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Puzzles, paradoxes and problem solving : an introduction to mathematical thinking

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Puzzles, Paradoxes, and Problem Solving: An Introduction to Mathematical Thinking Hardcover – 15 December 2014

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A Classroom-Tested, Alternative Approach to Teaching Math for Liberal Arts

Puzzles, Paradoxes, and Problem Solving: An Introduction to Mathematical Thinking uses puzzles and paradoxes to introduce basic principles of mathematical thought. The text is designed for students in liberal arts mathematics courses. Decision-making situations that progress from recreational problems to important contemporary applications develop the critical-thinking skills of non-science and non-technical majors.

The logical underpinnings of this textbook were developed and refined throughout many years of classroom feedback and in response to commentary from presentations at national conferences. The text’s five units focus on graphs, logic, probability, voting, and cryptography. The authors also cover related areas, such as operations research, game theory, number theory, combinatorics, statistics, and circuit design.

The text uses a core set of common representations, strategies, and algorithms to analyze diverse games, puzzles, and applications. This unified treatment logically connects the topics with a recurring set of solution approaches.

Requiring no mathematical prerequisites, this book helps students explore creative mathematical thinking and enhance their own critical-thinking skills. Students will acquire quantitative literacy and appreciation of mathematics through the text’s unified approach and wide range of interesting applications.

• ISBN-10 9781482227536
• ISBN-13 978-1482227536
• Edition 1st
• Publisher Chapman and Hall/CRC
• Publication date 15 December 2014
• Language English
• Dimensions 23 x 18 x 2 cm
• Print length 608 pages
• See all details

Product description

"… an interesting approach to the world of mathematics. … The layout is good, as is the coverage. … The reinforcing exercises are excellent. Using logic and other techniques, the text lays out methods to help students learn to think in a mathematical manner. Summing up: Recommended. General readers, lower- and upper-division undergraduates." ―M. D. Sanford, Felician College, Lodi, New Jersey, USA for CHOICE , November 2015

From the Publisher

About the author.

Marilyn A. Reba is a senior lecturer and Douglas R. Shier is a professor emeritus, both in the Department of Mathematical Sciences at Clemson University. The logical underpinnings of this textbook were developed and refined throughout many years of classroom feedback and in response to commentary from presentations at national conferences. Selected material from this book is currently being used in the Department of Mathematical Sciences’ liberal arts mathematics course and in a problem-solving course in the Honors College.

Product details

• ASIN ‏ : ‎ 1482227533
• Publisher ‏ : ‎ Chapman and Hall/CRC; 1st edition (15 December 2014)
• Language ‏ : ‎ English
• Hardcover ‏ : ‎ 608 pages
• ISBN-10 ‏ : ‎ 9781482227536
• ISBN-13 ‏ : ‎ 978-1482227536
• Dimensions ‏ : ‎ 23 x 18 x 2 cm

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Puzzles, Paradoxes, and Problem Solving: An Introduction to Mathematical Thinking

Reba, marilyn a., shier, douglas r., about the book, description:.

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About this title:

A Classroom-Tested, Alternative Approach to Teaching Math for Liberal Arts

Puzzles, Paradoxes, and Problem Solving: An Introduction to Mathematical Thinking uses puzzles and paradoxes to introduce basic principles of mathematical thought. The text is designed for students in liberal arts mathematics courses. Decision-making situations that progress from recreational problems to important contemporary applications develop the critical-thinking skills of non-science and non-technical majors.

The logical underpinnings of this textbook were developed and refined throughout many years of classroom feedback and in response to commentary from presentations at national conferences. The text’s five units focus on graphs, logic, probability, voting, and cryptography. The authors also cover related areas, such as operations research, game theory, number theory, combinatorics, statistics, and circuit design.

The text uses a core set of common representations, strategies, and algorithms to analyze diverse games, puzzles, and applications. This unified treatment logically connects the topics with a recurring set of solution approaches.

Requiring no mathematical prerequisites, this book helps students explore creative mathematical thinking and enhance their own critical-thinking skills. Students will acquire quantitative literacy and appreciation of mathematics through the text’s unified approach and wide range of interesting applications.

Marilyn A. Reba is a senior lecturer and Douglas R. Shier is a professor emeritus, both in the Department of Mathematical Sciences at Clemson University. The logical underpinnings of this textbook were developed and refined throughout many years of classroom feedback and in response to commentary from presentations at national conferences. Selected material from this book is currently being used in the Department of Mathematical Sciences’ liberal arts mathematics course and in a problem-solving course in the Honors College.

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Title: Puzzles, Paradoxes, and Problem Solving: An ... Publisher: Chapman and Hall/CRC Publication Date: 2014 Binding: hardcover Condition: New Edition: 1st Edition.

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The Path to Better Thinking Through Puzzles and Riddles

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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!

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:

A man has to transport a wolf, a goat, and a bunch of cabbages across a river. His boat could take only two of these at a time. How can he do this without leaving the wolf alone with the goat (as he might eat it) or the goat alone with the cabbages (as it might eat them)?

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 S es). 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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Comment Rules: Remember what Fonzie was like? Cool. That’s how we’re gonna be — cool. Critical is fine, but if you’re rude, we’ll delete your stuff. Please do not put your URL in the comment text and please use your PERSONAL name or initials and not your business name , as the latter comes off like spam. Have fun and thanks for adding to the conversation! (Thanks to Brian Oberkirch for the inspiration.)

