There are **30** NRICH Mathematical resources connected to **Fibonacci sequence**, you may find related items under Patterns, Sequences and Structure.

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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?

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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?

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

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Investigate Farey sequences of ratios of Fibonacci numbers.

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

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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?

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

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

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

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

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What have Fibonacci numbers got to do with Pythagorean triples?

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

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

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.

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

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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?

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

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

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

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

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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. . . .

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Explore the transformations and comment on what you find.

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

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

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

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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?

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

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