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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?

Paving Paths

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?

1 Step 2 Step

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?

Fibonacci Surprises

Age 11 to 14 Challenge Level:

You may have seen this sequence before: $$1, 1, 2, 3, 5, 8, 13, 21, ... $$

It is called the Fibonacci Sequence, and each term is calculated by adding together the previous two terms in the sequence.

Try adding together any three consecutive Fibonacci numbers.
What do you notice?
Can you explain it?

Choose any four consecutive Fibonacci numbers. Add the first and last, and divide by two.
What do you notice?
Can you explain it?

Add together any six consecutive Fibonacci numbers and divide by four.
What do you notice?
Can you explain it?

Can you discover any Fibonacci Surprises of your own?

With thanks to Don Steward, whose ideas formed the basis of this problem.