Fibonacci sequence

problem
Golden fractions

Age

16 to 18

Challenge level

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.
problem
Sheep talk

Age

7 to 11

Challenge level

In sheep talk the only letters used are B and A. A sequence of words is formed by following certain rules. What do you notice when you count the letters in each word?
problem
Golden fibs

Age

16 to 18

Challenge level

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!
problem
Fibonacci factors

Age

16 to 18

Challenge level

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
problem
Ordered sums

Age

14 to 16

Challenge level

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.
problem
Stringing it out

Age

14 to 16

Challenge level

Explore the transformations and comment on what you find.
problem
Spirals instead of sunflowers

Age

11 to 16

Challenge level

Using logo to investigate spirals
problem
LOGO challenge - circles as bugs

Age

11 to 16

Challenge level

Here are some circle bugs to try to replicate with some elegant programming, plus some sequences generated elegantly in LOGO.
problem
1 step 2 step

Age

11 to 14

Challenge level

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
problem
Gnomon dimensions

Age

14 to 16

Challenge level

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.