- Sorry, I haven't yet graded your quiz.
- You'll have a quiz this Friday over Pascal's triangle and Fibonacci Nim.
- Here are the rules:
- This is a two player game: Player 1 and Player 2.
- Player 1 goes 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.
- Last time we looked at some especially simple cases,
hoping to learn something about the winning
strategy. What did we conclude?
Number of sticks 1 2 3 4 5 6 7 8 Winner X P2 P2 P1 P2 P1 P1 P2 - We noticed the pattern 2,3,5,8,.... and from this we guess that the pattern goes up as 2,3,5,8,13,21,34,55,89,.... These are the so-called "Fibonacci numbers", obtained by adding the two preceeding Fibonaccis to get the next. So 55+89=144, the next Fibonacci number in the sequence. We traditionally start with 1,1, so the sequence is 1,1,2,3,5,8,....
We guess therefore that being player two when the number of counters is Fibonacci is a good idea.
- Now what's the winning strategy in Fibonacci Nim? It also involves
Fibonacci numbers.
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:
- 10=8+2
- 27=21+5+1
- 27=21+5+1
- 34=34 -- it's a Fibonacci!
- Let's play another game of nim, and watch my moves to see
if you can determine the winning strategy.
And let's use a tree, to keep track of the moves....
- 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!
Every Fibonacci has a name. Actually 1 has two names: and
. Every other Fibonacci number has a unique name.
is ...?
means the
Fibonacci number, where we then need to specify
. So it's a way of speaking about a generic Fibonacci number.
- Let's look at some objects from nature: a pine cone, and an ocotillo skeleton
- dandelions (Here's a scanned dandelion)
- Spirals in daisies:
- Similar spirals appear in artichokes, pineapples, etc.
