Problem Set 0

  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.  Can you produce number 26 by using digits 2,3,4,5 once and symbols +,-,×,/ at most once? Examples of permitted combinations are (34/2) + 5 = 22 and (3/1) - 2 + (4 × 5) = 21, but rather than 22 or 21 we would like to construct 26.


  4.  If   SAGE + SUAVE + SAGE = 46933, what word will correspond to the number 46933? Each number is represented with at most one letter.


  5.  A spaceship is to travel between 4 planets A,B,C,D along the edges of the tetra-hedron shown by dotted lines. Is it possible to plan the trip so as to traverse every edge exactly once?

    6.                                                   A
                                             B
                                             C
                                                               D
     

  6.  You have two egg timers (the kind with sand in an hourglass), one that measures seven minutes and one that measures eleven minutes.   You want to boil an egg for exactly 15 minutes.  How can you time it using these two timers?


  7.  How can you distribute 127 single dollar bills among 7 wallets so that any  integer amount from 1 through 127 dollars can be paid without opening the wallets?


  8.   A number of bacteria are placed in a glass. One second later each bacterium devides in two, the next second each of the resulting bacteria divides in two again, etc. After one minute the glass is full. When was the glass half-full?