math puzzles problem solving

Math and Logic Puzzles

If you REALLY like exercising your brain, figuring things 'round and 'round till you explode, then this is the page for you !

Whosoever shall solve these puzzles shall Rule The Universe!

... or at least they should ...

• Prodigy Math
• Prodigy English
• Is a Premium Membership Worth It?
• Promote a Growth Mindset
• Help Your Child Who's Struggling with Math
• Parent's Guide to Prodigy
• Assessments
• Math Curriculum Coverage
• English Curriculum Coverage
• Game Portal

20 Best Math Puzzles to Engage and Challenge Your Students

Written by Maria Kampen

Reviewed by Joshua Prieur, Ed.D.

Solve the hardest puzzle

Use Prodigy Math to boost engagement, offer differentiated instruction and help students enjoy math.

• 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!

• Algebra Calculator

Math Practice

Algebra lessons.

• Algebra Help
• Mobile App for iOS

Math Training

• Other Practice

Practice problems for algebra

• Algebra problems
• All problems (including fractions and negative numbers)

Math Training »

Number Balance

An algebra puzzle game for beginning algebra.

Number Balance »

Step-by-step lessons for learning to solve equations.

Algebra Lessons »

• Grades 6-12
• School Leaders

Free printable Mother's Day questionnaire 💐!

30 Thought-Provoking Math Puzzles for Middle Schoolers

Critical thinking, trial and error, and pure logic abound.

Tired of your tried-and-true math routine? Chances are if you’re feeling the itch to incorporate new activities into your math time, your students are as well. Mixing it up in math class can bring fresh perspectives to stale concepts or standards, and your students will enjoy stretching their brains in different ways with these middle school math puzzles. Critical thinking, trial and error, and pure logic abound in these 30 though-provoking puzzles. Get ready to reignite your middle schoolers’ excitement for math!

(Just a heads up, WeAreTeachers may collect a share of sales from the links on this page. We only recommend items our team loves!)

Sudoku is way more than just an activity to pass the time on long-haul flights. This math puzzle is actually a fantastic problem-solving activity for middle schoolers. Kick-starting your typical math class with a Sudoku puzzle will have your students thinking critically, practicing trial and error, and looking at math in a totally different way. Plus, you can differentiate by providing Easy, Medium, and Difficult puzzles.

Learn more: Sodoku Puzzles To Print

2. 5 Pirates Puzzle

Ahoy and shiver me timbers! This logic puzzle is perfect for a small-group activity to get your middle schoolers working together to solve the conundrum of how pirates plan to share treasure among themselves. Multiple scenarios will play out in this puzzle, so scaffolding with problem-solving strategies is a must.

Learn more:  5 Pirates Puzzles/Math Is Fun

3. Fives Challenge Puzzle

This puzzle is perfect for reviewing addition, multiplication, division, and subtraction and would be a great activity to do when gearing up to teach order of operations. Students could work in pairs or small groups to riddle out each target number.

Learn more:  Fives Challenge Puzzle/Math = Love

4. Beehive Puzzle

Perfect for a station during math rotation or for a rainy-day recess activity, this logic puzzle involves creating a beehive shape without having any squares of the same color touching each other. Students can practice trial and error as well as problem-solving.

Learn more:  Beehive Puzzle/Math = Love

5. Guess My Number

Guess My Number is just as much a riddle as it is a math puzzle. Students use their number sense to determine the number in question. As an extension activity, students can come up with their own clues and trade them with a classmate to solve.

Learn more:  Guess My Number/Education.com

6. Math Riddles

Perfect for a morning warmup, these middle school math puzzles activate all kinds of math knowledge. You can poll the class and have them show their work before clicking to reveal the correct answer. This site even has more challenging puzzles if your middle schoolers fly through the easier ones.

