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Math and Logic Puzzles

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20 Best Math Puzzles to Engage and Challenge Your Students

Written by Maria Kampen

Reviewed by Joshua Prieur, Ed.D.

Solve the hardest puzzle

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• Teacher Resources

1. Math crossword puzzles

2. math problem search, 3. math riddles.

It’s time for math class, and your students are bored.

It might sound harsh, but it’s true -- less than half of 8th grade students report being engaged at school according to this Gallup survey , and engagement levels only drop as students get older.

Math puzzles are one of the best -- and oldest -- ways to encourage student engagement. Brain teasers, logic puzzles and math riddles give students challenges that encourage problem-solving and logical thinking. They can be used in classroom gamification , and to inspire students to tackle problems they might have previously seen as too difficult.

Math puzzles for kids

Puzzles to Print

Take a crossword, and make it math: that’s the basic concept behind this highly adaptable math challenge. Instead of words, students use numbers to complete the vertical and horizontal strips. Math crossword puzzles can be adapted to teach concepts like money, addition, or rounding numbers. Solutions can be the products of equations or numbers given by clues.

Have students practice their addition, subtraction, multiplication and division skills by searching for hidden math equations in a word search-style puzzle . It can be adapted to any skill you want students to practice, and promotes a solid understanding of basic math facts.

My PreCalc students love riddles... can you figure out where the other dollar went?? #MathRiddles pic.twitter.com/BclqW9nq98 — Rachel Frasier (@MsFrasierMHS) January 8, 2019

Do your students love word problems ? Try giving them some math riddles that combine critical thinking with basic math skills. Put one up on the board for students to think about before class begins, or hand them out as extra practice after they’ve finished their work.

Prodigy is an engaging, game-based platform that turns math into an adventure! While it’s not a math puzzle in the traditional sense, Prodigy uses many of the same principles to develop critical thinking skills and mathematical fluency.

Students complete standards-aligned math questions to earn coins, collect pets and go on quests. Teachers can deliver differentiated math content to each student, prep for standardized tests and easily analyze student achievement data with a free account.

See how it works below!

KenKenKenKen

is a “grid-based numerical puzzle” that looks like a combined number cross and sudoku grid. Invented in 2004 by a famous Japanese math instructor named Tetsuya Miyamoto, it is featured daily in The New York Times and other newspapers. It challenges students to practice their basic math skills while they apply logic and critical thinking skills to the problem.

6. Pre-algebraic puzzles

Pre-algebraic puzzles use fun substitutions to get students ready to perform basic functions and encourage them to build problem-solving skills. They promote abstract reasoning and challenge students to think critically about the problems in front of them. As an added bonus, students who suffer from math anxiety might find the lack of complicated equations reassuring, and be more willing to attempt a solution.

7. Domino puzzle board

Games 4 Gains

There are hundreds of ways to use dominoes in your math classroom, but this puzzle gives students a chance to practice addition and multiplication in a fun, hands-on way. You can have students work alone or in pairs to complete the puzzle.

This online game and app challenges players to slide numbered tiles around a grid until they reach 2048. It’s super fun and not as easy as it sounds, so consider sending it home with students or assigning it after the rest of the lesson is over. It encourages students to think strategically about their next move, and it’s a great tool for learning about exponents.

Math in English

Kakuro , also called “Cross Sums,” is another mathematical crossword puzzle. Players must use the numbers one through nine to reach “clues” on the outside of the row. Decrease the size of the grid to make it easier for younger players, or keep it as is for students who need a challenge. Students can combine addition and critical thinking and develop multiple skills with one fun challenge.

10. Magic square

Magic squares have been around for thousands of years, and were introduced to Western civilization by translated Arabic texts during the Renaissance. While magic squares can be a variety of sizes, the three by three grid is the smallest possible version and is the most accessible for young students.

This is also a great math puzzle to try if your students are tactile learners. Using recycled bottle caps, label each with a number from one to nine. Have your students arrange them in a three by three square so that the sum of any three caps in a line (horizontally, vertically and diagonally) equals 15.

11. Perimeter magic triangle

This activity uses the same materials and concept as the magic square, but asks students to arrange the numbers one to six in a triangle where all three sides equal the same number. There are a few different solutions to this puzzle, so encourage students to see how many they can find.

Sudoku is an excellent after-lesson activity that encourages logical thinking and problem solving. You’ve probably already played this classic puzzle, and it’s a great choice for your students. Sudoku puzzles appear in newspapers around the world every day, and there are hundreds of online resources that generate puzzles based on difficulty.

