The Path to Better Thinking Through Puzzles and Riddles

The following is a guest post from A.J. Jacobs (@ajjacobs), a bestselling author, journalist, and human guinea pig. It is excerpted from his new book The Puzzler: One Man’s Quest to Solve the Most Baffling Puzzles Ever, from Crosswords to Jigsaws to the Meaning of Life. A.J. has written four New York Times bestsellers, including The Year of Living Biblically (for which he followed all the rules of the Bible as literally as possible) and Thanks a Thousand (for which he went around the world and thanked every person who had even the smallest role in making his morning cup of coffee possible). He has given four TED talks with a combined 10M+ views. He contributes to NPR and The New York Times and wrote the article “My Outsourced Life,” which was featured in The 4-Hour Workweek. He was once the answer to one down in The New York Times crossword puzzle. 

You can find my interview from 2016 with A.J here, and you can find last week’s interview with A.J. here.

Please enjoy!

Enter A.J…

My father was the one to introduce me to math puzzles.

He didn’t focus on the traditional kind. His were weirder than that, more homegrown. My dad’s greatest joy comes from baffling unsuspecting people—strangers, friends, family, whomever—and he often accomplishes this with math-based hijinks.

One time, when I was about eight years old, I asked my dad how fast race cars went. This was before Google, so my father was my version of a search engine.

“The fastest ones get up to about 50 million,” my dad said.

Even to my unschooled mind, 50 million miles per hour seemed off.

“That doesn’t sound right,” I said.

“Yes it is,” he said. “50 million fathoms per fortnight.”

I just stared at him.

“Oh, you wanted miles per hour?” my dad said. “I thought you meant in fathoms per fortnight.”

As you might know, a fathom equals six feet, and a fortnight is two weeks. My dad had decided that fathoms per fortnight would be his default way to measure speed, on the probably correct theory that no one else on earth had ever used that metric. I thanked him for this helpful information.

So, as you can see, I was exposed to recreational math early on, leaving me with a mixed legacy—a love of numbers, a healthy skepticism about numbers, and paranoia.

For this puzzle project, I’ve bought a dozen books with math and logic brainteasers. Reading these books often induces a mild panic. How would I know how many spheres can simultaneously touch a center sphere? I can’t even figure out where to start. What’s the entry point?

To remedy this problem, I decided to consult one of the world’s experts on math puzzles, hoping to learn some of her methods. Tanya Khovanova greets me on a video call. But before I’m allowed to ask her anything, she has a question for me.

“I have two coins,” she says, in a Russian accent. “Together they add up to 15 cents. One of them is not a nickel. What are the two coins?”

My palms begin to sweat. I did not expect a pop quiz.

Maybe she’s talking about foreign coins? Maybe rubles are involved, I say?

“Not foreign coins,” she says. “American currency.”

I employ one of the puzzle-solving strategies that I do know: Look closely at all of the words and see if you have fallen for any hidden assumptions.

Two coins.
Add up to 15.
One of them is not a nickel.

That last phrase is kind of ambiguous. She didn’t say “neither of them are nickels.” So . . . what if one is not a nickel, but the other one is?

“A dime and a nickel?” I say, tentatively. “Because the other one is a nickel?”

“Okay. You passed the test. So you can continue,” she says, smiling.

This is a relief. Because Tanya is a fascinating character. She is a Russian émigré who is now a lecturer at MIT. She writes a popular blog about the world’s twistiest math and logic puzzles (it’s called simply Tanya Khovanova’s Math Blog). And she has cracked pretty much every great math puzzle ever created. We’re talking coin puzzles, matchstick-arranging puzzles, river-crossing puzzles, math equation puzzles.

Tanya is on a mission. “I am very upset at the world,” she says. “There is so much faulty thinking, and puzzles can help us think better.”

Consider probability, she says. We are terrible at thinking probabilistically, and puzzles about odds can help us learn. They could teach us, for instance, the folly of playing the lottery. “The situation is unethical. I think that lottery organizers should spend part of the money they make on lotteries to educate people not to play the lottery.”

Tanya has been fascinated with math since her childhood in Moscow.

“The first thing that I remember, it wasn’t a puzzle, it was an idea. I remember that I was five years old and we were on a vacation in a village, and I was trying to go to sleep and I was thinking after each number there is the next number, and then there is the next number. At some point, I realized that there should be an infinity of numbers. And I had this feeling like I’m touching infinity, I’m touching the universe, just a euphoric feeling.”

Being a female Jewish math genius in 1970s Soviet Russia was not easy. She faced sexism and anti-Semitism. Tanya says the test for the prestigious Moscow State University—the Soviet equivalent of MIT—was rigged against Jews. Jewish students were given a separate and more difficult test. The problems were called “coffin problems,” which translates to “killer problems.” Tanya studied with other Jewish students and managed to pass the unfair test.

