In 1974, a geneticist named Marsha Jean Falco devised an ingenious research tool to help determine whether epilepsy in dogs was an inherited trait. She drew a series of symbols on index cards, where each card represented a dog and each symbol represented a DNA sequence, to create her own coding system. But as she shuffled and reshuffled the index cards over time, she began seeing the deck in terms of pure abstract patterns and combinations.
Eventually her personal coding system became the game of Set—just one of the many math-y games included in math teacher and bestselling author Ben Orlin's new book, Math Games with Bad Drawings. (You can read an excerpt and try your hand at a game of Quantum Go Fish here.)
Orlin's first book, Math with Bad Drawings, after his blog of the same name, was published in 2018. It included such highlights as placing a discussion of the correlation coefficient and "Anscombe's Quartet" into the world of Harry Potter and arguing that building the Death Star in the shape of a sphere may not have been the Galatic Empire's wisest move. We declared it "a great, entertaining read for neophytes and math fans alike, because Orlin excels at finding novel ways to connect the math to real-world problems—or in the case of the Death Star, to problems in fictional worlds."
In 2019, Orlin took on the challenge of conveying the usefulness and beauty of calculus with tall tales, witty asides, and even more bad drawings in Change Is the Only Constant: The Wisdom of Calculus in a Madcap World. That book is a colorful collection of 28 mathematical tales connecting concepts in calculus to art, literature, and all manner of things human beings grapple with on a daily basis. So a third book collecting a wide variety of math-y games was a natural progression—illustrated, as always, with Orlin's own distinctively bad drawings.
Math Games with Bad Drawings isn't really designed to be read cover to cover; rather, readers can browse randomly at their leisure to find the games best suited to their particular tastes and skills. (Wordle fans should enjoy Jotto, a similar logic-oriented word game invented in 1955.) There are 75.25 games in total, at least by Orlin's idiosyncratic count. "I wanted a non-integer number of games," he told Ars. Technically, there are around 51 full games, plus several he counted as 11/12s, because, even though they are full games, they didn't merit more than a brief mention in the book. Variations of existing games counted as one-quarter games, while very tiny rule variations counted as 1/57th of a game. Add it all up and you get 75.25.
The games can all be played with just a few common household items: pencil and paper, coins, colored pens, standard dice, Goldfish crackers, paper clips, your hands, and occasionally an Internet connection. There are five sections focused on five different categories of games: spatial games, number games, combination games, games of risk and reward, and information games. For each, Orlin outlines the basic rules of play, a few "tasting notes" about the subtleties of the gameplay, where the game came from, and why it matters. He also lists any variations and related games.
For instance, Ultimate Tic-Tac-Toe involves two players navigating nine smaller ("local") tic-tac-toe boards in a 3×3 grid, and the choices made determine the positions on a larger ("global") board. Quantum Tic-Tac-Toe is even trickier, whereby players must place their "entangled particles" (Xs or Os) in such a way that when the "wave function" collapses, the winning player will be left with three in a row. (The wave function collapses when various pairs of boxes form an "entangled loop": Box 1 is entangled with Box 2, which is entangled with Box 3, which is entangled with Box 1.)
"I tend to like the party games, the ones that actually work well for a group, more than the very intricate two-player, abstract strategy," said Orlin, although he admits to having a soft spot for Ultimate Tic-Tac-Toe, the game that inspired this book. Among his favorites is Sequencium, a game of pure strategy, which, like chess, tends to favor the player who moves first.
"You've got this little grid, and you start with some numbers coming out of the corner, and then you just keep adding one to the number and putting it into the next box," said Orlin. "You're trying to grow this little vine of numbers as you move around the board. It's a very simple ruleset but makes for a really surprisingly rich game." Bonus: Per Orlin, there's one surefire way for the second player to ensure a draw. Simply rotate the board 180 degrees and copy the first player's moves symmetrically.
Orlin is already hard at work on his next book: How to Speak Math: A Guide to the Universal Language for Native and Non-Native Speakers. Unlike his prior books, where he carefully avoided most mathematical notation, this one will teach readers how to read and interpret mathematical symbols, proofs, and mathematical diagrams. "Mathematical notation is off-putting for a lot of people, and you can discuss what is exciting and intellectually rich about mathematics without getting into the notation," he said. "But there are also certain things about math that you can only access through that notation."
