Fibonacci sequence

problem

Golden Fractions

Age

16 to 18

Challenge level

3 out of 3

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

Room Doubling

Age

7 to 11

Challenge level

2 out of 3

Investigate the different ways you could split up these rooms so that you have double the number.
problem

Colour Building

Age

11 to 14

Challenge level

1 out of 3

Using only the red and white rods, how many different ways are there to make up the other rods?
problem

Fibonacci Deduction

Age

11 to 14

Challenge level

1 out of 3

Leonard writes down a sequence of numbers. Can you find a formula to predict the seventh number in his sequence?
problem

Logo Challenge - Circles as Bugs

Age

11 to 16

Challenge level

2 out of 3

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

First Forward Into Logo 11: Sequences

Age

11 to 18

Challenge level

1 out of 3

This part introduces the use of Logo for number work. Learn how to use Logo to generate sequences of numbers.
game

Last Biscuit

Age

11 to 18

Challenge level

2 out of 3

Can you find a strategy that ensures you get to take the last biscuit in this game?
problem

Building Gnomons

Age

14 to 16

Challenge level

1 out of 3

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
problem

Gnomon Dimensions

Age

14 to 16

Challenge level

2 out of 3

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

Ordered Sums

Age

14 to 16

Challenge level

2 out of 3

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.