Learn more:  Math Riddles/Get Riddles

My seventh graders loved playing this puzzle as an early-finisher activity. Though the idea is simple (move the tiles until two of the same numbers touch), it’s actually great for recognizing exponents and also for thinking strategically.

Learn more:  2048/Prodigy

8. Magic Squares

Magic Squares have been around for thousands of years, and they come in all shapes and sizes. The 3×3 grid is a great size to introduce to your students and then work up to larger and more complex grids. You can even bring this puzzle off the paper and have your students write the grid out in sidewalk chalk, or write the numbers on water bottle caps to make a fun tactile activity.

Learn more:  Magic Squares/Prodigy

9. Impossible Domino Bridge

Using dominoes to build a seemingly impossible bridge is a perfect activity for the first day or week of a new school year. Your students can work together in small groups and get to know one another as they attempt to construct the bridge that looks like it could turn into a game of Jenga at any moment.

Learn more:  Impossible Domino Bridge/Math = Love

10. Math Picture Puzzles

Your students communicate through emojis anyway, so why not get math involved? This self-checking site allows them to work independently (on the honor system) and also choose between three levels of difficulty. Students can take this idea to the next level, create their own emojis, and arrange them in number sentences for their classmates to solve.

Learn more: Picture Puzzles/MathEasily.com

11. What Is the Weight?

Sometimes you just need a quick resource to get your students working on solving a math puzzle. This puzzle comes from an app, so you can have it downloaded on your students’ iPads or tablets. Middle schoolers will focus on determining the weights of different animals, which is good practice for estimating and working with customary/metric units of measurement.

Learn more: Brain Teasers/Mental Up

12. Colorku

Math doesn’t always have to be just about numbers. This board game uses colors and patterns to focus on analyzing sequences, and would be great to have on hand for those rainy-day recesses as well as for inclusion in a math station. Further, Colorku can be used as a calm-down tool or even a fidget tool.

Buy it: Colorku at Amazon

13. Rubik’s Cube

Rubik’s Cubes made a major comeback in popularity when I taught fifth grade. My students would happily sit together at recess to race each other to see who could solve the cube faster. Though entertaining, Rubik’s Cubes are also suited to teach students about growth mindset, spacial awareness, and 3D space.

Buy it: Rubik’s Cube at Amazon

14. SafeCracker

Though this puzzle looks like something out of an Indiana Jones quest, it’s actually a tactilely engaging tool that will delight even your most resistant math learners. The goal is to align the wheel into columns where the sum adds up to 40. You might need to get more than one of these middle school math puzzles for your classroom.

Buy it: SafeCracker at Amazon

15. “T” Brain Teaser Puzzle

In addition to sparking structural design creativity, this boxed wooden puzzle challenges middle schoolers to engage in trial and error as they work at fitting 50+ pieces into a cube. Much of math is learning how to persevere through tricky problems or procedures, and this puzzle definitely fosters that.

Buy it: T Brain Teaser at Amazon

16. Multistep Equation Puzzle

Solve-and-sort puzzles add flair to repeatedly solving different variations of a math problem for practice. In this free puzzle, students will need to not only solve the equations with variables on both sides, they will also need to sort the problem based on if their solution is positive or negative in order to uncover the secret word.

Get it: Solve-and-Sort Puzzle/Teachers Pay Teachers

In this variation of a classic Sudoku puzzle, students practice critical thinking and exercise their knowledge of how the four math operations work. The best thing about these types of puzzles is that the differentiation potential is endless. Students can solve smaller puzzles with addition, or use only prime numbers in a more complex multiplication problem.

Learn more: Yohaku

18. Jigmaze

One of the Standards for Mathematical Practices is perseverance, and all teachers know that this is a tough one to instill in students, even more so if students are struggling in foundational skills. This type of puzzle can be used to strengthen perseverance as students physically arrange and rearrange pieces of a broken maze.