Ars spoke with Orlin to learn more.
Ars Technica: Why math games?Ben Orlin: There's this thing that math does for people who enjoy math. You hear a setup to a problem and you just get pulled in. "Ooh, what is that going to create? Those choices of axioms or assumptions, what is going to emerge from that?" But other people don't respond to math that way. You give them the setup of a problem, and they shrug their shoulders. It's not an exciting prompt.
But most people want to try games. They want to see what happens when you start playing around in the space created by these rules. So I wanted to give people the experience of that kind of thinking, when you get into a ruleset and start messing around and seeing what possibilities it allows. Where do you have freedom, where are you constrained? It's all very logical thinking and to me, that's a very mathematical kind of thinking. Games are a great backdoor into that thinking because people just intrinsically enjoy it. With a more austere or abstract mathematical setting, they might not dive in as readily.
Ars Technica: You wrote this during the pandemic, when lots of people were isolated, so you deliberately picked games where you had to play it with at least one other person. What sets your book apart from all the other math puzzle books out there?Ben Orlin: I was surprised to find there wasn't a book like the one that I've just written out there already. There's lots of wonderful puzzle books; you can find shelves full of great books of mathematical puzzles and brain teasers and Sudokus and geometry problems. But not a book full of games like this.
Ars Technica: Certainly not one with such bad drawings.Ben Orlin: Exactly. The ones out there were much too well-illustrated, and nobody wants to read that. Games are constantly generating new puzzles. With a puzzle, you solve it, you're done. A game is like a fountain of puzzles that's constantly pouring out new puzzles for you. One thing I like about two-player games, or multiplayer games, is that you wind up creating puzzles for each other. Your cleverness in making this move is what necessitates the other person to be clever and make their move in response. You're playing a strategy game with someone else, setting each other a series of puzzles back and forth.
Ars Technica: You define a “math game” in your introduction as one that “makes you go, ‘Mmmm-mmmm, that’s math-y.’” Can you be a bit more specific?Ben Orlin: Everyone in the tabletop gaming community, for example, has very particular tastes. This book has my own aesthetic fingerprints on it, even though I draw the games from lots of different places. My preference is for games with very simple rulesets. I tend to like elegance. Some popular board games are like that. Some popular board games are a little more like very well-built Swiss watches where you have all these intricately balancing parts. It creates this beautiful game because you've got one mechanism that balances out another mechanism and forces you to make a decision.
The games in my book aren't really like that; they're much more elemental. I tried to pick rulesets where there's one idea behind each game: you're growing your vine of numbers, or you are trying to force the other person to pick too large of a number, or you are trying to take up as much space as you can, or you're trying to block the other person from being able to build any structure. I wanted the games to all be ones that you could pick up and read about and get playing within five minutes at the most.
The games here are functioning both as games and also as thought experiments or mathematical inspirations. For that to work, the main idea needs to be legible from the start. For example, Quantum Tic-Tac-Toe is giving you a little taste of what quantum mechanics is like. Ultimate Tic-Tac-Toe is giving you this sense of fractal structure and how the larger scale corresponds to the smaller scale. Another is a game of hidden information. You're going to draw a little map of regions and you're going to try to figure out your opponent’s map by probing for information. As a teacher, I want something where in 45 seconds at the front of the class, I can give you the overview of what we're learning, what we're exploring today. That'll create space where you can go lots of different directions and have lots of different fun experiences.
Ars Technica: As a teacher, what are your thoughts on the importance of incorporating play into learning?Ben Orlin: There are lots of great philosophers who will tell you that play is the essence of all thought, all activity, all behavior. There are times in our lives when we're just following the train tracks and doing what is prescribed. Those are not the parts of life that we love. Those are not parts of life that make life feel worth living. It's the moments of improvisation and play and the unexpected, when you're exploring possibilities. I think that's true in math and in life in general.
What we tend to do in math education is break off very discreet skills. So you're going to solve this kind of equation, with the constants set up in this way and the variables on this side, and then practice 10 of those. The next day we'll move on to a slight variant of that. And then the next week we'll try it with inequalities. You're practicing very isolated skills. So the equivalent to me would be like, if you were playing a sport or playing soccer, and you're going to do a passing drill on Monday. Tuesday is a slightly different kind of passing drill. Wednesday, you're doing shooting drills. You just do drill, after drill, after drill and you never actually play the games. You never get to see these skills put together into the fluid whole.
