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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Unraveling the Riddle of Math Puzzles!

Math puzzles are among the most intimidating in the world of puzzles. Many people will happily dive into a crossword or tackle a word seek at a moment’s notice, but drop some numbers into a puzzle, and they hesitate.

But there’s no reason to fear!

Math puzzles are certainly a different form of puzzling, but like all puzzles, there’s always a way in, if you know how to look for it. Today, we’re going to solve two math puzzles together in the hopes of demystifying this style of puzzle.

Let’s take a look at our first math puzzle, “Count the Votes.”

A problem developed at a recent election where 5,219 votes were cast for four candidates. The victor exceeded his opponents by 22, 30, and 73 votes, yet not one of them knew how to figure out the exact number of votes received by each. Can you?

Okay, where do we begin?

Let’s start with what we know. We know the total number of votes, 5,219. That will be one side of our equation.

We also know that the winner beat his three opponents by 22 votes, 30 votes, and 73 votes, respectively. Which means that the number of votes the winner received is the key to solving this puzzle. Let’s call that number of votes “x.”

The winner beat one opponent by 22 votes (x – 22), another by 30 votes (x – 30), and the last by 73 votes (x – 73).

We can build our simple equation from that information:

x + (x – 22) + (x – 30) + (x – 73) = 5219

Still a little daunting, but we can simplify it, because it doesn’t matter in which order we add or subtract things. So let’s look at that formula without the parentheses:

x + x – 22 + x – 30 + x – 73 = 5219

Now let’s reorganize it, putting the addition parts together and the subtraction parts together:

x + x + x + x – 22 – 30 – 73 = 5219

Subtracting those three numbers separately is the same as subtracting their total, so let’s simplify again:

x + x + x + x – 125 = 5219

Adding four x’s together is the same as multiplying one x by 4, so let’s express that:

4x – 125 = 5219

Now we’re getting somewhere.

And subtracting 125 from 4x is the same as adding 125 to 5219, so let’s do that:

4x = 5344

Finally, we divide 5344 by 4 to give us the value of x:

x = 1336

Which means that our victor got 1336 votes, one opponent got 1314 (x – 22), another opponent got 1306 (x – 30), and the last got 1263 (x – 73), totalling 5129 votes.

Now, that wasn’t so bad, was it? Let’s try another that’s a little bit harder.

This one is called “The Mathematical Cop.”

“Top of the mornin’ to you, officer,” said Mr. McGuire. “Can you tell me what time it is?”

“I can do that same,” replied Officer Clancy, who was known on the force as the mathematical cop. “Just add one quarter of the time from midnight until now to half the time from now until midnight, and it will give you the correct time.”

Can you figure out the exact time when this puzzling conversation took place?

Okay, this one isn’t as obvious about providing us with information, but the info is there if you look.

Since everything relates to the time “now,” we’ll make “now” our x.

Then we take each part of Officer Clancy’s statement in turn. “Just add one quarter of the time from midnight until now.”

“The time from midnight until now” is the same as “now,” x, so one quarter of that time is x/4.

And we’re meant to add that to “half the time from now until midnight.”

That’s a little bit tougher. After all, “the time from midnight to now” was easy, but “the time from now until midnight” covers the rest of a 24-hour day. So, if x covers the time from midnight to now, then “1440 – x” covers the time from now until midnight.

(There are 1440 minutes in a day, 60 minutes times 24 hours, and it’s easier to do all this in minutes, rather than hours and minutes.)

So “half the time from now until midnight” is (1440 – x)/2.

Okay, so what does our equation look like?

x/4 + (1440 – x)/2 = x

That’s pretty daunting, but we know what our goal is: to combine all those x’s and get them on the same side of the equal sign. And like the equation we built for “Count the Votes,” we can simplify it with some careful applied math.

The first step is to get rid of those pesky fractions.

Let’s multiply everything by 2 in order to remove the “/2” below “(1440 – x),” which gives us:

2x/4 + (1440 – x) = 2x

We can use the same trick to remove the “/4” below 2x:

2x + 4(1440 – x) = 8x

Now we’re getting somewhere! Let’s get rid of that 2x on the left by subtracting 2x from both sides:

4(1440 – x) = 6x

Let’s go a step further by multiplying both 1440 and x by 4:

5760 – 4x = 6x

One more step, and we’ve got all of those x’s combined on one side of the equation, as we’d hoped:

5760 = 10x

Divide 5760 by 10 and we’ve got x:

576 = x

If you recall, x represented the time “now,” but it’s still in minutes. To get the actual time, divide 576 by 60 to get the number of hours. 540 minutes = 9 hours, so 576 is 9 hours, 36 minutes.

It’s 9:36 AM, Officer, though to be honest, if you tell everyone the time this way, I imagine people stop asking you the time after a while.

I realize these are only two examples, and math puzzles come in all shapes and sizes, but hopefully, they don’t seem quite so intimidating, now that you know how to pick them apart for the important information.

Good luck! And if you find any math puzzles you need help with, send them our way! They could end up the subject of a future blog post!

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