along one of the straight lines on it into to pieces. We cannot break two pieces at a time. What is the smallest number of breaks needed for completing the process? 
  7. You have 12 coins, one of which is counterfeit. You do not know whether the counterfeit coin is lighter or heavier than the others. Use three weightings on a balance with two pans to find the counterfeit coin. (Try it for 8 coins instead of 12 at first.)