Learn more: Jigmaze/Math = Love

19. Flexagons

Flexagons, octaflexagons, and dodecaflexagons (say that one 10 times fast!) are a mathematical take on traditional origami. Through constructing these paper creations, your students will get exposure to geometrical terms such as faces ,  equilateral triangles , and all manner of types of 3D shapes.

Get it: Flexagons/Medium

20. Möbius Strip

Though the high-level mathematical equation may be well above your students’ heads (and mine too, if I’m being honest), the STEAM-centered concept of a Möbius strip can be a fun one to explore and create (no need to go into cosines and conversational belts). Middle school math puzzles for the win!

Get it for free: Make a Möbius/STEAMsational

In this complex-looking puzzle, the goal is for the sum of each vertical or horizontal line to match the number given at the beginning of the row or column. This site comes with a great explanation on exactly what that means and how to achieve it. A Kakuro puzzle would be a great “learn as you go” activity for students where they really must pay close attention to the instructions to be able to understand the goal.

Learn more: Kakuro/Braingle

22. Number Searches

This school district’s site has tons of grade-specific number puzzles that would be perfect for when you need to be out of the classroom and have a substitute teacher. They are ready to be printed and contain easy explanations for your students. Check out the number searches, patterns, and 3D riddles.

Learn more: Number Searches/Cranbury School District

23. Two Truths and One Lie

The tried-and-true icebreaker used at many a staff meeting and the first week of school, Two Truths and One Lie can also be used to review and practice tons of mathematical concepts. These middle school math puzzles cover concepts such as negative numbers, fractions, and a ton more.

Buy it: Two Truths & One Lie Math Edition at Amazon

24. Adding Integers Puzzle

The objective of this cuttable resource is for students to solve the integer problem and match up expressions that end up having the same sum. The multiple size options are great for differentiation or to make this independent activity into a small-group collaborative activity.

Buy it: Adding Integers at Teachers Pay Teachers

25. Perfect Square Roots

For upper middle school students, this square-roots puzzle helps with the recognition of perfect square roots. Rather than simply memorizing the perfect square roots, students work to identify and spell out the specific square root and ensure that it fits within the crossword. In this way, the puzzle is self-checking as well.

Buy it: Square Roots Crossword at Teachers Pay Teachers

26. Factor Tree Challenge

Factor trees are an effective way to visually show students the factors of numbers. Trees allow a chain of multiple factors, so you can start with a large number and end up with “branches” that show all of the factors. Once your middle schoolers are familiar with this concept, have them explore this self-checking challenge (and many others as well) that will test their knowledge of abstract factors.

Learn more: Prime Challenges/Transum

27. Ludicross

Another take on Sudoku, Ludicross is interactive in that students can drag and drop the number into position with the goal of making the sum of the numbers in both diagonals the same. Like several of the other puzzles mentioned in this list, students can take this number puzzle to the next level by creating their own and swapping with a classmate to solve.

Learn more: Ludicross/Transum

28. Interactive Mobiles

These colorfully shaped mobiles are a unique way for students to make pattern associations. Because these puzzles are self-paced, students can begin with a simple puzzle and work their way up to complex mobiles with three or more shapes.

Try it: Mobiles/SolveMe Puzzles

29. Deleting Sheep

This logic puzzle is a doozy! The objective is to remove only two numbers in each row with the result being that each horizontal and vertical line equals 30. Trial and error and problem-solving skills abound in this puzzle, and it will keep your middle schoolers engaged for quite some time.

Get it: Deleting Sheep/Dover Publications

30. Pips Puzzle

Have any spare decks of cards lying around your classroom? This inexpensive item provides a different take on a Magic Square. Students can work in small groups, and maybe you can ignite a little class competition to see which groups can complete the challenge the fastest.

Buy it: Pips Puzzle/Math = Love

Looking for more engaging math resources? Try these Magical Math Puzzles and Number Tricks To Wow Your Students .

Plus, get all the latest teaching tips and tricks when you sign up for our free newsletters .

