Math and Puzzle Fans Find Magic in Martin Gardner’s Legacy | Scientific American

October 29, 2024

7 min read

Math and Puzzle Fans Find Magic in Martin Gardner’s Legacy

Scientific American columnist Martin Gardner started a long mathematical conversation that continues today

By Sarah Lewin Frasier

Cutting a donut three times to get three pieces.

A. Martin UW Photography/Getty Images

Martin Gardner’s Mathematical Games column in Scientific American fascinated and mystified readers for decades—and his legacy continues to bring mathematicians, artists and puzzlers together.

Gardner had no formal mathematical training, and his path to science and math writing was a strange one. “He started out as a child magician, and the last thing he published was also a magic trick, about a month before he died,” says Colm Mulcahy, a professor emeritus of mathematics at Spelman College. “He started and finished with magic.” But in between Gardner had an 80-year publishing career as a writer and journalist and published more than 100 books. He became an expert at explaining math, science and skepticism to the public and perplexed people all over the world with his puzzles and paradoxes.

Gardner “didn’t have the high profile of his contemporary Richard Feynman or his friend Isaac Asimov, and he lacked the PR instincts of Salvador Dalí, who sought him out [to discuss four-dimensional shapes], or Steve Jobs,” Mulcahy says. “But his legacy might yet exceed theirs.” And Mulcahy would know—he is chair of the Gathering 4 Gardner Foundation, which runs a biennial conference that draws mathematicians, magicians and puzzle fanatics from around the world to talk about what fascinates them, as well as monthly talks and presentations on recreational mathematics, puzzling, science, and more.

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To celebrate the launch of our new Games section, as well as what would have been Martin Gardner’s 110th birthday on October 21, Scientific American talked with Mulcahy about Gardner’s impact, intriguing math puzzles, and the role puzzles and games play in creative and mathematical thought.

[An edited transcript of the conversation follows.]

Like many people, you first found out about Martin Gardner by reading his Mathematical Games columns—is that right?

Many dozens, if not hundreds, of people have told me over the years that they bought Scientific American for one reason: to read his column. A lot of the people who wrote in to him as teenagers or adults later became very successful people in the fields of mathematics and logic, philosophy, even computer science. His influence was enormous. And I saw his stuff when I was a teenager, but then I went off and did adult stuff [such as getting an advanced mathematics degree]. Then I found out later in life that it was okay to have fun as well. Full circle—in the past 20, 25 years, I’m reexploring the wealth of stuff that he brought to the public’s attention.

[Read Gardner’s 1956 column on strange folding flexagons here]

Do you remember the first puzzle or column of his that you saw as a teenager in Ireland?

I remember a couple. One [Gardner puzzle book] had a picture on the cover of what mathematicians call a torus, the shape of a doughnut or a bagel. Martin’s question was: How many pieces could you get by cutting it three times? [To see for yourself], you’d have to get a donut or a bagel—neither of which were available in Ireland at that time—and a short knife, then cut carefully. We now know the answer in terms of a general formula [for any number of cuts], but when he asked that in the 1950s, it was a totally new question.

Another fantastic puzzle of his I remember vividly involved an unusual shape. When I do recreational math with kids, the way I present it is the following: A cylinder looks round from one side but looks like a square (or rectangle) from another side. A cone is triangular from one perspective and circular from another. A square-based pyramid is triangular from one side and square from the other. The question is, can you think of a single shape that looks like a triangle from one side, a circle from a second side and a square from the third side? The shape doesn’t have a common name—and today in the age of “I can look it up on the Internet,” if something doesn’t have a name, it’s hard to Google it.

But the funny thing is, when I presented it to children in recent years, I’ve had kids say to me, “Oh, that’s like—” And they named something: a household item! So now I say to kids, “I bet you’ve got one in your bathroom” or “I bet there’s one in your house somewhere.”

You’ve said that Gardner’s version of the puzzle described the shape and asked to find the volume of one particular version of it—the one with triangular vertical cross-sections.

He had some great brain teasers. They weren’t all his own, but he was very good at giving credit for ideas from other people, and as he got famous, people would send him puzzles, and he would publish them, sometimes for the first time.

And people started sending him a lot more than just puzzles, right?

When he got respectable in the math community, which took him a few years, mathematicians would send him things that were news to the world, such as fractals invented by a man [mathematician Benoit Mandelbrot] who actually lived very near him, just outside Manhattan..., and Penrose tiles—he got the scoop on that. He got the scoop on so many things: RSA cryptography, [John Conway’s] Game of Life. These are all seminal things that launched little micro industries.

