Numberphile’s “Pebbling the Chessboard” game and Mr. Honner’s square

Yesterday I saw an amazing math post on Twitter by Dan Anderson:

I love seeing math games with surprising outcomes that are simple to explain.  NumberPhile’s video on the problem is a masterpiece:

Solving the problem in the game involves summing a fairly complicated infinite series:

3/4 + 4/8 + 5/16 + 6/32 + 7/64 + . . . . .

The Numberphile video shows one way to sum that series, and eariler this year Patrick Honner published a nice visual proof showing how to sum (nearly) the same series:

Here is his beautiful picture that 1/4 + 2/8 + 3/16 + 4/32 + 5/64 +  . . . . = 1

I thought the game in the Numberphile video would be a super fun project to work through with kids.  I spent a little time last night trying to figure out how to talk about it with my boys and then spent the morning today going through it.

The first thing we talked about was Mr. Honner’s visual proof.  I wanted to do that at the beginning so that we wouldn’t get too distracted by the series when it came up during the game:

Finally, introducing the concept of invariants and connecting the game with Mr. Honner’s series:

The “Pebbling the Chessboard” game is such an amazingly fun and instructive exercise for kids.  Wish I would have known about this game back when I was teaching!