- Bunnies and Leonardo Pisano Fibonacci
- 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!
- The Fibonacci numbers as a recurrence relation
(do it again, do it again, do it again!). - Fibonacci
Numbers and Nature:
- Let's look at some pine cones....
- dandelions (Here's a scanned dandelion)
- Spirals in daisies:
- Similar things happen in artichokes, pineapples, etc.
- Fibonacci number decomposition and Fibonacci Nim
- 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.
- Rules:
- More
details on Fibonacci numbers and nim. It turns out that
Every natural number is either
- Fibonacci, or
- can be written as a sum of non-consecutive Fibonacci's in a unique way.
Examples?
- Fibonacci number decomposition and Fibonacci Nim
- 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
- If you looked at some special cases, what can we
conclude?
- As we move toward the so-called "golden mean" and "golden rectangle", we'll start with a nice logarithmic spiral (see this site)
