Celebrating the Puzzly Legacy of John Horton Conway

The worlds of puzzles and mathematics overlap more than you might think. I’m not just talking about word problems or mathy brain teasers like the Birthday Puzzle or the jugs of water trap from Die Hard with a Vengeance.

For twenty-five years, Martin Gardner penned a column in Scientific American called Mathematical Games, adding a marvelous sense of puzzly spirit and whimsy to the field of mathematics, exploring everything from the works of M.C. Escher to visual puzzles like the mobius strip and tangrams. He was also a champion of recreational math, the concept that there are inherently fun and entertaining ways to do math, not just homework, analysis, and number crunching.

And on more than one occasion, Gardner turned to the genius and innovative thinking of John Horton Conway for inspiration.

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[Image courtesy of Wikipedia.]

Conway was best known as a mathematician, but that one word fails to encapsulate either his creativity or the depth of his devotion to the field. Conway was a pioneer, contributing to some mathematical fields (geometry and number theory among them), vastly expanding what could be accomplished in other fields (particularly game theory), and even creating new fields (like cellular automata).

Professor of Mathematics, Emeritus, Simon Kochen said, “He was like a butterfly going from one thing to another, always with magical qualities to the results.” The Guardian described him in equally glowing terms as “a cross between Archimedes, Mick Jagger and Salvador Dalí.”

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[Image courtesy of Cornell.edu.]

His most famous creation is The Game of Life, a model that not only visually details how algorithms work, but explores how cells and biological forms evolve and interact.

Essentially, imagine a sheet of graph paper. In The Game of Life, you choose a starting scenario, then watch the game proceed according to certain rules:

  • Any live cell with fewer than two live neighbors dies, as if by underpopulation.
  • Any live cell with two or three live neighbors lives on to the next generation.
  • Any live cell with more than three live neighbors dies, as if by overpopulation.
  • Any dead cell with exactly three live neighbors becomes a live cell, as if by reproduction.

The process plays out from your starting point completely without your intervention, spiraling and expanding outward.

It’s the ultimate if-then sequence that can proceed unhindered for generations. It is a literal launchpad for various potential futures based on a single choice. It’s mind-bending and simple all at once. (And you can try it yourself here!)

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[Image courtesy of Sign-Up.To.]

But that’s far from Conway’s only contribution to the world of puzzles.

Not only did he analyze and explore puzzles like the Soma cube and Peg Solitaire, but he created or had a hand in creating numerous other puzzles that expanded upon mathematical concepts.

I could delve into creations like Hackenbush, the Angel Problem, Phutball/Philosopher’s Football, Conway’s Soldiers, and more — and perhaps I will in the future — but I’d like to focus on one of his most charming contributions: Sprouts.

Sprouts is a pencil-and-paper strategy game where players try to keep the game going by drawing a line between two dots on the paper and adding a new dot somewhere along that line.

The rules are simple, but the gameplay can quickly become tricky:

  • The line may be straight or curved, but must not touch or cross itself or any other line.
  • The new spot cannot be placed on top of one of the endpoints of the new line. Thus the new spot splits the line into two shorter lines.
  • No spot may have more than three lines attached to it.

Check out this sample game:

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[Image courtesy of Fun Mines.]

It’s a perfect example of the playfulness Conway brought to the mathematical field and teaching. The game is strategic, easy to learn, difficult to master, and encourages repeated engagement.

In a piece about Conway, Princeton professor Manjul Bhargava said, “I learned very quickly that playing games and working on mathematics were closely intertwined activities for him, if not actually the same activity.”

He would carry all sorts of bits and bobs that would assist him in explaining different concepts. Dice, ropes, decks of cards, a Slinky… any number of random objects were mentioned as potential teaching tools.

Professor Joseph Kohn shared a story about Conway’s enthusiasm for teaching and impressive span of knowledge. Apparently, Conway was on his way to a large public lecture. En route, he asked his companions what topic he should cover. Imagine promising to do a lecture with no preparation at all, and deciding on the way what it would be about.

