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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?

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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?

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Farey Sequences

There are lots of ideas to explore in these sequences of ordered fractions.

Growing Surprises

Age 11 to 14 Challenge Level:
Take a look at the pattern below:
 

Can you describe the next few patterns in the sequence?

Can you work out how many squares there are in each pattern?

Is there a quick way to work out the number of squares in the 20th pattern? Or the 50th pattern? Or...


Here's another pattern made from squares:
 
Can you describe the next few patterns in the sequence?

Can you work out how many squares there are in each pattern?

Is there a quick way to work out the number of squares in the 20th pattern? Or the 50th pattern? Or...


Can you describe any relationships between the two patterns?


Here are two similar linked patterns:


Can you find a quick way to work out how many squares there are in these patterns?
Can you describe any relationships between these two patterns?


You may wish to look at some similar problems, Seven Squares, Attractive Tablecloths and Steel Cables.



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