# Following up yesterday’s probability problem with one of my favorite ideas involving high school algebra

Yesterday we did a neat project on problem #1 from Frederick Mosteller’s probability challenge book:

Going through problem #1 from Frederick Mosteller’s probability challenge book with kids

Today my older son is off mountain biking so the follow up project is with my younger son who is in 8th grade. I thought it would be fun to look at additional solutions to yesterday’s puzzle and show him how we could write down a formula for those solutions.

We started by look at some of the small solutions to yesterday’s problem and looking for patters. My son noticed a connection to $\sqrt{2}$ that made me really happy!

Next I showed him the Internet Sequence Database. I wanted to show him that sometimes when you are looking at a sequence of integers, it is something that other people have studied before:

Now we returned to the whiteboard to study the sequence more carefully. Our starting point was the recurrence relation that we learned about in the last video:

Finally, and this is one of my favorite high school algebra examples, we took a first step at solving the recurrence relation. This step is a nice application of factoring and using the quadratic formula: