We were unpacking a bunch of books today and ran across James Tanton’s *Solve This* and Theoni Pappaas’s *The Adventures of Penrose The Mathematical Cat*. I asked the boys to find a problem they’d like to talk about and they found a gem from Tanton’s book:

16.1 A Fair Game?

Peter has ten coins, Pennelope has nine. Peter and Pennelope agree to toss all their coins simultaneously. Whoever receives the largest number of heads will win. In case of a tie Pennelope will be declared the winner, so as to offset the advantage Peter has to begin with. Given this agreement, who is most likely to win?

We began by talking about the problem and the boys had some ideas right away. Their initial guess is that Peter will win, but they want to explore the problem by looking at an easier case first.

After discussing the problem we started looking at the easier case in which Peter has one coin and Pennelope has 1 coin. One of the nice things about this simple version of the problem is that you can actually talk through all of the cases. Talking through these cases is a really nice introduction to counting and probability. The other nice thing about the simple case, of course, is the surprising answer!

Given the surprising answer from the last video, we decided to check out the next easiest case – Peter has 3 coins and Pennelope has 2. Fortunately we’ve already understood the cases that Pennelope will encounter here, so we just have to understand the situations that can come up for Peter.

Again, this exercise is a nice exercise in counting and also this time the snap cubes we are using for heads and tails help show the symmetry involved in the various cases. The boys do a nice job here working through this slightly more complicated example:

Finally we decided to try to tackle the original problem. My younger son has a great symmetry idea what we talk about for a bit, but it seemed that approaching the problem this way was still a bit out of reach for them.

They wanted to calculate, but I didn’t đź™‚ As sort of an on-the-fly compromise, I thought working a little bit with Pascal’s triangle would still be a good exercise and also show them that the calculation they were looking to do would be pretty ugly!

So, maybe not the most satisfying end to the project since we didn’t completely solve the problem. Nonetheless, I like all of the great math practice that we got in this project – lots of good ideas from introductory probability and counting, and lots of good arithmetic practice, too.

Also, it was nice to see the boys starting to think about using symmetry to help solve problems. Even if they aren’t yet quite able to put those ideas to work in their solution yet, I’m glad those ideas are starting to come to the surface.

Tomorrow we are starting to work through Art of Problem Solving’s *Introduction to Counting and Probability*. Today’s project was a great way to motivate that summer project!