13. Flexagon

There’s a pretty good chance that by now, fidget spinners have infiltrated your classroom. If you want to counter that invasion, consider challenging your students to create flexagons. Flexagons are paper-folded objects that can be transformed into different shapes through pinching and folding, and will keep wandering fingers busy and focused on the wonders of geometry.

14. Turn the fish

This puzzle seems simple, but it just might stump your students. After setting up sticks in the required order, challenge them to make the fish swim in the other direction -- by moving just three matchsticks.

15. Join the dots

Cool Math 4 Kids

This puzzle challenges students to connect all the dots in a three by three grid using only four straight lines. While it may sound easy, chances are that it will take your class a while to come up with the solution. (Hint: it requires some “out of the box” thinking.)

16. Brain teasers

While they don’t always deal directly with math skills, brain teasers can be important tools in the development of a child’s critical thinking skills. Incorporate brain teasers into a classroom discussion, or use them as math journal prompts and challenge students to explain their thinking.

Bonus: For a discussion on probability introduce an older class to the Monty Hall Problem, one of the most controversial math logic problems of all time.

17. Tower of Hanoi

This interactive logic puzzle was invented by a French mathematician named Edouard Lucas in 1883. It even comes with an origin story: According to legend, there is a temple with three posts and 64 golden disks.

Priests move these disks in accordance with the rules of the game, in order to fulfill a prophecy that claims the world will end with the last move of the puzzle. But not to worry -- it’s going to take the priests about 585 billion years to finish, so you’ll be able to fit in the rest of your math class.

Starting with three disks stacked on top of each other, students must move all of the disks from the first to the third pole without stacking a larger disk on top of a smaller one. Older students can even learn about the functions behind the solution: the minimum number of moves can be expressed by the equation 2n-1, where n is the number of disks.

18. Tangram

Tangram puzzles -- which originated in China and were brought to Europe during the early 19th century through trade routes -- use seven flat, geometric shapes to make silhouettes. While Tangrams are usually made out of wood, you can make sets for your class out of colored construction paper or felt.

Tangrams are an excellent tool for learners who enjoy being able to manipulate their work, and there are thousands of published problems to keep your students busy.

Similar to Sudoku, Str8ts challenges players to use their logic skills to place numbers in blank squares. The numbers might be consecutive, but can appear in any order. For example, a row could be filled with 5, 7, 4, 6 and 8 . This puzzle is better suited to older students, and can be used as a before-class or after-lesson activity to reinforce essential logic skills.

20. Mobius band

Is it magic? Is it geometry? Your students will be so amazed they might have a hard time figuring it out. Have them model the problem with strips of paper and see for themselves how it works in real life. With older students, use mobius bands to talk about geometry and surface area.

Why use math puzzles to teach?

Math puzzles encourage critical thinking.

Critical thinking and logic skills are important for all careers, not just STEM-related ones. Puzzles challenge students to understand structure and apply logical thinking skills to new problems.

A study from the Eurasia Journal of Mathematics, Science and Technology Education found that puzzles “develop logical thinking, combinatorial abilities, strengthen the capacity of abstract thinking and operating with spatial images, instill critical thinking and develop mathematical memory.”

All these skills allow young students to build a foundation of skills they’ll draw on for the rest of their lives, no matter what kind of post-secondary route they pursue.

They help build math fluency

Math games can help students build a basic understanding of essential math concepts, and as another study shows, can also help them retain concepts longer .

In the study, early elementary students gradually moved from using the “counting” part of their brains to complete math problems to the “remembering” part that adults use, suggesting math puzzles and repeated problems can help build the essential skill of math fluency .

Many of the math puzzles above allow students to practice essential addition, subtraction, multiplication and division skills, while advanced or modified problems can be used to introduce pre-algebraic concepts and advanced logic skills.

Math puzzles connect to existing curricula

No matter what curriculum you’re using, there’s a good chance it emphasizes problem-solving, critique and abstract thinking. This is especially true of Common Core math and similar curricula.

How Math Skills Impact Student Development

Math puzzles allow students to develop foundational skills in a number of key areas, and can influence how students approach math practically and abstractly. You can also tie them into strategies like active learning and differentiated instruction.

Instead of just teaching facts and formulas, math puzzles allow you to connect directly with core standards in the curriculum. You can also use them to provide a valuable starting point for measuring how well students are developing their critical thinking and abstract reasoning skills.

