# Walking down the path to the surreal numbers with kids

For reasons that are somewhat mysterious to me, I’ve seen a bunch of media coverage about John Conway in the last couple of months.

Of course, there’s Siobhan Roberts’s awesome biography:

Which Jordan Ellenberg reviews in the Wall Street Journal here:

Jordan Ellenberg’s review of Genius at Play

Roberts also has a great Quanta Magazine article:

John Conway: A life in games

One other item that caught my attention last week was a Jim Propp blog post I saw tweeted out by Jordan Ellenberg:

Jim Propp’s “The Life of Games”

Propp actually wrote some of my graduate school recommendation letters back in . . . oh why did I bring this up . . . 1992, so it was cool to learn that he had a math blog!

Anyway, Propp’s piece describes the game of checker stacks so well and so simply that I thought it would be fun to try to talk through some of the ideas with the boys. We didn’t get to some of the more unusual features of the game today – and so didn’t really get to the surreal numbers – but even showing the boys how the number 1/2 shows up in the game was really fun. We’ll take a few more steps down the path to the surreal numbers tomorrow.

Here’s how today’s project went (oh, and sorry that the camera angle is so bad in the last two videos, I didn’t notice that the tripod got bumped until I was publishing the videos.):

The first step was a quick introduction the game of checker stacks and a few thoughts from the boys about the game:

Next we studied what happens when we stack checkers on top of each other. First we studied the relatively simple situation – a game with a single red vs. a stack with two blues. Then we moved on to a more complicated situation of a single red vs. a red blue stack. The boys were able to get their arms around these two situations, which was nice.

I realized at the end of the last movie that I wasn’t following Propp’s presentation correctly – he was using blue red stacks rather than red blue stacks. The difference becomes important when you are trying to find a game when it matters which color goes first. So for the third movie we we now studied the blue red stacks.

Also, the boys remembered in between these two videos that we had some 3d-printed red and blue action figures!

One interesting bit of math from the kids in this video was that they assumed that all of the stacks would have values that were represented by integers. Because of that assumption, they think the value of the blue red stack must be 0 or lower. We’ll explore that idea a little more in the last video.

The last game we study is a game with a single red vs two blue red stacks. Propp uses this game to illustrate a surprising property of the blue red stack. The boys were indeed surprised by the result 🙂

So, a really fun start to our journey to the surreal numbers. It was neat to see that even some fairly simple positions from the checker stacks game gave the kids some challenging things to think about – and even a few surprises!