In 1990, Tanya left Russia. She moved to the United States and married a longtime American friend. She worked for a defense contractor near Boston but hated it because “I thought it destroyed my karma.” She started teaching as a volunteer at MIT before they hired her as a full-time lecturer.

Her philosophy: puzzles should be used more often in teaching math. First of all, they entertain us while teaching us how to think rigorously. And second, puzzles can lead to genuine advances in mathematics—topics such as conditional probability and topology were originally explored in puzzle form.

Math Puzzles 1.0

The very first math puzzles—at least according to some scholars—date back to Egypt’s Rhind Papyrus, about 1500 B.C.E. They’re closer to problems than puzzles, since they don’t require much ingenuity. But the unnamed author did try to spice them up with some whimsical details, such as in Problem 79.

Problem 79. There are seven houses.
In each house there are seven cats.
Each cat kills seven mice.
Each mouse has eaten seven grains of barley.
Each grain would have produced seven hekat (a unit of measurement).
What is the sum of all the enumerated things?

Arguably the first book with actual twisty and turny math puzzles came several centuries later. The ninth century Holy Roman Emperor Charlemagne was a puzzle addict, and he hired a British scholar named Alcuin of York to be his official puzzlemaker. Alcuin’s book Problems to Sharpen the Young introduced, among other things, the first known river-crossing problem. Here it is:

For river-crossing problems, you need to realize that you must take a counterintuitive step backward before continuing forward. You must think outside the box.

Way Outside the Box

Tanya reminds me that “thinking outside of the box” wasn’t always a cliché. The origin of the phrase is an actual puzzle: Connect all the dots in this diagram using just four straight lines:

The answer:

Nowadays the phrase is overused and is often a punchline, as in the cartoon of the cat thinking outside its litter box. But it’s still an important concept: to find a solution, you often have to break expectations.

“My students have taught me as much as I have taught them about this,” she says.

“How do you mean?” I ask.

She tells me to think about this puzzle: “You have a basket containing five apples. You have five hungry friends. You give each of your friends one apple. After the distribution, each of your friends has one apple, yet there is an apple remaining in the basket. How can that be?”

The traditional answer is: you give four friends an apple, and then hand the fifth friend the basket with the apple still in it. So each friend has an apple, and there’s still one in the basket.

“For that answer, you have to think out of the box,” says Tanya. “But my students have come up with answers that are even farther out of the box.”

Their suggestions include:

One friend already has an apple.
You kill one of your friends.
You are narcissistic and you are your own friend.
The friend who didn’t get an apple stops being your friend.
An extra apple falls from the tree to the basket.
And Tanya’s favorite: The basket is your friend. We should not discount people’s emotional connection with inanimate objects.

“The lesson my students taught me is that I’m good at thinking outside the box. But I realized, I’m inside my own bigger box. And maybe we all are.”

How to Solve Problems

But how do you get yourself to think outside the box? How do you approach a math problem? I know how to start a jigsaw puzzle (the edges, usually) and a crossword (look for plurals and fill in the Ses). But how do you approach a math problem?

After talking to Tanya and another great math puzzle expert, Dartmouth professor Peter Winkler, I’ve come up with a list of tools for math and logic problems. Here are three of my favorites.

1) Reverse it.

When confronted with a problem, try reversing it. Turn it upside down.

Sometimes quite literally, turn it upside down.

Such as this problem:

What number belongs in the blank in this sequence:
16 06 68 88 __ 98
(It’s 87. Turn the page upside down to see why.)

There are other puzzles that require you to reverse your thinking in a slightly less literal way. Like this one:

A man is imprisoned in a ten-foot by ten-foot by ten-foot room. The walls are made of concrete, the floor is made of dirt, and the only openings are a locked door and a skylight. The man has a small shovel and starts to dig a hole in the floor. He knows that it is impossible to tunnel out of the prison cell, but he continues to dig anyway. What is the man’s plan?

Pause here if you want to figure it out yourself.

The solution is: The man wasn’t just digging a hole. He was also doing the opposite: building a little mountain of dirt. And his plan was to climb the mountain and get to the skylight.

I love reversing my thinking. Earlier this week, I was cleaning up the trail of clothes left by the males in our family (including me) that littered our apartment. I picked up an armload of clothes, then went to the hamper in my bedroom and dumped the clothes, then went back out. But wait. What if I . . . took the hamper with me. If I bring the hamper to the clothes. I’d save myself several trips. As Will Shortz once suggested, I took a bow.

2) Figure out the real goal.

One of my favorite brainteasers comes from Martin Gardner, who wrote a famous monthly column about math puzzles in Scientific American for three decades, starting in 1962. He died in 2010, but he still has tons of devotees, hundreds of whom attend a biannual event, the Gathering 4 Gardner, where they talk puzzles, paradoxes, and the genius of Martin.

