## Teaser 2981: Faulty pedometer

**From The Sunday Times, 10th November 2019** [link]

Judith is a keen walker who uses a five-digit pedometer to record her number of steps. Her pedometer is inaccurate as some of the counters consistently move on to 0 early by missing out one or more digits. For instance, one of them might roll over from 7 to 0 every time instead of from 7 to 8, missing out digits 8 and 9. She is, however, well aware of this and can work out the correct number of steps.

After walking her usual distance, the pedometer shows 37225 steps but she knows that the true number is 32% less than this. A second distance she walks requires a 30% reduction in the number displayed to give the true number of steps.

How many steps is the second distance?

[teaser2981]

## Jim Randell 7:38 am

on9 November 2019 Permalink |Originally I thought the pedometer would be accumulating steps (so the second reading would add some steps to the first reading), but I could not find a solution where the second reading should be reduced by 30% (although I did get an answer where it was reduced by 28%).

So I considered the problem where the pedometer is reset to zero before the start of each walk, and doesn’t wrap around. This gives a unique solution.

This Python program finds possible bases for the 32% reduction, and then uses those bases to look for values with a 30% reduction. It runs in 161ms.

Run:[ @repl.it ]Solution:The second distance was 10080 steps.The units digit skips 9.

The tens digit operates normally.

The hundreds digit skips 9.

The thousands digit skips 8 and 9.

The ten thousands digit must be able to read as far as 3 correctly (in order for the initial reading to be 37225).

So, initially, 25313 steps gives a reading of 37225.

and:

And then a value of 10080 steps gives a reading of 14400.

and:

LikeLike