Tips for using math puzzles in the classroom

View this post on Instagram A post shared by Sarah Werstuik (@teach.plan.love)

Now that you’ve got some great math puzzles, it might be tricky to figure out how to best incorporate them into your classroom. Here are some suggestions for making the most of your lesson time:

Make sure the puzzles are the right level for your class

If the problems are too easy, students will get bored and disengage from the lesson. However, if the problems are too difficult to solve, there’s a good chance they’ll get frustrated and give up early.

There’s a time and a place

While fun math puzzles are a great way to engage your students in developing critical thinking skills, they’re not a tool for teaching important math concepts. Instead, use them to reinforce the concepts they’ve already learned.

Kitty Rutherford , a Mathematics Consultant in North Carolina, emphasizes that math puzzles and games shouldn’t be based solely on mental math skills , but on “conceptual understanding” that builds fluency over time. Math puzzles help build the essential balance between thinking and remembering.

Give them space to figure it out

Rachel Keen , from the Department of Psychology at the University of Virginia, conducted a study about problem-solving skills in preschoolers. She found that “playful, exploratory learning leads to more creative and flexible use of materials than does explicit training from an adult.”

Give your students space to struggle with a problem and apply their own solutions before jumping in to help them. If the problem is grade-appropriate and solvable, students will learn more from applying their own reasoning to it than just watching you solve it for them.

Model puzzles for your students

Use problems like the mobius strip to awe and amaze your students before drawing them into a larger discussion about the mathematical concept that it represents. If possible, make math puzzles physical using recycled craft supplies or modular tools.

Afterward, have a class discussion or put up math journal prompts. What methods did your students try? What tools did they use? What worked and what didn’t? Having students explicitly state how they got to their solution (or even where they got stuck) challenges them to examine their process and draw conclusions from their experience.

Final thoughts on math puzzles

Be aware that it might take a while to get all your students on board -- they could be hesitant about approaching unfamiliar problems or stuck in the unenthusiasm that math class often brings. Consider creating a weekly leaderboard in your classroom for the students that complete the most puzzles, or work through a few as a class before sending students off on their own.

Instead of yawns and bored stares , get ready to see eager participants and thoughtful concentration. Whether you choose to use them as an after-class bonus, a first day of school activity or as part of a targeted lesson plan, math puzzles will delight your students while also allowing them to develop critical skills that they’ll use for the rest of their lives.

What are you waiting for? Get puzzling!

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VR - Logic Problems

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Sometimes questions in Verbal Reasoning papers are one-offs – they don’t have a particular style and they are simply about making sense of a lot of information. Logic Problems are one of these.

You may have come across logic puzzles in certain magazines. If so, you’ll know all about them. They exercise the brain by making us use our deductive reasoning skills to find the answer. In fact, the answer is already there in the question – you just have to be able to dig it out!

How Are Logic Problems Used In The Exam?

They usually take to form of some information expressed in words which the child must use to work out the answer to a question. If you’ve never seen a logic problem before then the best way to show you is through an example.

Example Question One

Let’s make sense of this. It is often worth reading through the answers one-by-one and applying them to the information given. When practising, allow your child to speak out loud. I encourage a child to do this as it enables me to check his or her logic and it should be something you are capable of doing as well. As ever, make sure that you’ve worked out the answer to any question you’re trialling with your child so as to give pointers as they are working them out.

So, statement A mentions Simon but not the hockey club. Therefore, we cannot know whether or not he has joined the general knowledge club - it isn’t possible to prove.

Statement B mentions Jessica and, although we are told that everyone has joined a club, it could be absolutely any. It could even be the football club – there’s no reason why not! Again, her joining the hockey club is not provable.

Statement C is possibly true – perhaps Henry is good at chess and that’s why he joined the club. Careless students will go for this as the first statement which seems reasonable. However, it is not a certainty and definitely can’t be proven using the information we are given.

Statement E suggests that Jessica is the only one to dislike football and is also a reasonable suggestion. However, we are given nothing provable in this regard and it is impossible to know for certain. Just because it doesn’t mention that Simon and Henry don’t like football doesn’t mean that they do or don’t like it. It’s speculation, it may be true, it mustn’t deflect from the logical certainty.

This leaves statement D, ‘Simon has joined two clubs’. Of course, it says in the preamble that Simon has joined the hockey club and one other, which means he has joined two clubs. He cannot have joined more, nor could he have joined fewer, otherwise the wording would have to be different. Statement D is therefore the correct answer.