You Might Also Like

Digital Escape Rooms for Fun and Formative Assessment

Escape the monotony while racing the clock. Continue Reading

Copyright © 2024. All rights reserved. 5335 Gate Parkway, Jacksonville, FL 32256

Page No. 1049

Math Games > Math puzzles

100+ Free Math Puzzles ONLINE + Printables

I want to share a fun and easy way to give kids a chance to practice their math skills: math puzzles.

Kids can solve our maths puzzles online or you can download printable maths puzzles as PDF files.

Each math puzzle game contains also at least three math puzzle worksheets which can be printed out or you can browse our best printable math puzzles here .

These mathematical puzzles games are great for use in classroom to help students practise their math skills, or as a time-filler exercise if you've got a spare few minutes to fill up. Enjoy these free math puzzles for kids!

Free math puzzles for kids

Math picture puzzles

Emoji math puzzles are loved by students and teachers. These online math puzzles are great for home schooling. Play now

Addition puzzles

Learn addition up to 20 with our free maths puzzles. Kids are looking for pairs of tiles: the addition problem and the corresponding result. Download printable math puzzles. Play now

Add and subtract within 100

A crossword type math puzzles where the answers are from addition and subtraction calculations within 100. Play now

Math puzzles with matchsticks

Move 1 match to fix the equation. A new matchstick puzzle every day. Solve matchstick math puzzles. Play now

Math brain teasers puzzles

Solve emoji math brain teasers online. You can choose the difficulty of these brain math puzzle games. Play now

Math WORDLE game

Drag and drop the numbers and signs into the correct positions to make the math equation correct. Play now

Math multiplication puzzles

Learn multiplication easily with this online math puzzles game. After completing all 6 multiplication tasks kids will uncover a nice pictures of animals. Play now

Magic squares puzzles

Learn add and subtract within 100: play magic squares math puzzles online. Download magic squares worksheets. Play now

Online math puzzle games

Solving math puzzles online is a super easy way for your kids to practice their skills, even if you're on the go. Or when your students become bored with routine and predictable lesson plans, you can include a few new and fun activities into your upcoming lesson plans: these simple mathematical puzzles games.

Thanksgiving math puzzles

Maths logic puzzles for Thanksgiving. Free math puzzles to enjoy the festive time.

Halloween math puzzles

Solve Halloween math puzzles online or download Halloween math puzzles in PDF.

Christmas math puzzles

Logic math puzzles for Christmas. Free printable maths puzzles in PDF to enjoy the festive time.

Math crossword puzzles

Play crossword math puzzles online. Fill in the missing numbers so the operations are correct.

Christmas brain teasers

Christmas math brain teasers. Logic math puzzles with answers to enjoy Christmas time.

Thanksgiving brain teasers

Thanksgiving math brain teasers. Online math puzzles to enjoy the festive time.

Halloween brain teasers

Halloween math brain teasers are Halloween themed online math puzzles for kids.

Addition pyramids

Pyramid math puzzle game teaches the addition up to 100. Play mathematical puzzles online.

Number matching games

Count the dots and match with the correct number.

Subtraction math puzzles

Learn to subtract within 20, reveal a hidden picture.

Subtraction pyramids

Subtraction pyramid puzzle game. Play online math puzzles and Download math puzzles in PDF.

Multiplication pyramids

Learn to multiply within 500. Play math puzzle games online. Free worksheets are included.

Division puzzles

Solve division math puzzles online.

Multiplication crossword

Solve multiplication math puzzles online. Download multiplication crossword puzzles as PDF files.

Addition BINGO

Play addition BINGO game online. Generate free addition BINGO cards in PDF.

Multiplication BINGO

Play multiplication BINGO game ONLINE. Download multiplication BINGO sheets.

Factor BINGO game

Find the missing factor in this online BINGO game. Get 5 stars in row, column or diagonal.

Simplifying fractions game

Play simplifying fractions games. Simplifying fraction BINGO is a fun math puzzle game.

