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Broad Topics > Patterns, Sequences and Structure > Fibonacci sequence

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?

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?

Fibonacci Surprises

Age 11 to 14 Challenge Level:

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

Farey Fibonacci

Age 16 to 18 Short Challenge Level:

Investigate Farey sequences of ratios of Fibonacci numbers.

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.

Simple Train Journeys

Age 5 to 11 Challenge Level:

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

First Forward Into Logo 11: Sequences

Age 11 to 18 Challenge Level:

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

Whirling Fibonacci Squares

Age 11 to 16

Draw whirling squares and see how Fibonacci sequences and golden rectangles are connected.

Plus or Minus

Age 16 to 18 Challenge Level:

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

Pythagorean Fibs

Age 16 to 18 Challenge Level:

What have Fibonacci numbers got to do with Pythagorean triples?

Fibonacci Fashion

Age 16 to 18 Challenge Level:

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

Golden Mathematics

Age 16 to 18

A voyage of discovery through a sequence of challenges exploring properties of the Golden Ratio and Fibonacci numbers.

The Golden Ratio, Fibonacci Numbers and Continued Fractions.

Age 14 to 16

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.

Golden Fractions

Age 16 to 18 Challenge Level:

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

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?

Got a Strategy for Last Biscuit?

Age 11 to 16 Challenge Level:

Can you beat the computer in the challenging strategy game?

Leonardo of Pisa and the Golden Rectangle

Age 7 to 16

Leonardo who?! Well, Leonardo is better known as Fibonacci and this article will tell you some of fascinating things about his famous sequence.

Fibonacci's Three Wishes 2

Age 7 to 14

Second of two articles about Fibonacci, written for students.

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!

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?

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. . . .

Stringing it Out

Age 14 to 16 Challenge Level:

Explore the transformations and comment on what you find.

Age 11 to 16 Challenge Level:

Using logo to investigate spirals

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.

Continued Fractions I

Age 14 to 18

An article introducing continued fractions with some simple puzzles for the reader.

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.

Paving Paths

Age 11 to 14 Challenge Level:

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?

Golden Powers

Age 16 to 18 Challenge Level:

You add 1 to the golden ratio to get its square. How do you find higher powers?

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.