I have recently acquired one of those little devices you attach to your belt that counts the number of steps you make in a day. The result of this is that you get obsessional about maximising your number of steps in every situation. There is even a temptation to mince along with very short steps rather than my usual manly stride, just to increase the step-count!

Today, I had to go up an escalator in a shopping centre. I never stand still on an escalator, unless it is totally blocked with other people. So, I found myself thinking about how many steps might I manage going up this escalator. It seemed intuitive to me that the faster I went the more steps I would get in before I reached the top. After all, if my speed was zero (i.e. standing still on one step) then my step-count for the escalator ride would also be zero.

Intuitively I imagined that there would be a linear relationship here. So, if I went twice as fast I would get in twice as many steps before I reached the top. But, wait a minute! That can't be true. Because I could never get more than the total number of visible steps on the escalator, however fast I went. In fact, to get in a number of steps equal to the number of visible steps on the escalator I would have to travel at an infinite speed!

This was getting so interesting, I forgot to do my shopping and just walked home puzzling it out.

It's a really nice mathematical problem. I managed to calculate, for example, that ...

- if there are 30 visible steps on the escalator (i.e. the number of steps you would see if the escalator were stationary), and
- the steps are disappearing at the top at the rate of 90 steps per minute, and
- I go up the escalator at the rate of 120 steps per minute (pretty fast),

If I went twice as fast up this escalator, doing 240 steps per minute, I calculated that I would increase my step-count to 21 or possibly 22 steps. And at half the speed, i.e. 60 steps per minute, I would do 12 steps before reaching the top of the escalator.

I'll leave this as a challenge for readers ... can you produce a formula for the number of steps I add to my step-counter (

*m*) if there are

*p*visible steps; the steps are disappearing at the top at the rate of

*s*steps per minute; and I go up at the rate of

*n*steps per minute?

*m*= ....?

And what would the graph look like, if you plotted

*m*against

*n*?

Solution in my next post.

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