1.  7   
   2.  5   
   3.  4

  1. The following seven weights are sufficient:
        1, 2, 4, 8, 16, 32, 64.
     For example, to weigh 77 g,
         77 = 64 + 8 + 4 + 1

  2. The following five weights are sufficient:
        1, 3, 9, 27, 81.
     For example, to weigh 52 g,
         52 = 81 - 27 - 3 + 1

  3. At his first move A leaves 496 counters.
     Then whatever B removes, at his next move A can leave 485.
     Then whatever B removes, at his next move A can leave 474.
     (And so on, coming down in steps of 11)
           ...
     At his final move A leaves 1 counter, forcing B to lose.

      (a) the highest point, A, of the flange, and
      (b) the lowest point, B, of the flange ?

2. A ship's propeller shaft is turning at one revolution per second. It is supported in a roller bearing, with cylindrical rollers. The inner race has diameter 20 cm and the rollers each have diameter 2 cm. How long does a roller take to make one revolution and return to its original position ?

3. In question 3, if the inner race were fixed and the outer race were turning at one revolution per second, how long would a roller take to make one revolution ?