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example 1Exsample 2

A walk is made up of diagonal steps, starting at the bottom left and ending up back on the bottom line (x-axis). You can move diagonally up and down towards the right but you cannot move towards the left.

A diagonal must go from a left-hand corner of a square to the opposite right-hand corner of the same square.

The examples above all show 10-step walks. So let's look at a 2-step walk in more detail:

2-step walk

There is only one way to make a 2-step walk - from A up to x and down to B:

2-step walk with labels

Here are two 4-step walks:

4-step example 24-step example

Are there any more 4-step walks?

How do you know you have them all?

How many 6-step, 8-step walks are there?

Can you find a general rule?

See also the problems One Basket and Counting Binary Ops