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.


[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.”


[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!)


[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:


[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.


[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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The Value of Recreational Math

There was a wonderful opinion piece in The New York Times a few weeks ago about the importance of recreational math.

Now, as author Manil Suri said, I’m sure that to some people, the idea of recreational math sounds like an oxymoron. But it’s everywhere! From poker players calculating their odds based on the cards dealt to the number crunching in role-playing games in order to complete certain tasks (or develop a character’s skills), math is built into many recreational activities.

It’s certainly a part of many kinds of puzzles, including brain teasers. Heck, previous brain teasers featured here in the blog like Mystery Number, the Birthday Puzzle, and the jugs of water trap from Die Hard with a Vengeance would all easily fall under the umbrella of recreational math.

The article goes on to mention the wonderful work of Martin Gardner, whose column “Mathematical Games” in Scientific American was a mainstay of recreational math and puzzly whimsy for over twenty-five years.

From Suri’s article:

In his final article for Scientific American, in 1998, Mr. Gardner lamented the “glacial” progress resulting from his efforts to have recreational math introduced into school curriculums “as a way to interest young students in the wonders of mathematics.”

Indeed, a paper this year in the Journal of Humanistic Mathematics points out that recreational math can be used to awaken mathematics-related “joy,” “satisfaction,” “excitement” and “curiosity” in students, which the educational policies of several countries (including China, India, Finland, Sweden, England, Singapore and Japan) call for in writing.

In contrast, the Common Core in the United States does not explicitly mention this emotional side of the subject, regarding mathematics only as a tool.

This is an excellent counterpoint to the regular argument that the primary value of puzzle-solving and other activities (like recreational math) is to stave off brain health issues later in life.

In a previous post, we discussed the inconsistent reports about the effects of puzzle-solving on the brain, leaving it unclear if regular doses of puzzles and recreational math are beneficial for other aspects of brain health over time, like memory retention, neuroplasticity, and concentration.

That may well be the case, but Suri’s point stands. The idea of instilling a sense of fun and wonder into the field of math, especially for younger minds? That’s one worth pursuing.

Show them that math can be about more than fulfilling homework or graphing parabola. Show them that mathematical concepts can help you crack a diabolical seesaw brain teaser, save a village with a grain of rice, or find an alternate solution to a PSAT question and prove the testers wrong.

It has been championed in the past by television shows like Square One TV and MythBusters, but sadly, examples like that are few and far between.

And if we can instill recreational math as a key facet of math itself, then we’d be one step closer to ensuring that STEM courses will have plenty of participants in the future.

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Picture this…

There are all kinds of puzzles. That’s one of the best things about being a puzzle fan: the fact that there are so many varieties of puzzle out there waiting to challenge you. From word searches and crosswords to Sudoku and cryptograms, from brain teasers and sliding-tile puzzles to pattern puzzles and rebuses, the possibilities are endless.

One of the most engaging (and surprisingly challenging, I’ve found) fall under what I’d call picture puzzles.

Picture puzzles demand keen visualization skills, since they often involve manipulating shapes in your head. You might be asked to deduce what letter is on the hidden side of a six-sided die, or how to divide farmland with only three lines and separate every sheep from each other. It’s visual trickery and topnotch puzzle-solving wrapped up in one.

And one of my favorite Picture puzzle wizards is Sam Loyd.

I was introduced to Mr. Loyd’s puzzles in Mathematical Puzzles of Sam Loyd — a collection assembled by puzzle aficionado Martin Gardner — a book I stumbled upon in the library one day while strolling the shelves.

The collection offers math puzzles and brain teasers, deduction problems and tracing games, but the ones that most intrigued me were his Picture puzzles.

Now, adding pictures to puzzles is hardly a new idea. David L. Hoyt’s Jumble puzzle features one, and plenty of Matching / Spot the Difference puzzles — like the Match-Up and The Shadow puzzles offered by our pals at Penny/Dell Puzzles — rely heavily on art. But there’s a marvelous charm to Loyd’s drawings, creating a scene and telling a small story as he offers one brain teaser after another.

I’ll post a few of my favorites, so you can get a sense of not only his artistic stylings, but the extreme cleverness involved in creating these puzzles.