Example Question Two

John has a cat and a dog. Raj has a goldfish. Sara has two rabbits, two goldfish and a rat. Daryl has twice the number of cats as John while both he and Raj own a snake. Sara doesn’t like either of John’s pet lizards but likes both his rabbits. Raj would like to have more than his two rats but is not allowed.

Who has the most pets?

This is a very wordy question but is easily broken down visually. Encourage your child to draw out a very simple table to store information and then read off it for the answer. Use ticks or crosses, numbers and abbreviations, but don’t write out full names as this will waste time.

Here’s a version of the table that could be written out for this question, with full names written to help explain what is happening:

The easiest way to arrange the information is to deal with one sentence at a time and that way you never get caught out. Taking pieces of information from different sections of the paragraph, for instance finding out everything to do with John first, is not a great idea. It means that some elements of a sentence must be ignored and may not get reviewed later. Encourage your child to be systematic.

The answer, using the table, is clearly John. He has six pets and that is more than the other children.

Once your child gets good at this, the table produced may look a bit like this:

There is no reason to differentiate between rats and rabbits so the letter ‘r’ is fine; if the question needed it, ensure your child could make clear abbreviations to differentiate between them. You could use a tally chart instead of letters in this question but I would be reluctant, as if you lose your train of thought you cannot go back and start from where you were as you only have lines or numbers and don’t know whether you’ve dealt with all the cats, Raj’s rats or whatever!

It’s always good to be brief but abbreviate too much and you could take time to restart if you make an error.

Sample Tests

So, now we understand what Logic Problems are and, more importantly, the quickest way to solve them. It’s time to put your child’s powers of deduction to the test.

There are four quizzes on the Education Quizzes site devoted to logic problems. Go through them with your child, offering advice if necessary, gleaned from the techniques we’ve shown you in this lesson.

You’ll find the quizzes in our Eleven Plus Verbal Reasoning section or, alternatively, you can follow these links:

Reading Information 1

Reading Information 2

Reading Information 3

Reading Information 4

Let’s see if your child has what it takes to be the next Sherlock Holmes! Good luck.

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• Number Theory
• Probability
• Everyday Math
• Classical Mechanics
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Take a guided, problem-solving based approach to learning Logic. These compilations provide unique perspectives and applications you won't find anywhere else.

What's inside

• Introduction
• Puzzles and Riddles
• Multi-Level Thinking
• The Rational Detective
• Syllogisms and Sets
• Logic Machines
• Arithmetic With Logic Gates
• Propositional Logic
• First-Order Logic

Joy of Problem Solving

• Intro to Problem Solving
• Coin Rearrangements
• Truth Tellers and Liars
• Operator Searches
• Matchstick Puzzles

Community Wiki

Browse through thousands of Logic wikis written by our community of experts.

• Truth-Tellers and Liars
• Cryptogram - Problem Solving
• Solving Propositional Logic Word Problem
• Mind Reading with Math
• Information Compression
• K-level thinking
• Chess Puzzles
• Arithmetic Puzzles - Operator Search
• Arithmetic Puzzles - Fill in the Blanks
• Elimination Grids
• Grid Puzzles
• Combinatorial Games - Definition
• Combinatorial Games - Winning Positions
• Tic Tac Toe
• Sprague Grundy Theorem
• Chess Puzzles - Reduced Games
• Chess Puzzles - Opening Strategies
• Chess Puzzles - Rook Strategies
• Rook Polynomial
• Game Theory
• Nash Equilibrium
• Zero-Sum Games
• Prisoner's Dilemma
• Braess' Paradox
• Utility Functions
• Cognitive Bias
• Monty Hall Problem
• Birthday Problem
• Two-Envelope Paradox
• Simpson's Paradox
• Berkson's Paradox
• Newcomb's Paradox
• Benford's Law
• Mathematics of Voting
• Survivorship Bias
• Russell's Paradox
• Zeno's Paradox
• Gabriel's Horn
• Truth Tables
• Proof by Contradiction
• Mathematical Logic and Computability
• Mathematical Logic and Computability II (continuation)
• Propositional Logic Using Algebra
• Venn Diagram
• Predicate Logic

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KS2 / 11+ / UKMT

11+ advice, puzzles & challenges.

