# A nice counting problem from James Tanton

Sow another super problem from James Tanton yesterday:

Seemed like a great thing to talk through with the boys this morning. We started off with a quick overview of the problem, and both kids had really nice initial reactions. I was afraid that my older son’s idea had a lot of overlap with my younger son’s idea, but part (d) shows that my younger son’s idea was actually totally different (and really cool!)

The boys decided to follow my older son’s approach first. They wrote down a first couple of numbers looking for a pattern. They pick up on the pattern in the numbers fairly quickly and see that groups of 100 three digit integers have 19 numbers with at least one zero. It was fun to hear my younger son begin to discuss how to count up to the 100th number

We cut off the last video when we found the 28th number in the list, which was 190. Below we continue counting up to the 100th number and eventually find that 550 is the correct answer.

Next we went back to the beginning to see my younger son’s idea for solving the problem. He sees a neat pattern that unfortunately isn’t quite right, but turns out to be relatively easy to adjust and make correct. It is a way of solving this problem that I never would have seen. I think this is a great way to see that a “wrong” pattern can be pretty close to “right”!

Finally, I wanted to show one alternate method that would help solve the problem – counting what you don’t what. Although the solution here really isn’t that different from what the boys did already, the technique of counting the numbers without zeros is something that I wanted them to see.

So, another great problem from James Tanton. Because the problem allows for some many different approaches, it is a really nice problem to use with kids to talk through different counting techniques. Also, as my younger son shows, sometimes you’ll even see a counting technique that you really weren’t expecting! Fun morning.