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!

This is great, Mike. I created that representation for fun, mostly because that series had been haunting me and I wanted to figure out a different way to understand it. I never imagined it being used as an instructional tool.

Thanks so much for sharing! It’s a cool feeling seeing your work being used in unexpected, and authentic, ways!

I wanted to respond to this last night, but instead got distracted by an oh-so-exciting plumbing issue. yuck.

For some reason I remember quite clearly my first encounter with the series 1 + 2 / 2 + 3 / 4 + 4 / 8 + 5 / 16 + . . . . My approach to the problem was essentially identical to what the Numberphile video outlines, though without their geometry. As a first introduction to the problem, this solution is actually quite satisfying. You take a complicated problem, reduce it to something that you already know how to do, and then see a nice result. That process itself is an important lesson.

Later I learned two other proofs. The first was to view the series 1 + 2x + 3x^2 + 4x^3 + . . . as the derivative of 1 + x + x^2 + x^3 + . . . , and the second was take the series 1 + 2x + 3x^2 + 4x^3 + . . . . and multiply it by (1 – x), which reduces it again to 1 + x + x^2 + . . . .

All three of these proofs are satisfying computations, but lack the depth and insight that your proof gives. I was quite happy to share your proof with my 7 year old who has only been looking at fractions for a few weeks because unlike the other computation proofs, in your proof the fractions fade into the background while the geometric insight takes center stage. I think it is quite a powerful instructional tool.

Glad to see that it made your top 5 or 6 posts for 2013.

## Comments

This is great, Mike. I created that representation for fun, mostly because that series had been haunting me and I wanted to figure out a different way to understand it. I never imagined it being used as an instructional tool.

Thanks so much for sharing! It’s a cool feeling seeing your work being used in unexpected, and authentic, ways!

I wanted to respond to this last night, but instead got distracted by an oh-so-exciting plumbing issue. yuck.

For some reason I remember quite clearly my first encounter with the series 1 + 2 / 2 + 3 / 4 + 4 / 8 + 5 / 16 + . . . . My approach to the problem was essentially identical to what the Numberphile video outlines, though without their geometry. As a first introduction to the problem, this solution is actually quite satisfying. You take a complicated problem, reduce it to something that you already know how to do, and then see a nice result. That process itself is an important lesson.

Later I learned two other proofs. The first was to view the series 1 + 2x + 3x^2 + 4x^3 + . . . as the derivative of 1 + x + x^2 + x^3 + . . . , and the second was take the series 1 + 2x + 3x^2 + 4x^3 + . . . . and multiply it by (1 – x), which reduces it again to 1 + x + x^2 + . . . .

All three of these proofs are satisfying computations, but lack the depth and insight that your proof gives. I was quite happy to share your proof with my 7 year old who has only been looking at fractions for a few weeks because unlike the other computation proofs, in your proof the fractions fade into the background while the geometric insight takes center stage. I think it is quite a powerful instructional tool.

Glad to see that it made your top 5 or 6 posts for 2013.

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