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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.
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?
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?
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!
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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!
NRICH is part of the family of activities in the Millennium Mathematics Project.