Martin posed this puzzle in his book Entertaining Mathematical Puzzles:

Two boys on bicycles, 20 miles apart, began racing directly toward each other. The instant they started, a fly on the handlebar of one bicycle started flying straight toward the other cyclist. As soon as it reached the other handlebar, it turned and started back. The fly flew back and forth in this way, from handlebar to handlebar, until the two bicycles met.

If each bicycle had a constant speed of 10 miles an hour, and the fly flew at a constant speed of 15 miles an hour, how far did the fly fly?

Pause here if you want to try it yourself, spoilers ahead.

So how to solve this? Most people’s first instinct—including mine—is to trace the back-and-forth path of the fly and try to add up the distance.

With this method, you’d try to calculate the distance from Biker 1’s handlebars to Biker 2’s handlebars. Then the fly would make a U-turn, so you’d calculate the next distance, from Biker 2 to Biker 1. And so on until the bikes met.

This turns out to be a highly complex computation involving the speed of the bikers, the speed of the insect, and time and distance. The operation is called “summing an infinite series.”

This calculation is impossible to do in your head. Well, practically impossible. Legend has it that the brilliant Hungarian mathematician John von Neumann was once asked this brainteaser at a party, and, to the amazement of the quizzer, gave the correct answer by summing the “infinite series” in his head, no calculator needed.

Von Neumann was too smart for his own good. If he had paused for a moment, he might have realized there’s a much easier way to solve this problem.

Which brings me back to the strategy: What is the real goal?

You want to phrase the problem in the simplest possible way. Strip the problem to its basics, and you’ll realize you are looking for one thing: the distance the fly can fly in an allotted amount of time.

You can ignore the fly’s back and forth switch of directions. You can ignore the handlebars. They’re irrelevant. You just need to know how far the insect can go in the time it takes the bikes to meet.

Which turns out to be a pretty easy calculation:

If each bike was going at 10 miles per hour, and they were 20 miles apart, then it would take the bikes one hour to reach each other.

So the fly was buzzing around for one hour. What is the distance the fly can cover in one hour? Well, it’s going 15 miles per hour. So the answer is fifteen miles.

We often complicate problems when there’s an easier method right in front of us. I think this is true in more than just math puzzles.

I’m not sure if this is exactly analogous, but it’s staring me in the mirror, so let me tell you about one example. Recently, I was faced with the puzzle of how to cut my own hair. During quarantine, I couldn’t go to the barber, and Julie claimed she wasn’t qualified. I had to do it myself using YouTube tutorials.

My first attempt to cut my own hair had mixed results. The front turned out okay, but the harder-to-reach back of my head was a disaster, filled with uneven patches.

So I paused. I rephrased the problem. The goal is not to cut my hair flawlessly. The goal is to look respectable on Zoom. And on Zoom, no one ever sees the back of my head.

So the simplest solution: Just cut the front of my hair and leave the back alone to grow wild and free. Puzzle solved! Though for the first time in my life, I do have a mullet.

3) Break it down into manageable chunks 

One type of logic puzzle—often called Fermi Problems—provides excellent training for solving some real-life problems. A Fermi Problem is one like this: “How many piano tuners are there in New York City?” You have to estimate the size of something about which you are totally ignorant. 

If you just take a wild guess without reflecting, you’ll probably be off by orders of magnitude. Instead, as David Epstein explains in the psychology book Range: Why Generalists Triumph in a Specialized World, the best method is to break the problem down into parts you can reasonably estimate.

As Epstein writes: “How many households are in New York? What portion might have pianos? How often are pianos tuned? How long might it take to tune a piano? How many homes can one tuner reach in a day? How many days a year does a tuner work?” 

You won’t guess it exactly, but you’ll be much more likely to be in the ballpark. As Epstein writes, “None of the individual estimates has to be particularly accurate in order to get a reasonable overall answer.”

Epstein calls it an important tool in his “conceptual Swiss Army knife.” I too find it helpful when reading statistics from dubious media sources, or listening to wild cocktail party speculation. 

Breaking problems into chunks even works when trying to motivate yourself. Take the puzzle of how I can get my lazy butt to walk the treadmill for a few minutes a day. If I say to myself, “You have to walk on the treadmill for an hour today,” I will delay this task forever. So I break it down. I put the big picture out of my mind. First, I tackle the subgoal of putting on my sneakers. I can do that. Then the subgoal of turning the treadmill on. I can do that. And just step onto the rubber belt for just five minutes. I can do that. And eventually, I’m walking and realize this isn’t so bad. I can do this. I stay on for the full hour.

Excerpted with permission from THE PUZZLER: One Man’s Quest to Solve the Most Baffling Puzzles Ever, from Crosswords to Jigsaws to the Meaning of Life by A.J Jacobs.  

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