Making careful lists, counting and looking for a pattern were useful strategies for solving this problem, which is nicely explained by Christopher from Burpham Primary School, Guildford, Surrey:

"Following the hint to try one step, then two and so on, I listed all the combinations of answers for the first few numbers of steps.

Once I had got to 7, I thought about finding a pattern to help me work out the bigger numbers. I noticed that the next number in the sequence was the sum of the two previous numbers. This meant I could work out the answer, which is 233. "

Alex from The Abbey School, Woodbridge, Suffolk sent in a well made list of all the possibilities up to 8 steps, as well as finding the pattern:

I noticed that the number you are trying to get is always the sum of the 2 numbers before it i.e. the Fibonacci series, so I think the next ones will be:
9 Steps:55 ways (34+21)
10 Steps:89 ways (55+34)
11 Steps:144 ways (89+55)
12 Steps:233 ways (144+89)

Excellent solutions were also sent in by: Olivia from St Ives School, Haslemere; Melis Onalan , Umur Ta Demir and Altay Alpgut from Private IRMAK Primary School, Istanbul, Turkey; Daniel from Anglo-Chinese School, Singapore; and Year 6 Maths Club, St Francis School, Maldon, Essex.

This last solution has been chosen because it tells the story of a team effort in finding a good solution. It is from: Emily , Caroline , Rebecca , Clare , Camilla and Gabriella from The Mount School, York.

"We started by drawing diagrams, like this

But Emily had the idea of just using the numbers, so

At this point Camilla came up with 'Fibonacci' and the cry was taken up by the others !

2 +3 = 5
3 + 5 = 8
5 + 8 = 13 so ..... 21, 34, 55, 89, 144, 233

The answer is 233"

Here is another way of looking at the problem which may make it easier to see why the Fibonacci series occurs:

Liam's staircase has 12 steps, and he must start EITHER taking just a single step OR by taking a double step. If he takes a single step to start with, then he has 11 steps left. If instead he takes a double step to start with, he only has 10 steps left. So the number of ways he can go down all 12 steps is equal to the the number of ways to go down 11 steps PLUS the number of ways to go down 10 steps. But that's just the same as the way the Fibonacci numbers are defined - each number is the sum of the two numbers that came before it. So the answer is just the 12th Fibonacci number!

Let's call the number of ways that Liam can go down a staircase with n steps $S_n$. (What we eventually want to know is $S_{12}$, the number of ways he can go down his 12-step staircase.)

Liam must start off taking EITHER a single step OR a double step.

If he takes a single step to start with, then he has n-1 steps left and there are $S_{n-1}$ ways he could do those.

If instead he takes a double step to start with, he only has n-2 steps left and there are $S_{n-2}$ ways he could do those.

So the total number of ways he can do all n steps is the sum of these two:

${\displaystyle S_n=S_{n-1}+S_{n-2}.}$

But that's just the definition of the Fibonacci numbers! So the answer is the 12th Fibonnaci number.