Click here to download 2 pages for free.

Math games puzzles

50 multiplication puzzles in PDF

Fun printable math puzzles for kids to practice multiplication. 50 different pictures, many multiplication tasks.

🛒 GET YOUR PDF

Digital Download: 50 pages

Free printable maths puzzles PDF

Looking for some fun ways to practice number recognition, addition, subtraction, multiplication and logical thinking? Try these free printable math puzzle worksheets in PDF. Use the links below to automatically generate kids math puzzles in PDF.

Most of these math puzzle worksheets have two pages, the first one is the math puzzle game and the other page contains also the answer key. You can download more free math puzzles with answers in PDF format under individual games or browse all our free math worksheets here .

100 Math Puzzles

Number puzzle games

Math puzzle games are super engaging and loved by students. All these online math puzzles for kids are differentiated based on grade and ability levels.

Engaging in mathematical puzzles at any point of a math lesson is a great way to ignite student thinking, spark creativity, and build anticipation - especially with students who typically struggle in math class and are reluctant to participate.

Kids math puzzles online

If you are looking for a great activity for a long journey or a rainy day try our online math puzzles games.

The great thing about our math puzzle games is that you no longer have to print out maths puzzles and continue to use paper each time you want to play again. But of course, if you want to reduce the screen time you can download printable math puzzles in PDF too.

Free math puzzles online

Math puzzles are fun and can be enjoyed by people of all ages, providing fun and intellectually stimulating entertainment. I created these math puzzle games and math BINGO games with the goal to provide real and practical examples of how you can help students learn math by exploring interesting problems.

Logic math puzzles with answers

So if you're looking for a simple way to squeeze in a little extra math practice to keep your kids' minds sharp over the holiday break, or if you're looking for something productive yet fun for your early finishers, be sure to check out our printable math puzzles and games!

Even if students are not directly engaged in mathematical problem-solving, their curiosity and interest will carry on throughout the math’s lessons.

Free math puzzles printable

In addition to these online math puzzles, I have created also printable math games for kids . They also teach basic math facts in an engaging way.

These make a fun alternative to printable math puzzles and can also be helpful for kids who are struggling or math anxious, as it takes some of the pressure off. I hope your kids enjoy these all math puzzles games!

Printable math puzzles in PDF

Download the collection of our best free math puzzle worksheets for Grades 1-5.

Download now

What are maths puzzles?

Math puzzles are engaging and thought-provoking challenges that involve mathematical concepts and problem-solving skills. Our maths puzzles come in various forms, including math picture puzzles , logic puzzles, geometric puzzles, and word problems with mathematical elements:

• Number puzzles often require manipulating and arranging numbers to reveal a hidden pattern or solve a mathematical equation. Examples include math wordle game , Sudoku, magic squares , Kakuro or math crosswords . These number puzzles usually involve placing numbers in specific positions or filling in the missing numbers using deductive reasoning.
• Logic math puzzles focus on logical reasoning and deduction to arrive at a solution. They often represent a set of clues or conditions that need to be analyzed and combine to determine a unique answer. Popular examples of logic puzzles include grid-based puzzles such as these emoji math brain teasers , matchstick puzzles and the Zebra puzzle.
• Geometric puzzles involve shapes, angles and spatial relationships. They usually require visualization and manipulation of geometric shapes to solve them given problem. Examples include tangram puzzles, dissection puzzles, and folding puzzles.
• Word problems with math elements challenge the solver to translate a real-world situation into a mathematical equation or model. These maths puzzles require comprehension, analytical thinking and mathematical reasoning. They cover a wide range of topics such as algebra, geometry, probability and arithmetic.

Math puzzles generator

I have a variety of math puzzles available, covering a lot of math skills: from logic math puzzles and brain teasers to crosswords and everything in between, encouraging people of all ages to challenge their minds and enjoy the fun and satisfaction of maths puzzle-solving.