Each of these puzzles requires a keen ability to visualize shapes in your head, as well as some serious craftiness to conquer the challenge Loyd sets before you.  Have you figured them out?

I’ll post the solution to the first one and leave the others for you to puzzle out.

[I didn’t line it up perfectly, but you get the idea.]

Loyd is part of a marvelous tradition of inventiveness and creativity that has helped contribute to the rich and vibrant puzzle world we enjoy today.

[To check out more of Sam Loyd’s puzzles, you can visit this website dedicated to his puzzly legacy (including several puzzles for sale!), a treasure trove of Picture puzzle fun.]

Thanks for visiting the PuzzleNation blog today! You can like us on Facebookfollow us on Twitter, cruise our boards on Pinterest, check out our Tumblr, download our Classic Word Search iBook (recently featured by Apple in the Made for iBooks category!), play our games at PuzzleNation.com, or contact us here at the blog!

Celebrating a true elder statesman of puzzles

Yesterday was Martin Gardner’s birthday. (And unfortunately, I wasn’t feeling well, so I apologize for not getting this post up in a more timely manner.)

If you don’t know Martin Gardner, you absolutely should.

For twenty-five years, he 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.

His affection for magic, puzzles, and mathematics was infectious, and events known as Gatherings 4 Gardner began springing up. After his passing on May 22, 2010, his legacy now lives on thanks to an annual global event known as the Martin Gardner Celebration of Mind.

Magic tricks, puzzles, recreational math problems, and stories about Martin are shared by admirers and devotees, all in the hopes of maintaining and spreading the wonderful spirit of playful mathematical experimentation that Gardner embodied so brilliantly.

So today, we here at PuzzleNation invite our fellow puzzle fiends and math lovers to join us in celebrating not only Martin’s life and love of all things mathy and puzzly, but the marvelous tradition of sharing those interests with others.

And in the spirit of sharing similar puzzle joy, I proudly give you our own little contribution… the mobius bagel.

[For a complete bibliography and breakdown of the fifteen printings of Gardner’s Mathematical Games columns, click here. To check out some of his many delightful columns, click here. And to solve a number of Martin’s most popular puzzles and brain teasers, click this link provided by our friends at ThinkFun.]

Thanks for visiting the PuzzleNation blog today! You can like us on Facebook, follow us on Twitter, cruise our boards on Pinterest, check out our Classic Word Search iBook (recently featured by Apple in the Made for iBooks category!), play our games at PuzzleNation.com, or contact us here at the blog!

5 Questions with Tanya Thompson of ThinkFun

Welcome to the fifth edition of PuzzleNation Blog’s interview feature, 5 Questions!

We’re reaching out to puzzle constructors, video game writers and designers, writers, filmmakers, and puzzle enthusiasts from all walks of life, talking to people who make puzzles and people who enjoy them in the hopes of exploring the puzzle community as a whole.

And I’m excited to have Tanya Thompson as our latest 5 Questions interviewee!

Tanya is Head of Inventor Relations at ThinkFun, literally traveling the world to meeting with inventors and puzzle innovators in order to create new puzzle game products under the ThinkFun brand. She was part of the team that made Laser Maze a reality — check out our review of Laser Maze — and in the photo, she’s playing around with their latest puzzle game, WordARound (another coup for her and the ThinkFun team).

Tanya was gracious enough to take some time out to talk to us, so without further ado, let’s get to the interview!

5 Questions for Tanya Thompson

1.) What in your estimation makes for a truly great puzzle or puzzle game?

Good question. It depends on the context. If you’re asking me personally, as an avid gamer and puzzler, I like puzzles that are new or innovative. I have over 1000 puzzles in my collection and I attend the International Puzzle Party (IPP) where some of the brightest puzzle minds (creators and solvers) gather. I love it when I see something truly new, whether it be a mechanism, a component or a solving technique.

If you’re asking me what I look for regarding a ThinkFun puzzle or puzzle game, that’s different. For ThinkFun it needs to be commercially viable. It has to be interesting to the masses, not just puzzle nuts like myself. If it can sit on the shelf and cause people just walking by to pick it up and play with it, then it might be right for ThinkFun.