11+ Maths advice to help gain entry into top UK independent and grammar schools.  This includes:

• Interview advice
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• Top UK schools linked 4 commmon exam test providers - Durham CEM Select, ISEB Commmon Pre Test, London Consortium and GL Assessment)
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Past Papers and Schools

For everything you could ever need to want to know about 11+ schools, past papers and exam boards clck 11+ Maths Past Papers

Brainteasers

For hundreds of fun brainteasers organised by difficulty level click the following button:

Highly recommended fun and challenging books that very able students love are:

• Can You Solve My Problems? - Alex Bellos
• So You Think You've Got Problems? - Alex Bellos
• The Ultimate Mathematical Challenge - The UK Mathematics Trust
• Murdle Mystery Logic Puzzles - G. T. Karber
• Fantastic Book of Logic Puzzles - Muriel Mandell
• My Best Mathematical and Logic Puzzles - Martin Gardner
• The Book of Numbers -  John Conway and Richard Guy
• What is Mathematics - Richard Courant and Herbert Robbins
• The World of Mathematics - James R. Newman
• The Symmetries Of Things - John Conway 
• Maths with Bad Drawings - Ben Orlin
• Manga Guide Series - These books offer a fun insight into advanced mathematics

Anything by Martin Gardner, Ian Stewart, or Keith Devin is good.

Further Challenges

This website has endless challenges, but if you'd like to see others then visiit the following websites:

• Matt Parker - Think Maths
• Matt Parker - Stand Up Maths
• Ted Ed Riddles
• The Art Of Problem Solving
• MathWorks Math Modeling Challenge
• Project Euler . This has great math and coding puzzles.
• Euclidea App - This is a fun game which becomes surprisingly difficult
• Geogebra and Desmos . These are good visualisation tools. Desmos is more friendly than geogebra, but is a bit more limited especially since it cannot plot in 3D). 
• Wolfram Αlpha  - This is fun to play around with

Depending on where you live, there might be a nearby maths circle which are fantastic. Find out if there are any near you!

Video Games

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• Human Resources
• Opus Magnum

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Puzzles and games to exercise your mind, puzzles and logic problems - stretch, word problems, extension activities, challenges - stretch, maths revision and resources for all maths courses.

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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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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”

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Best Games That Reward Creative Problem-Solving

• As video game technology advances, puzzles have become more open-ended, allowing for creative solutions and multiple outcomes.
• Games like Portal 2 and Baldur's Gate 3 reward players for thinking creatively and finding unique ways to solve problems.
• Undertale and Scribblenauts Unlimited offer players the freedom to approach puzzles in their own way, showcasing the importance of player choice.

In the early days of video games, puzzles and their solutions were pretty cut and dry. Get from Point A to Point B, and accomplish A, B, and C using a very specific method. There was only one way to beat the game; players just needed to find it. Part of this was due to hardware and scripting limitations. Consoles had such limited memory that there simply wasn't room for more creative solutions.

11 Best Puzzle Games On Steam That Deserve More Attention

But as video games and consoles become more advanced, puzzle-solving in games has become more open-ended. Developers are now usually more than happy to hand players the reins and let them run wild with their creativity. All the pieces are there; players just need to put them together. There are no right or wrong answers, just what works and what doesn't, and for those looking to flex their creative puzzle-solving muscles and get rewarded for doing so , there are plenty of gaming options available.

Peace is Always An Option

Platform(s) PS Vita, Switch, PS4, PC, Xbox One

Released September 15, 2015

Developer(s) Toby Fox

Genre(s) RPG

This 2015 smash hit revolutionized puzzle-solving and story progression in RPG adventure games, being one of the first where players could complete the entire game without killing a single enemy. Instead, they could choose the Pacifist route and befriend NPCs instead (or go from killing a single enemy to all of them in the notorious Genocide route ).

Beyond the multiple endings, Undertale is filled with puzzles to overcome, not all of which have a single solution. Is a dastardly skeleton placing a maze trap before the fallen human? They might be able to solve it, or they can opt to simply walk around it. New dialogue options and story branches open and close based on who the player chooses to spare and how. The reward for creative puzzle-solving is that players get to explore more of this colorful world and iconic characters.

Scribblenauts Unlimited

The noun's the limit.