These puzzle math games can be helpful for kids who are struggling or math anxious, as it takes some of the pressure off (being able to match the answers together). I hope your kids enjoy these maths puzzles!

Our mathematical puzzles serve multiple purposes. These maths puzzles can be used not only as educational tools to teach math, but also maths puzzle games stimulate children's interest and overall engagement.

As kids successfully solve math logic puzzles, they may develop a greater sense of confidence and accomplishment in their math abilities. Math puzzle games make learning more enjoyable and engaging.

Please, share it:

SolveMe Puzzles

Frequently Asked Questions

• Problem-solving skills
• Analytical thinking skills
• Reasoning skills
• Logical approach
• Relating math to the actual world
• Applying math on practical situations
• Develops child’s reasoning skills and makes them a logical thinker
• Makes them a problem solver by relating math to a practical situation
• The child starts thinking analytically which helps them to get different approaches to a particular problem
• Learns to understand the 'why' behind the 'what'
• Puzzle cards also help the child to keep up the engagement level and develop their interest in maths
• By working on puzzle cards, a child’s brain develops to a much higher extent when compared to their grade level
• Brain teasers
• Math riddles
• Picture puzzles
• Logic puzzles
• Number puzzle
• Crossword puzzle
• Geometry puzzles

• Home   |  
• About   |  
• Contact Us   |  
• Privacy   |  
• Newsletter   |  
• Shop   |  
• 🔍 Search Site
• Easter Color By Number Sheets
• Printable Easter Dot to Dot
• Easter Worksheets for kids
• Kindergarten
• All Generated Sheets
• Place Value Generated Sheets
• Addition Generated Sheets
• Subtraction Generated Sheets
• Multiplication Generated Sheets
• Division Generated Sheets
• Money Generated Sheets
• Negative Numbers Generated Sheets
• Fraction Generated Sheets
• Place Value Zones
• Number Bonds
• Addition & Subtraction
• Times Tables
• Fraction & Percent Zones
• All Calculators
• Fraction Calculators
• Percent calculators
• Area & Volume Calculators
• Age Calculator
• Height Calculator
• Roman Numeral Calculator
• Coloring Pages
• Fun Math Sheets
• Math Puzzles
• Mental Math Sheets
• Online Times Tables
• Online Addition & Subtraction
• Math Grab Packs
• All Math Quizzes
• 1st Grade Quizzes
• 2nd Grade Quizzes
• 3rd Grade Quizzes
• 4th Grade Quizzes
• 5th Grade Quizzes
• 6th Grade Math Quizzes
• Place Value
• Rounding Numbers
• Comparing Numbers
• Number Lines
• Prime Numbers
• Negative Numbers
• Roman Numerals
• Subtraction
• Add & Subtract
• Multiplication
• Fraction Worksheets
• Learning Fractions
• Fraction Printables
• Percent Worksheets & Help
• All Geometry
• 2d Shapes Worksheets
• 3d Shapes Worksheets
• Shape Properties
• Geometry Cheat Sheets
• Printable Shapes
• Coordinates
• Measurement
• Math Conversion
• Statistics Worksheets
• Bar Graph Worksheets
• Venn Diagrams
• All Word Problems
• Finding all possibilities
• Logic Problems
• Ratio Word Problems
• All UK Maths Sheets
• Year 1 Maths Worksheets
• Year 2 Maths Worksheets
• Year 3 Maths Worksheets
• Year 4 Maths Worksheets
• Year 5 Maths Worksheets
• Year 6 Maths Worksheets
• All AU Maths Sheets
• Kindergarten Maths Australia
• Year 1 Maths Australia
• Year 2 Maths Australia
• Year 3 Maths Australia
• Year 4 Maths Australia
• Year 5 Maths Australia
• Meet the Sallies
• Certificates

Math Puzzles for 1st Grade

Welcome to our Math Puzzles for 1st grade page. Here you will find our range of 1st Grade Math Brain Teasers and Puzzles which will help your child apply and practice their Math skills to solve a range of challenges and number problems.

