Question 19.50

image 20. The game of Fibonacci Nim begins with counters. Two players take turns removing at least one counter, but no more than twice as many as the opponent just did. The winner is the player who takes the last counter. One other rule: The first player may not win immediately by taking all the counters on the first turn. (Adapted from Martin Gardner, Mathematical Circus, Knopf, New York, 1979.)

  1. Play this game taking turns with an opponent and starting with different numbers of counters and try to come up with a strategy for one player or the other to win. (Hint: The key is that any positive integer can be represented uniquely as a sum of Fibonacci numbers.)
  2. Proceed as in part (a), but with the rule changes that the player who takes the last counter loses and the first player may not take all but one counter.