Platform(s) Wii U, Switch, PS4, PC, iOS, Android, 3DS, Xbox One

Released November 13, 2012

Developer(s) 5th Cell

Genre(s) Puzzle, Sandbox

Scribblenauts proves that the pen is indeed mightier than the sword. No, seriously. The titular heroes in this popular puzzle RPG series use the power of words to solve all their problems. Players can write down a noun and watch it appear in the world. Giving it an adjective results in the object or creature transforming to match its new description. Giving a pig wings will allow it to fly, and manifesting some glue out of the ether will permit players to stick things together. The possibilities are literally endless.

Scribblenauts Unlimited builds on its predecessors with an expansive list of nouns, adjectives, and adverbs that players can use and provides multiple ways to solve each puzzle. The game doesn't care how players accomplish their goals, so long as they clear the level's main objective. The only real limit is the player's imagination (and their vocabulary).

Aperture Science Tests One's Creativity

Platform(s) PS3, Switch, Xbox 360, PC

Released April 18, 2011

Developer(s) Valve

Genre(s) Puzzle, Platformer

The Portal games make no secret that they want to test players' problem-solving abilities. The game's central antagonist, GLaDOS , makes that abundantly clear. In fact, the entire plot is to use those same skills to escape the testing facility once and for all. Each test follows the same simple premise: get to the end of the testing chamber in one piece. How the player does it is up to them - and the game's physics engine. Jump through portals, avoid deadly turrets and bottomless pits, propel through the air -- it doesn't matter. As the saying goes, "If it's stupid and it works, it isn't stupid."

Portal 2 takes this engaging premise and amps it up to eleven. Puzzles are bigger, deadlier, and more complicated than ever before. While some answers are laid out more obviously than others, players are free to ignore those options and think outside the box as much as they want. So long as they reach the final door, GLaDOS doesn't mind how they did it. After all, data is data.

The Legend of Zelda: Tears of the Kingdom

Zonai devices help players reinvent the wheel.

Platform(s) Switch

Released May 12, 2023

Developer(s) Nintendo

Genre(s) Adventure

The Legend of Zelda: Breath of the Wild was a massive breath of fresh air for the franchise. Linear adventures with clear-cut solutions were swapped out for a massive open world and a fun new physics engine that led to infinitely creative gameplay. Many fans of the series felt the game essentially reinvented the wheel of a franchise that had started to get a bit formulaic. However, its direct sequel, Tears of the Kingdom , didn't just reinvent the wheel; it allowed players to invent their own.

8 Hardest Zelda Games To 100% Complete

On top of the many new mechanics introduced in Breath of the Wild , Tears of the Kingdom introduced several new mechanics for players to tinker around with. The best by far is the introduction of Zonai devices, allowing players to use various moving mechanical parts in endless combinations to solve puzzles. Gamers can build siege engines that breathe fire, gliders powered by fans, or massive catapaults to launch enemies (or themselves) off into the horizon.

Baldur's Gate 3

Any solution is just a dice roll away.

Platform(s) Xbox Series X, PS5, PC, macOS, Stadia

Released August 3, 2023

Developer(s) Larian Studios

This award-winning RPG action-adventure game from 2023 famously adapted the core rules of Dungeons & Dragons 5th Edition into a video game format. But the game didn't just take the basic stats and mechanics; it went above and beyond to allow players to think just as creatively as they would while playing real D&D . Any D&D campaign can go off the rails at any moment. One bad dice roll or truly chaotic player choice, and there will be consequences. That's what makes the game so fun, and Baldur's Gate 3 understands this.

No two playthroughs of Baldur's Gate 3 are the same, thanks to the infinite array of choices , dice checks, and consequences. Players are rewarded for seeing just how many different ways they can solve the same problem. Do they exterminate the goblin camp or side with them? Can they pickpocket the vendor for powerful items to avoid paying? Can they talk their way out of being arrested, either through smooth-talking or saber-rattling? Every decision opens doors and closes others. Each interaction can forge new allies or enemies. Every quest has multiple endings, good or bad. Players can play the campaign however they want and will be rewarded for their creative thinking along the way.

Puzzle Games You Have To Be A Genius To Complete

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I won the puzzle yesterday but didn’t see the 100 points when finished

I won the puzzle yesterday and never got my 100 points. Are we suppose to do more in order to get 100 points. Thank you

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I would like a response without either of these repetitive answers: clear cache and cookies and change browsers. So far, I have not been even able to get ANY transactions in the first place with Microsoft rewards ignoring half the points I put into searching/ puzzle points. Is there any way of fixing these ignored points or is this a dead end?

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Just submitted my request to have this issue resolved but it's annoying because several people in my family all try to do the same thing.

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