For full functionality of this site it is necessary to enable JavaScript.

Here are the instructions how to enable JavaScript in your web browser .

Using puzzles is a great way to learn Math facts and develop mental calculation and thinking skills in a fun and easy way.

Most children are much more engaged and motivated solving puzzles than working out pages of traditional math facts.

Using these math puzzles below will help your child to develop their Math fact skills as well as their strategic thinking and reasoning.

There are different versions of each puzzle from 1st to 5th grade, so it is easy to find an easier or harder version of the same puzzle. Each puzzle comes complete with answers.

Quicklinks to ...

• 2nd Grade Math Puzzles
• 3rd Grade Math Puzzles
• 4th Grade Math Puzzles
• 5th Grade Math Puzzles

Here you will find a range of challenges and puzzles to develop your child's math skills and number facts in a fun way.

These puzzles have been designed to support the first grade skills of adding and subtracting, as well as problem solving.

Using these math puzzles for 1st grade will help your child to:

• learn and practice their addition facts to 12+12;
• practice their subtraction facts to 12;
• develop problem solving skills and reasoning.

Salamander Line-Up

Salamander Line-up involves placing salamanders on a square grid so that no salamander is in the same row or column as another salamander.

It is a good puzzle for developing logical and reasoning skills.

• Salamander Line Up Puzzle 1
• PDF version
• Tricolor Circles Puzzle

Tricolor Circles puzzle is a simple puzzle involving coloring nine circles linked together so that no two touching circles are the same color.

It is a good activity for developing reasoning and logic!

Five Circles Puzzle

Five Circles puzzle is a simple puzzle involving placing numbers in five circles so that the totals of the rows and columns are the same.

It is a good activity for adding together several small numbers and for developing perseverance!

• 5 Circles Puzzle

Captain's Square Puzzle

Captain's Square puzzle involves working out the totals of rows and columns made up of different salamanders. Each salamander has its own value.

Great for learning to add together several small numbers, and it is very easy to extend by changing the values of the salamanders.

• Captain's Square Puzzle 1

Newton's Crosses Puzzle

Newton's crosses puzzle is a challenge which involves placing numbers in the correct place to get the same total in each row and column in the cross.

It is a similar puzzle to the 5 Circles puzzle and a good activity for practicing addition facts.

• Newton's Crosses Puzzle 1
• The Two Chests Puzzle 1

The Two Chests Puzzle is a puzzle which involves placing 5 bags of coins into the chests so that one of the chests contains double the amount of the other chest.

It is a good activity to use for practicing addition facts, and also to develop trial-and-improvement strategies.

Quadra's Square Puzzle

Quadra's Square Puzzle is a puzzle which involves placing digits in the correct places so that each row and column adds up to the same number.

It is a good activity to use for practicing adding, and also to develop perseverance.

• Quadra's Square Puzzle 1

Quadra's Addition Square Puzzle

Quadra's Square Puzzle is a puzzle which involves placing numbers in the correct places so that each hexagon is the sum of the two circles that join it.

There are 4 missing digits to place so that the sums are correct.

• Quadra's Addition Square Puzzle 1

Magic Square Puzzle

Magic Square Puzzle is a puzzle which involves placing digits in the correct places so that each row, column and diagonal adds up to the same number.

• Magic Square Puzzle 1
• More Magic Square Worksheets

Sally's Hexagon Puzzle

Sally's hexagon number puzzle is a challenge which involve accurate adding of two numbers together.

Each number in the hexagon pyramid is made by adding up the 2 numbers below it.

• Sally's Hexagon Puzzle 1

Arithmogon Triangle Puzzle

The Arithmogon triangle puzzle is a math puzzle to help develop adding and subtracting numbers and is also useful for developing logical thinking and pre-algebra skills at a higher level.