A.J. Jacobs is one of my favorite guests. If you liked this topic you might enjoy the Remote Associates Test. (remote-associates-test.com)

Puzzles and riddles have been used for centuries as a way to improve thinking skills. They force you to think outside the box and look at problems in new and different ways. In addition, puzzles and riddles can help improve your memory and attention span. While there are many different ways to improve your thinking skills, puzzles and riddles offer a fun and unique way to challenge yourself. So next time you’re feeling stuck, try working on a puzzle or riddle. You might be surprised at how much better you think after.

Re the “16 06 68 88 __ 98” puzzle – I think it’s rather flawed, as you’d have to write this upside down to fit the puzzle, hence it wouldn’t be “87”

New psychology research reveals the hidden benefits of hitting a mental wall

H ave you ever felt stuck while solving a problem? That frustrating feeling of hitting a mental wall is called an impasse. But what if these moments of feeling stuck are more than just obstacles? In recent work published in Cognition , researchers Wendy Ross and Selene Arfini find that these impasses play a crucial role in problem-solving, especially when they lead to sudden moments of insight—those “aha!” breakthroughs.

They argue that understanding when and how people feel stuck can shed light on cognitive processes that are essential not only for solving puzzles but for learning and innovation. This research also addresses the paradox of how impasses, though seemingly detrimental, can sometimes enhance problem-solving abilities.

Experiment 1 involved 125 participants tackling a “stumper” riddle within 60 seconds to elicit impasses. See the riddle below.

“Dame Dora owns an Old Masters painting in a heavy gilded frame. The cord for hanging the painting, as old as the painting itself, is made of thick 3- ply hemp, and is somewhat frayed. Dame Dora was thinking of replacing it. But before she could, a couple of hungry little mice invaded her mansion. Sneaking behind the painting, they chewed right through the cord. For a while nobody noticed because the painting didn’t budge. Explain the painting’s stability briefly.”

Participants also completed phenomenological measures gauging their experiences (e.g., feeling stupid, awareness of time) via free-text responses.

The researchers found that 69% of participants reported experiencing an impasse during the problem-solving tasks. Interestingly, those who did not report an impasse had a problem-solving success rate of 32%, which exceeded the 15% success rate among those who felt stuck. Feeling stuck was negatively associated with successful problem resolution, and experiencing moments of insight. Only 23% of those who felt stuck reported an insight experience, compared to 83% among those who did not feel impasse.

While the average latency to arrive at a correct solution was higher for those who felt stuck (57.1 seconds) compared to those who did not (51.9 seconds), this difference was not statistically significant, suggesting that feeling stuck does not always lead to delays in finding solutions. There were varied emotional responses to feeling stuck, ranging from frustration and anxiety to cognitive looping, highlighting the multidimensional nature of impasse in problem-solving.

Experiment 2 introduced 247 participants to similar riddles within 90 seconds. They completed a scale measuring impasse as unresolved, resolved, or not experienced. A substantial 74% of participants experienced impasse during the task. Those who resolved their impasse showed significantly better results, with 53% solving the problems correctly, compared to only 11% who did not resolve their impasse, and 52% who never experienced it. Those resolving impasses took about 84 seconds, slightly longer than those who never experienced impasses at around 69 seconds.

Participants who resolved their impasses reported higher levels of insight (67%) compared to those who did not experience impasse (45%). Those unable to resolve impasse saw drastically lower insight rates (7%). Resolved impasse was associated with more active and potentially motivating emotions like anxiety or happiness, while unresolved impasse was associated with more passive emotions such as sadness or hopelessness.

Experiment 3 included 119 participants and assessed the impact of time pressure by manipulating problem-solving time across several groups with time constraints of 30, 60, 90, and 120 seconds. Two parallel experiments were conducted: one with easier stimuli to increase the likelihood of resolving impasses, and another with stimuli matching the difficulty of previous experiments.

In the easier set of stimuli, correct solution rates improved with increased time—from 56% at 30 seconds to 72% at 120 seconds—though the increase was not statistically significant. However, with harder problems, time significantly impacted performance, improving from 36% at 30 seconds to 66% at 90 seconds. Most participants experienced impasse, but those who resolved it matched the success rates of those who never felt stuck. Longer times generally led to higher latency but not necessarily better outcomes.

Experiment 4 broadened the scope to different types of cognitive tasks, including a verbal insight puzzle, a verbal analytical puzzle, a word problem from the UK GCSE mathematics curriculum, and the original stumper used in earlier experiments. Participants were given 4 minutes per task.

The researchers observed higher solution rates and levels of insight in trials where impasse was resolved compared to those where it was not, indicating that resolving impasses can significantly enhance problem-solving performance. However, unresolved impasses drastically reduced insight levels. Consistent with prior experiments, there were longer times in unresolved cases. Resolved impasses were associated with higher affect and motivation levels, reinforcing the positive effects of overcoming impasses.

Interestingly, trials characterized by fast impasse reported higher solution rates and insights, showing that the subjective experience of speediness was beneficial, although it did not necessarily correlate with actual faster solution times.

Further research in more naturalistic settings will be important to better understand the real-world dynamics of feeling stuck.

This work challenges the traditional view of impasses as mere obstacles, proposing instead that they can act as catalysts for deeper learning and innovation.

The research, “ Impasse-Driven problem solving: The multidimensional nature of feeling stuck ”, was authored by Wendy Rossa and Selene Arfini.