He was the go-to—if you were a mathematician and you had a question about a funny shape, you might write to Martin, and he would pass it on to somebody in Chicago who might know the answer. And that might bounce to Japan or New Zealand and come back a month later. He was the watercooler for intellectual ideas of that nature at a time way before the Internet.

And he drew readers into a field that previously seemed off-limits to those without advanced training.

The Gathering 4 Gardner conference tries to continue some of that watercooler spirit by hosting talks and activities focused on math, music, magic, puzzling, sculpture, and much more. What is that like?

The conference was started in 1993 in Atlanta [and has been held there ever since]. It was such a secret thing. It was kind of insiders—friends of Martin Gardner. It started off very small, with 30 or 40 people. [Gardner himself] was very shy, and he didn’t like being in a room with 30 people saying, “We worship you,” so he begged out pretty early. It’s now several hundred people from all over the world, and there’s always something that happens there that just blows your mind, something that nobody saw coming. You get a great mix of people. And luckily, as time goes by, we’re getting more diverse people, because that’s always been a challenge. The Gardner community has traditionally been older white guys, and as we die off, it’s very healthy to have a more representative cross section of the world population. So I’m glad to say that’s happening nowadays.

What kinds of events, conversations and interactions happen at these gatherings?

We have a series of very short talks, and the basic idea is the elevator pitch: “This is what I’m interested in; this is what I’ve discovered; if you want to know more, see me at the coffee break.” So it comes fast and furious. And there are a few plenary talks. [Inventor of the Rubik’s Cube Ernő] Rubik was there a few years ago—we brought in some big names. There are social events; there’s a lot of sculpture building and puzzles.

A lot of people are not mathematicians at all, [such as] art people who [work in] design and have a particular interest that overlaps. Some of them discover things that the mathematicians missed because they have better 3D instincts. And there’s some fantastic new puzzles.

Can you tell me about those types of 3D puzzles?

[Mathematician and sculptor] George Hart designs a lot of our activities, such as the sculpture builds, and he has designed some extraordinary puzzles. He has some videos that are very famous—I mentioned before that if you slice a bagel in the usual way, you get two half-bagels, each of which is a loop. If you just slam a knife straight down, you get two half-bagels of a different sort. But there’s another way to cut a bagel, which is extraordinary, so that you get two interlocked, looped half-bagels. And that sounds nuts. George Hart has a video of that.

[Read Gardner’s 1957 column on Möbius strips and other strange structures here]

George Hart has a puzzle that’s a 3D-printed pyramid that has been cut into two [winding] pieces of the same shape. The goal is to reassemble the pyramid. I’ve got one, and I tried for four or five years; I was totally convinced that there was something wrong, like the plastic was too stiff or something. I ran into George once, and he assembled it in a microsecond. There’s a twist, literally a twist—you have to twist it at exactly the right angle for it to slip together. I love minimalist puzzles like that. Which is better, 100-piece or 1,000-piece? Well, how about a two-piece? It messes with your brain.

To you, what’s the link between math and puzzling?

Some mathematicians like puzzles. Some, I would say more than half, do not like them. I’d say a lot of people hate them. They find them intimidating, just as some of the general public does, because some people do mathematics in a very rigid way. People think mathematics is very down-the-middle, with no choices to be made. Well, the breakthroughs are made by creative people; even [Albert] Einstein couldn’t solve all the equations with his bare hands, but he had ideas that nobody else had, and that’s often the case with mathematics. We’ve seen the examples of [amateurs, including Marjorie Rice in the 1970s, who collaborated with Gardner, and David Smith’s “einstein tile,” more recently] who came up with ideas that mathematicians had missed. And it just blew the socks off the math community. Mathematicians who are not creative, they’re just going to be teaching calculus and doing adult stuff, and I don’t want to party with them either.

Sarah Lewin Frasier is Scientific American's assistant news editor. She plans, assigns and edits the Advances section of the monthly magazine, as well as editing online news. Before joining Scientific American in 2019, she chronicled humanity's journey to the stars as associate editor at Space.com. (And even earlier, she was a print intern at Scientific American.) Frasier holds an A.B. in mathematics from Brown University and an M.A. in journalism from New York University's Science, Health and Environmental Reporting Program. She enjoys musical theater and mathematical paper craft.