Working through a challenging counting problem from James Tanton

I saw a neat problem from James Tanton during the week:

If I roll a die five times how many distinct values should I expect to see?

— James Tanton (@jamestanton) June 1, 2017

A follow up post from Amy Hogan made me realize that I should use the problem for a project:

A couple of my math team students decided to explore this today with a simulation first. pic.twitter.com/ltFBHrqYPl

— Amy Hogan (@alittlestats) June 2, 2017

So, I decided to make looking at this problem our weekend project. Today I wanted to try a few simulations. Tomorrow we’ll count the cases. That case work is pretty challenging, but I think we’ll be able to get through it.

Here’s the introduction to the problem:

Next I had the boys roll 5 dice 50 times and record the numbers they saw each time. Here are those results as well as our first estimate of the answer to Tanton’s question:

Finally we moved on to do a simulation. Normally I would have used Mathematica, but I’ve got a program related to one of our old Goldbach Conjecture videos running right now. This problem was easy enough to deal with on Excel, though, so I looked at a simulation there.

Here’s what they saw in the numbers:

Definitely a nice start to this project. Counting the different cases tomorrow will be a bit more difficult, but we’ll also get to talk about some pretty interesting mathematical ideas like partitions and choosing numbers. I’m excited to see if we are able to get all the way to the end.