How would you teach this problem?

I’m doing a slow walk through Art of Problem Solving’s Introduction to Counting & Probability with the boys this summer. Mainly just for fun, but also because I thought that it would be neat to see them working together.

Peeking a little bit ahead in the book last night I found this problem in the Challenge Problem section at the end of chapter 2 (problem 2.33 on page 47). It is from an old ARML:

How many zeroes do we write when we write all the integers from 1 to 256 in binary?

I like this problem a lot and am planning on using it for one of our Family Math activities this weekend. Before we dive into the Family Math project, though, I thought it would be fun to ask around – how would you approach going through this problem with kids?

Any and all ideas are welcome!