Numbers for Words People

The Easter Bunny is very polite. He isn’t one to enter people’s homes through roof cavities in the middle of the night! Instead, he knocks on the front door during daytime. Often, this involves climbing up a set of stairs to get to the front porch.

The Easter Bunny can hop either 1 or 2 steps at a time. Faced with 10 steps, in how many ways can the bunny reach the top of the stairs?

Attempting to solve the problem

Our bunny can hop up the stairs in the following ways:

One step at a time: 1, 2, 3, 4, 5, 6, 7, 8, 9 , 10

Two steps at a time: 2, 4, 6, 8, 10

One step at a time till the final two: 1, 2, 3, 4, 5, 6, 7, 8, 10

Maybe two-steps at a time, then all ones: 2, 3, 4, 5, 6, 7, 8, 9, 10

Those were 4 ways in which the bunny can hop up a staircase with 10 steps. There seem to be a lot of ways of getting to the top of the stairs. It will be difficult to find all of them and to trust that we haven’t missed any.

Solving a smaller version of the problem

As we have done before, we will use this problem to think like mathematicians. Let’s solve smaller versions of this problem and see what we find out.

Let’s start with one step to the porch: There is one way to get there, a single hop.

If there are two steps, then the bunny can hop one at a time (1, 2), or hop two steps, straight to the top. That’s two ways of reaching the porch.

If there are three steps, the bunny can scale them in these ways: One step at a time (1, 2, 3), one step then two (1, 3), or two steps then one (2, 3).

Four steps, the bunny can go: 1, 2, 3, 4 or 1, 2, 4 or 1, 3, 4 or 2, 3, 4 or 2, 4:

What about five steps? Let’s start thinking differently:

The bunny can get to step 5 through steps 3 or 4:

• There are 3 ways for the bunny to get to step 3. He can then hop up to 5.
• There are 5 ways for the bunny to get to step 4. He can then hop up to 5.

In total, there are $3 + 5 = 8$ ways for the bunny to get to step 5

Summarising what we know

The table below shows the data we have gathered so far:

Number of steps	1	2	3	4	5
Ways of getting to the top	1	2	3	5	8

Can you spot the pattern? Every time we add one step, the number of ways becomes the sum of the numbers in the last two rows. Those who have read the very first post on this blog, will recognise this as the Fibonacci Sequence!

Solving the original question

Let’s keep our pattern going to work out the number of ways for the bunny to reach step 10:

Steps	1	2	3	4	5	6	7	8	9	10
Ways	1	2	3	5	8	$5+8$ $= 13$	$8+13$ $=21$	$13+21$ $=34$	$21+34$ $=55$	$34+55$ $=89$

Final answer: The bunny can hop up a staircase of 10 steps in 89 different ways!

Acknowledgements

This was a reframing of a famous problem, Leo the Rabbit, on the youcubed mathematics website. I was first presented with this problem by an exceptional maths teacher, Mr Gregory Breese when we both taught VCE Algorithmics.

Your turn

Maths teachers always set homework, haha! To give you a sense of how fast the number of ways grows, why don’t you work out how many ways the bunny can hop up a staircase with 15 steps? Let me know your answer in the comments.