Now, to excel at soccer, you actually do need to do some skill practice. But you also need to do a lot of playing the game. If someone had never seen soccer before, it would be crazy to make them wait a decade through drill after drill after drill before they get any glimpse of what the sport is like. A lot of math education feels that way. You get very rare glimpses of what the game of mathematics is like, trying to find abstract structure beneath this messy reality.
That's the cycle of mathematics: You go to the world, you get some abstract idea, you deal with the abstraction, and then you transport the insight back into reality. That cycle is so rarely completed in school. It is very much like a piano student who only ever gets to practice their scales and never gets to play a song, or a soccer player who just is doing passing drills all week long and never gets to play a game.
Ars Technica: Can these games be incorporated into classroom use?Ben Orlin: I probably should be saying, "Oh yeah, these are perfect pedagogical tools." But they're not particularly curricular. I conceived of them as logical thinking experiences. I didn't pick them to be mathematical lessons. I picked them to be fun games and then tried to find a little mathematics to tie into them. But I think they give you a slightly more authentic mathematical experience than something that might look more like traditional mathematics. They give you a chance to explore and try different things and to navigate a logical space, which is what a lot of mathematics is about.
One of the things I like doing when introducing something new is to just play a game of 'which one is larger?' I was doing exponents with the middle schoolers recently. Seven to the eight versus eight to the seven: which one do you think is larger? You have a little discussion, take a vote, you do a little calculation, and then you come back and see if you can reach a consensus as a class. That's not a random example. A lot of what we're doing in mathematics is asking which is bigger— comparing sizes of things.
The insight that games offer is that the goal is often very narrow. You're trying to score more points than your opponent, or you're trying to get the ball in the net. But the methods you might use to achieve the goal are incredibly wide-ranging, and you have almost infinite degrees of freedom. Math education could do a little bit more of that—making clear what the goals are—because often the intellectual goals of math class aren't particularly clear to students.
Over history, we've discovered very, very efficient processes for achieving certain goals: computational algorithms, ways of transforming a certain problem into a certain kind of equation that's very easy to solve. So we leap right into teaching these very efficient problem-solving techniques without motivating the problems that we're trying to solve.
Ars Technica: Games can also feed into mathematical research in some interesting ways.Ben Orlin: AlphaGo is the best example of a game sitting right at the frontier of research into computer science or mathematics. There's a lot of healthy interplay between games and mathematical research. That's because a lot of the people who do mathematical research and are looking for new ideas also love games. When Claude Shannon and Alan Turing were sitting together having their lunches, they couldn't really talk about their work because they were both doing top-secret work. So what would they talk about?
They'd talk about a chess-playing computer, because they both love chess and they both were founders of this computer revolution. That was just a natural thing for them to think about. What would you do if you wanted to make a computer be a good chess player? So, in some sense, you can look through the history of computer science and artificial intelligence and see the changing regimes and the new ideas coming in, in terms of how people are thinking about these kind of classic game-playing tasks.
Ars Technica: Was there anything surprising that you discovered while you were writing the book?Ben Orlin: The idea that mathematically rich and complicated objects could actually captivate a tremendous mass audience. It doesn't happen when you just present the math as math. But if you package the math as an interesting game or a puzzle, something they can get their hands on, it can become a viral hit. Those who were around in 1980 remember that the Rubik's cube was a wild, crazy, viral phenomenon. It's hard to imagine a toy like that now. But almost exactly a hundred years before the Rubik's cube, the same thing happened with another toy that was an interesting group theory problem.
The "15 puzzle" is a four-by-four grid. There's one spot missing and the other ones have the numbers one through 15. To play, you keep sliding pieces into the empty spot. Often you'll see it now in the form of these little plastic-y puzzles. It was invented by a postmaster in Canastota, New York, named Noyes Palmer Chapman around 1880, and it became this viral hit. Thousands of men were losing entire evenings and abandoning their families because they were obsessed with this little box that they couldn't solve. And if someone handed you a board that was all in correct order, except for switching 14 and 15, it's actually impossible to solve. You can get infinitesimally close, where it feels like you can be just one transposition away. But you can never solve it.
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