Giant Steps

Outside spiral staircase with blue sky

In 13 Steps, you may have found out how many different ways there are to climb up 13 steps, taking one or two steps at a time. 

Suppose I had really long legs, so I could climb the stairs one, two or three steps at a time.

Is there a quick way to work out how many different ways there are to climb the stairs if I know how may steps there are?

What if I could also go up four steps at a time?


Suppose I really like taking big steps, so I never want to climb just one step at once.

Is there a quick way to work out how many different ways I can climb a flight of stairs, if I only climb up two or three steps at a time?



We started at a staircase of 1 step and then moved on upwards...

We had proposed that instead of the Fibonacci sequence being generated by adding the two previous numbers in the sequence to generate the next term, that in this case maybe we might see a sequence develop by adding the 3 previous terms. We got the following results by listing out all possible combinations of steps: 1, 2, 4, 7, 13, 24 These numbers satisfied our initial hypothesis so we decided that we were most likely correct. The 13th number in this list yields 1705.

We found it a bit too difficult to justify the hypothesis but we felt that the numbers spoke for themselves.


Can anyone help George justify why his hypothesis might be correct?

Can anyone build on insights from 13 Steps?