SEARCH HOME
 Math Central Quandaries & Queries
 Question from Kate, a parent: My daughter was given a math question, which we did together. However, when her class took it up in school, there were different answers among the students. The question is: A small toad is sitting at the bottom of the stairs. He wants to get to the 10th step, but is having some difficulty. He jumps up 2 steps and then back 1 each time. How many jumps will he take to reach the 10th step? The 4th grade students came up with answers such as 10, 26 etc. My daughter and I got 17. How would you interpret this question?

We have two responses for you

Kate,

you need to focus on what happens every time he jumps (except possibly the last time). When he first jumps he goes up two but slides back one for a net gain of one step. Thus after his 1st jump he's on step 1. Similarly he will be on step 2 after his 2nd jump. And so on for a while. The crux of the matter is what happens when he makes a leap on the 9th day - he'll be starting from the 8th step. He will leap and reach the 10th step and slide back one - now semantics come into the problem, has he reached the 10th step at this point or not? In this sense the problem is a bit vague but I would tend to think he has 'reached' the 10th step. Otherwise it would take him a 10th leap to make it to the 10th step, i.e. to finish up on the 10th step. This problem is often stated in terms of jumping out of a well in which case I would certainly say that on the 9th day he's out of the well. If it's a long staircase in your problem then I would listen to an argument that reaching the 10th step means ending up on it.

Hope this helps,

Penny

Hi Kate.

The way I read this is that the toad starts below step #1 and each time he moves between two steps (either going up or going down) is a jump.

So I'd say the following sequence happens:

On the first jump, the toad reaches the 2nd step.
(On the second jump, he jumps to the 1st step).
On the third jump, the toad reaches the 3rd step.
(On the fourth jump, he jumps back to the 2nd step).

...

On the fifteenth jump, the toad reaches the 9th step.
(On the sixteenth jump, he jumps back to the 8th step).
On the seventeenth jump, the toad reaches the 10th step.

So you are right: 17 jumps.

Think of it as two interwoven sequences:

Odd numbered jumps end on floors 2, 3, 4, 5, 6, 7, 8, 9, 10
Even numbered jumps end on floors 1, 2, 3, 4, 5, 6, 7, 8, 9.

So the mixed sequence is 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8, 10.

Cheers,
Stephen La Rocque.

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.