As for my favorite puzzles – That’s a tough question. I have a lot of favorites. I love Rush Hour because I think ThinkFun revolutionized the toy and game industry with it. This was an entirely new way of doing puzzles. Of the classics, I love the Soma Cube, both for its elegance and its mathematical completeness.

I must mention designers here too. I love Iwahiro because he always surprises me with his puzzles. One year he’ll do wood, then aluminum, then cloth. He is so creative. Vesa Timonen and Timo Jokitalo develop wonderful single player games as well as aha-style. Oskar van Deventer has revolutionized twisties. Akio Kamei creates the most beautiful puzzles that look like everyday objects and the key to their solutions lie in those objects. Kagen Schaefer does exquisite puzzle furniture. I could go on and on!

2.) What ThinkFun products are your favorites, and which are you most proud of? Is it difficult to walk the tightrope of producing challenging, educational, AND fun products?

I love ThinkFun and what we stand for. We produce addictively fun games that sharpen and challenge your mind. We want to make the world better thinkers. I think our products do just that. I’m especially proud of this year’s releases Laser Maze and Word Around because I championed them into the company from the inventors. However kudos need to be given to the amazing Product Development team that took the ideas and made them into exceptional products as well as our incredible sales and marketing teams who knew just how to get them out into the world. You asked if it is difficult to walk the tightrope of producing challenging, educational AND fun products, it’s not so much that it’s difficult, it’s more that it is who we are. We’re awesome tightrope walkers!

3.) What’s your relationship to puzzles and games, and how did you come to be an integral part of ThinkFun?

I’ve always loved puzzles and games. My first career was a mathematics teacher and I used puzzles and games in my classroom to inspire my students. One of ways I did this as a teacher was to organize a SNAP Math Fair in my school. It brought puzzles into the classroom.

Through my work with SNAP I met Bill Ritchie, the CEO of ThinkFun. Bill soon hired me, and now I travel the world meeting inventors to bring in the new ideas for ThinkFun, and then work as part of a team that develops the ideas into products. Bill is also very passionate about reaching communities of people that want activities and games to exercise their brains. So I am also on a team of people working on programs that do just that! I love what I do and I’m blessed to have such a great job with ThinkFun!

4.) What’s next for Tanya Thompson and ThinkFun?

What’s next for me? Who knows. I’d love to attend TED someday, I’d love to travel to Asia and I’d love to sit and play a board game with Wil Wheaton. More imminent, I have been asked to launch a puzzle/game newsletter this fall. It will focus on great puzzle/game people in our industry, and will include a puzzle/game takeaway.

I am also excited to soon be off to find our next big thing! I’ve built an incredible network of inventors and creatives and I look forward to the fall when our submission cycle opens back up and I’ll be out there seeing great new ideas again! What’s next for ThinkFun? You’ll have to wait and see but you can be certain that it will be more products and programs that will sharpen and challenge your mind!

5.) If you could give the readers, aspiring inventors, and puzzle fans in the audience one piece of advice, what would it be?

Live passionately and get involved. I am passionate about mathematics, puzzles, games and education. When I was a teacher, I taught with passion. It brought me to ThinkFun and I now have a job that allows me to work within the fields I’m passionate about. 

Also, get involved with other people who share similar passions. I chair a committee called the Gathering for Gardner – Celebration of Mind (G4G – CoM) that promotes Martin Gardner’s life’s work. Martin wrote a column for 25 years in Scientific American called Mathematical Games. He was passionate about math, magic and puzzles.

Around his birthdate of October 21st parties/events occur around the world. Big or small, it doesn’t matter. The important thing is that people gather to share math, magic and puzzles. I was lucky enough to call Martin a friend and I am honored to be a part of this organization.

I am also on the Executive Advisory Board of the Chicago Toy and Game Week, a great series of events held in November for anyone wanting to know more about or to network within the Toy and Game Industry.

Many thanks to Tanya Thompson for her time and ThinkFun for their puzzly camaraderie! With so many interesting and innovative puzzle games in their arsenal, I’m sure we’ll be talking about them again soon.

Thanks for visiting the PuzzleNation blog today! You can like us on Facebook, follow us on Twitter, check out our Classic Word Search iBook (recently featured by Apple in the Made for iBooks category!), play our games at PuzzleNation.com, or contact us here at the blog!