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Fibonacci sequences are named after a merchant, one Leonardo of Pisa who had the nickname Fibonacci. On his travels, around 1200 AD, he learnt a lot of mathematics (particularly algebra) from the Arabs.
The Arabs had developed the study of mathematics for about 800 years after the fall of the Greek and Roman civilisations. The story behind the methods in this problem spans this whole period.
Equations in which one seeks whole number solutions, are called Diophantine equations after Diophantus (250 A.D) who developed the method and a special notation for recording it.
For more about the stories of Diophantus and Fibonacci see the History of Maths website .
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?
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?
Here are some circle bugs to try to replicate with some elegant programming, plus some sequences generated elegantly in LOGO.