Day 17, MAT115

• Rules:
  • There are two players, Player 1 and Player 2. Player 1 will go first.
  • An arbitrary number of counters is thrown down (e.g. toothpicks, pennies).
  • Player 1 must pick up at least one counter, but may not pick up all the counters.
  • Player 2 must then pick up at least one counter, and not more than twice the number of counters picked up on Player 1's turn.
  • The players continue taking turns, subject to these same conditions: pick up at least one, no more than twice what the preceding player took.
  • The player who picks up the final counter wins.
• Let's play a sample game: Suppose that there are 27 counters on the table.
• You play a sample game with a partner: Suppose that there are 43 counters on the table.

• Start with a new-born pair of "immortal" bunnies (they never die).
• A new-born pair requires a month to mature.
• The following month, the pair produces a new pair, which is subject to the same rules.
• Do it again, do it again, do it again!

• If you looked at some special cases, what can we conclude?

  Number of sticks	1	2	3	4	5	6	7	8
  Winner	X

• What's the winning strategy in Fibonacci Nim?
• Ratios of Fibonacci numbers

• Let's look at some pine cones....
• dandelions (Here's a scanned dandelion)
• Spirals in daisies:
  • 21 spirals
  • 34 spirals
  • together
• Similar things happen in artichokes, pineapples, etc.

Every natural number is either

1. Fibonacci, or
2. can be written as a sum of non-consecutive Fibonacci's in a unique way.

Examples?