Fibonacci sequence

problem
Plus or minus

Age

16 to 18

Challenge level

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.
problem
Room doubling

Age

7 to 11

Challenge level

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

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

Age

11 to 14

Challenge level

Play around with the Fibonacci sequence and discover some surprising results!
problem
Fibs

Age

11 to 14

Challenge level

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?
game

Last biscuit

Age

11 to 18

Challenge level

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

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

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

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.