Naturally, after choosing a topic in the car, the lecture went off without a hitch. He improvised the entire thing.

Of course, you would expect nothing less from a man who could recite pi from memory to more than 1100 digits? Or who, at a moment’s notice, could calculate the day of the week for any given date (employing a technique he called his Doomsday algorithm).


Conway unfortunately passed away earlier this month, due to complications from COVID-19, at the age of 82.

His contributions to the worlds of mathematics and puzzles, not to mention his tireless support of recreational math, cannot be overstated. His work and his play will not soon be forgotten.

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[Image courtesy of Macleans.]

If you’d like to learn more about Conway, be sure to check out Genius at Play: The Curious Mind of John Horton Conway by Siobhan Roberts.

[My many thanks to friend of the blog Andrew Haynes for suggesting today’s subject and contributing notes and sources.]


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Farewell, Keith.

The puzzle world is a relatively small one, and sadly, it grew smaller last week, as friend of the blog Keith Yarbrough passed away.

On more than a few occasions, I’ve asked my fellow constructors and cruciverbalists for their help on blog posts, whether the topic was advice for solving crosswords, constructing puzzles, or writing dynamic clues. Keith was often the first person I would turn to for help.

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A long-time member of the Penny Dell Puzzles family, Keith specialized in crossword/variety magazines, blending a knack for charming and clever cluing with eclectic themes for the many variety puzzles he crafted and edited. His varied interests ensured that he never ran out of ideas for interesting themes or intriguing twists on worn-out crossword tropes.

Keith was equally at home in a classroom, in an orchestra, and in a crossword tournament, a true lover of both the arts and the sciences. His affection for music was well-known — many bands, including The Optics and The Gene Gnomes, can attest to his skill playing the tuba — and yet, he could unravel a deduction problem or a fiendish math puzzle as easily as he could read the notes on sheet music.

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Having worked together for more than a decade, I struggle to recall all of the topics we covered in conversation together. We talked Nobel Prize nominees and quantum physics one week, then Looney Tunes cartoons and silent comedy film shorts the next.

We would dissect the minutiae of Breaking Bad and Twin Peaks, share stories about our dogs (I prefer labs and retrievers, Keith loved his poodles), or recount entire George Carlin routines from memory. When he found out I was a wrestling fan, he laughed and told me about the time he met Mighty Igor outside a sandwich shop. (He grew up in North Carolina, a hotbed for wrestling in decades past.)

And, of course, we talked about puzzles. Keith was a pro at whipping up new clues for crosswords, and they were often as immensely clever as they were completely inappropriate for a family-friendly audience. Several of the funniest clues that I’ve featured in previous blog posts were his — and he was always striving to find new ways to clue tiresome words or to push the boundaries of humdrum constructing and “appropriate entries”. (Just last week, he had to re-edit a puzzle of one-word film titles because he tried to sneak “Caligula” past the censors. *laughs*)

One time, while trying to fix a submitted crossword, he turned to me and asked with a smile, “I know this is a long shot, but is there a non-offensive way to clue ‘witch hunt’?” (The best we could come up with was “pressing engagement.” And no, that would never ever make the cut.)

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I’ll miss his humor, his wit, and his friendship. And I know that I’m far from the only one who feels that way.

Kind words have been pouring in for days now, as people share photos and memories of Keith. The Winston-Salem Symphony dedicated a concert in his memory a few nights ago. (I pilfered several of the photos in this post from those I’ve seen shared.)

It was my privilege to work with Keith for over a decade, and I’ll miss him very much. I’ll miss the fascinating, weird, unexpected bits of trivia he’d throw my way. I’ll miss the music references he would gamely try to explain to me. I’ll miss the way he was always a half-step faster than me on brain teasers and word puzzles. I’ll miss the sly ways he pushed the creative envelope. I’ll miss him a different way every day.

Farewell, my friend. Farewell, Keith.


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