The Fibonacci Numbers

Adapted from  Fibonacci Numbers Worksheet - Ron Knott

Additional reference:
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html where you can find a much fuller resource and back-up information for these questions.

A.  The Fibonacci Numbers

The Fibonacci numbers are defined as follows.
The "first" is 1, so we write Fib(1)=1.
The second is 1 also , so Fib(2)=1.
After that, "the next is the sum of the last two" or, in maths notation,
Fib(n)=Fib(n-1)+Fib(n-2) if n>2

According to this formula, we could have begun with Fib(0)=0 and you will often see this in other books. Fill in this table of Fibonacci numbers so you can refer to it in later investigations: 

n:     0   1   2   3   4   5   6   7   8   9   10     11     12	

Fib(n):0   1   1   2

B.  Bees, Fibonacci numbers and Family trees

Yours answers here will be needed in a later problem, so don't throw them away yet!
Each female bee has a TWO parents, a male and a female, but each male bee has ONLY ONE - a female.
So what is the family tree of a male bee?
He has 1 parent: a female.
He has 2 grand-parents (the parents of his mother) which are a male and a female.
He has 3 great-grand-parents and so on.
Represent this information in a Bee Family Tree for up to 6 generations. There is more on the Fibonacci Numbers and Nature page.
  








C.  Seed heads

Here is a picture of the centre of a flower without its petals. You can see this pattern in a daisy (but it is much smaller), or a sunflower where it is much larger and on other flowers too. The circles represent seeds.
On the "seed head" picture, count the number of spirals at the edge that turn in one direction.
How many turn in the other direction?

 Try counting the spirals on pinecones and pineapples, sunflowers, even the "florets" on cauliflowers and brocolli - they all have a Fibonacci number of spirals in each direction. See this Web page for more.

D.  Fibonacci Divisors

Look at your table of Fibonacci numbers. What about multiples of 5? Now try multiples of 3. What about multiples of 8?
 
You may have noticed that the multiples we looked for were of 2, 3, 5 and 8. Can you guess why? Try some more numbers and find the pattern of where the Fibonacci numbers are that are exact multiples of your number.

There is a general rule for divisibility of a Fibonacci number by another. So, if I asked "Where will the multiples of 13 be in the Fibonacci series?", can you give me a rule without looking at the Fibonacci Table first?

  What about multiples of other numbers, such as 4, or 6 or 7?
      



E.  Fibonacci Puzzles

You can adapt this puzzle to 1p and 2p coins in the UK, or US nickels and dimes (5 and 10 cents) or any other denominations of coin or note where one is twice the other. Here we use US dollar bills for 10 and 20 dollars.

 A shop is selling a product for $100.
If a customer only has $10 and $20 bills, in how many different ways can they pay for the product. Ten $10 bills will do, or five $20's or - there are clearly a lot of ways. We want to count the number of different sequences (ways of laying out their bills on the counter). There are a lot, so try writing down a few first toget a feel for the problem.

Now let's investigate this with a smaller amount and see if we can come up with a principle and a pattern to we know we will not have missed some combinations.
Let's first try a smaller price - say $20. The customer can give $10, $10 or else $20 - two ways.

 If the price was 30, then we have 10,10,10 or 10,20 or 20,10 which, since the bills are in a different order, is a different solution. In total there are three different sequences in which we could pay 30 using just 10 and 20 notes.
How many are there for 40? For 50? And 60? Can you spot a pattern? Can you guess from your pattern how many ways you think there might be for 100?
 
 

An interesting connection

Arrange your solutions according to how many bills are used in each solution for each total amount: for example: for 30 we had 1 solution with three bills (10,10,10) and 2 solutions each with two bills (10,20 and 20,10). These are already entered into the table for you:
Total $= $10 $20 $30 $40 $50 $60 $70
1 bill              
2 bills
  2 
       
3 bills     1        
4 bills              
5 bills              
6 bills              
7 bills              
You will find that many entries are 0 and the rest form a triangular shape. Using the table, how would I find out the total number of solutions for $30? or for $40? Have you seen this pattern of numbers before? Can you guess how it continues?
 
  



F.  A Fibonacci Rectangles Jigsaw Puzzle

Here is a series of jigsaw puzzles where all the pieces are squares where the sides are Fibonacci numbers. Also, the sides of each rectangular jigsaw are also Fibonacci numbers! For instance, the largest one is 8-by-5 and haspieces which are 1x1, 1x1, 2x2, 3x3 and 5x5. So we can write the whole area in two ways giving us the single equation:

8 * 5 = 1*1+1*1 + 2*2+3*3+5*5

Write down the evaluations you get for the other 4 diagrams and check that each side gives the same number.
 
 

What would be the next size of square to add on to the largest diagram? How do the jigsaws continue?