## Genius at Play

The last time I visited John Horton Conway, he showed me a book, *Genius At Play: The Curious Mind of John Horton Conway* by Siobhan Roberts. It had nothing to do with him wanting to brag about the book. Nothing, nothing at all.

He just wanted to show me a picture from the book. While he was browsing this book about himself, I took a photo of him (featured on the left). The picture he was looking for was on page 314. I am reproducing it here (Photo by Michael A. Stecker, courtesy of Stephen D. Miller).

When we finally found the picture, he asked me how many tennis balls were in it. I smelled a trick question right away and didn’t bite. I shrugged. It appears that one of the balls at the far corner at the base of the pyramid had rolled away. So there was one less ball than I would have calculated. So, how many balls are there?

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## tomislater:

1662?

28 July 2015, 5:14 pm## Naveen Dankal:

down vote accepted

There are two different kinds of layers here, as it alternates between centered hexagonal numbers (e.g. the top layer is the first centered hexagonal number, i.e. 1, the third layer from the top is the second centered hexagonal number, i.e. 7, etc.), and layers where the sides alternate between lengths n and n+1 (e.g. the second layer from the top of the pyramid is a ‘hexagon’ with sides 1 and 2, the fourth layer is a hexagon with sides 2 and 3, etc.

So, the numbers of tennis balls in all the odd layers of this pyramid is 9^3=729

For the other layers: The formula for the figure with sides n and n+1 is 3n^2.

Since you have 8 of those layers, you have 3 times the sum of the first 8 square numbers, and the general sum of the first n square numbers is n(n+1)(2n+1)/6, so that gives you 3⋅8(8+1)(2⋅8+1)/6=612 more tennis balls, for a total of 729+612=1341

18 January 2018, 11:15 pm## 魔大农:

Hi Tanya,

I was wondering if you’d be willing to contribute your photo(s) of John Conway to the public domain, by assigning a creative commons license. If you could upload them to commons.wikimedia.org yourself, that would be perfect, but I would also be happy to do that. The latest photos of JHC now available in this way are from 2005.

Kind regards,

31 May 2020, 5:30 pmFrode 魔大农 Lindeijer

## tanyakh:

I tried to upload this picture to wiki. I am not sure it worked. Also, there is a typo in the title. I can’t fix it. It would be great if someone can remove the text after the period.

9 July 2020, 9:21 pm