Extreme Math
Home Study Assignment #2

	1	2	3	4	5	6	7	8	9	10	11	12
Odd numbers:
Primes:
Factorials:
Fibonacci numbers:
Powers of 2:
Mersenne numbers:

2. A certain rabbit — we’ll just call her C to protect her privacy — is very active and can easily hop up the stairs two steps at a time. (We’ll call the double hop a “jump.”) Of course C can also hop one step at a time like any other bunny. (We’ll just call the single hop a “hop.”) The question is, in how many different ways can she go up a staircase containing 12 steps? Or 14 steps...or 15...or any number at all?

As always, let’s look at a simpler example or two first. Then maybe you can find a pattern and can extend it to 12 steps...or even more. For example, suppose C had to go up a very short staircase with only five steps:

• She could do two jumps and one hop. Let’s write this as JJH.
• Or she could do five hops (HHHHH).
• Or a hop followed by a jump and two hops (HJHH).
• Or a hop followed by two jumps (HJJ).
• Or a hop between two jumps (JHJ)
• Or...

Gee, even this is getting complicated. I’m not being very organized, am I? Have I left any out? Oh, yes, I’ve left out JHHH and HHJH and even HHHJ.

There, I think I’ve finally found all of them: eight ways in all. Or do I have any duplicates? Or have I accidentally left any out? We’ve got to be more organized!

Your task is to do all this in a much more structured way. Even five steps turned out to be complicated, so maybe you should start by looking at 3-step and 4-step staircases. Then try to find a pattern, and answer the original question — the one for 12 steps — without actually counting all the ways for 12 steps. Finally, see if you can explain why your pattern works!