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Ordered Sums

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.

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Fibs

The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?

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Paving Paths

How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?

Spirals Instead of Sunflowers

Age 11 to 16 Challenge Level:

Beginners to LOGO programming may want to start by working through the FIRST FORWARD series of introductory articles before tackling this problem.

 

Below are some examples of recursion. I had been trying to replicate the spirals within the head of a sunflower!

 

               

The program used was:

TO SUNF :S :A
FD :S RT :A
SUNF :S + 1 :A
END

You might like to replicate the spirals above by considering suitable values for the variables :S and :A.

The real challenge is to simulate the head of a sunflower with more spirals that are much more tightly packed and look distinctly more 'Fibonacci' in proportion!