# Ghost the Bunny

Laura’s pet bunny, Ghost, hops up a flight of 12 stairs. Ghost can hop up one or two steps at a time, and never hops down. How many ways can Ghost reach the top?

(note, if Ghost goes up one at a time, and ends with a double hop, or if he starts with a double hop, then goes up the rest one at a time, those count as two different ways) (iow, order matters.)

If 12 steps is too challenging, solve for a smaller number.

And if you solve this easily, try to find a second or a third method. (I have three)

Finally, I believe this puzzle has been around in various forms for years, but I believe I ran into the bunny version about ten years ago at a talk by Al Posmentier, Professor of Mathematics Education (and now Dean of Education) at CCNY.

I only found 3 bunnies on stairs via google (excluding playboy bunnies and girls named bunny).

I like teachers who grow beyond their subject. My success in teaching science is to incorporate history into the curriculum. By using storytelling it keeps the students interested and involved.

Then you probably appreciated today’s other post.

When I do “Ghost the Bunny” with kids, it is not part of a unit.

I intersperse such out-of-curriculum problem solving throughout my teaching. Once every two, or three or four weeks I will throw away all or most of a period on something other than “the next topic.”

It gives the kids an occasional break. It keeps them from assuming there is an algorithm for everything. It lets them learn techniques for attacking unfamiliar situations.

In the case of Ghost, I’d like them to think about notation. I’d like some of them to try to draw the stairs (and start to learn to evaluate and discard strategies that are not working). I’d like some of them to try to solve smaller problems instead. I’d like some to look for patterns. I’d like some to subdivide the problem by cases. And I’d like to spend a lot of “looking back” time evaluating what went right, what went wrong, and finding yet another approach.

Anyway, I’m off topic. You said you like the story-telling. I give talks on this stuff (problem solving for middle and high school students); I’ve taught courses in it. And I firmly believe that, especially for an extended problem, a little bit of cute context is very important to catch some initial interest.

At some point I was planning on writing about an article from 25 years ago that I like: Posing Problems Properly by Thomas Butts. I like it for other reasons, but Butts did (I have no idea what he thinks today) prominently recommend giving a little bit of setting or context…

Could you share your strategies? I am looking for a variety of ways to solve this problem and relay those strategies to others. I would appreciate any help! Thanks

There are a number of things you could look for. I recently ran this problem with a class, and here are some of the strategies (I call them heuristics) that came into play in the various solutions:

Make an organized list

Make a complete list

Choose appropriate symbols

Solve a smaller problem

Look for a pattern

Look for sub-problems

Recall a useful formula

Divide the problem into cases

If you like, write down what you have tried before, and someone can help direct you to which strategies would be most helpful from where you are.

Could you please tell the answer I need it right away. for math class.

There are some suggestions, just above your comment. Here’s more help than I usually give my own students:

Solve the problem for the smallest possible staircase. Then pick the next biggest staircase, and keep working up. You will either get good at counting, or find a useful pattern, or both.

Good luck. And if you start, and get stuck, feel free to write about your progress so far in this space. It is likely that someone will try to help.

Okey thanks!

please help! im having trouble

What have you done so far? I can make suggestions, but you’ll have to indicate what you already tried.

And two and a half years later, I have more like 4 or 5 ways to attack this.

i like

been trying this problems for hours and know the answer and fibonacci but do not know how to explain the reasoning behind it.

Think about getting to the 100th step. Where was Ghost just before?

Either on 99, and he took a single hop.

Or on 98, and he took a double hop.

I think that’s the most Fibonacci explanation possible.

when u think u found way to easily solve a problem but realize that your “helper” is your teacher. ;-;