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!

2 thoughts on “How would you teach this problem?

  1. I assume that you are going to ignore leading zeros.
    1. Let them get started
    2. When they are getting fed up I would suggest that there might be some structure or pattern appearing which could be useful.
    3. To go any further I would have to have cracked the problem myself beforehand!

  2. Also a wonderful opportunity to introduce hexadecimal notation, which pops up now and again, for example you get #20 or %20 in file names and urls in place of the space character.

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