The numbers in the two circles are added together to give the number in the linking rectangle.

Puzzle 1b is a harder puzzle for more able mathematicians.

• Arithmogon Triangle Puzzle 1a
• Arithmogon Triangle Puzzle 1b

Newton's Number Track Puzzle

Newton's number track puzzle is a math puzzle to help develop adding of small numbers together.

Each number in the number track is made by adding the 2 previous numbers together.

• Newton's Number Track Puzzle 1

Quadra's Operation Puzzle

Quadra's operation puzzle involves choosing the correct operations to make the math fact correct.

Great for developing adding and subtracting skills, and getting children to experiment with numbers and develop a number sense.

• Quadra's Operation Puzzle 1

Tree Adding Puzzle

This adding puzzle involves using addition to work out the missing numbers on the trees.

• Tree Adding Puzzle 1
• Number Maze Target 15 Puzzle

The aim of this puzzle is for you to finish the maze with a total of 15. Choose your path carefully!

The maze involves adding and subtracting up to 20.

Number Grid Challenge: Target 20 Puzzle

The aim of this grid puzzle is for you to find a path through the grid with a total of 20. Choose your path carefully!

The maze involves adding a series of 4 to 5 numbers (with values up to 9) together to try to get a total of 20.

• Number Grid Challenge 1

Looking for a harder puzzle...?

If these Math Puzzles for 1st grade are not at the right level for you, try some of our harder 2nd grade math puzzles.

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these math puzzles for 1st grade.

Number Search Puzzles

Number Search Puzzles are a great way to get children looking for numbers and developing number recognition skills.

They are also a good resource for developing short term number memory skills, and can be a good way to take the fear out of large numbers.

We have a range of different number search puzzles - from easier puzzles to trickier ones to work out.

With the easier puzzles, the numbers only go horizontally (left to right) or vertically downwards.

The numbers get progressively larger on the trickier puzzles, and the grids get larger.

• Number Search Puzzles (Easier)
• Number Search Puzzles (Medium)
• Number Search Worksheets (Hard)

First Grade Math Games

Here you will find a range of free printable First Grade Math games. All children like to play Math games, and you will find a good range of 1st Grade Math Games here for your child to play and enjoy.

• 1st Grade Math Games

1st Grade Math Word Problems

Here you will find a range of math word problems aimed at first grade level. Each problem sheet is based on an interesting theme such as parties or the seaside.

• Math Problems for Children 1st Grade

Longer Math Problems

• First Grade Math Problems

Math Riddles

Here is our collection of free math riddles from 1st grade to 5th grade.

You will find a range of number riddles which will help your child to develop their place value skills, as well as developing their problem solving and reasoning.

The riddles are also useful for developing understanding of mathematical language.

• Place Value Riddles

Please give us feedback on any of our math puzzles for 1st grade at the bottom of the page.

How to Print or Save these sheets 🖶

Need help with printing or saving? Follow these 3 steps to get your worksheets printed perfectly!

• How to Print support

Subscribe to Math Salamanders News

Sign up for our newsletter to get free math support delivered to your inbox each month. Plus, get a seasonal math grab pack included for free!

• Newsletter Signup

Return to First Grade Math Worksheets

Return to Printable Math Puzzles Hub page

Return from Math Puzzles 1st Grade to Math Salamanders Homepage

Math-Salamanders.com

The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.

We welcome any comments about our site or worksheets on the Facebook comments box at the bottom of every page.

New! Comments

TOP OF PAGE

© 2010-2024 Math Salamanders Limited. All Rights Reserved.

• Privacy Policy
• Copyright Policy

Reset password New user? Sign up

Existing user? Log in

Logical Puzzles

Already have an account? Log in here.

• Andrew Hayes

A logical puzzle is a problem that can be solved through deductive reasoning. This page gives a summary of the types of logical puzzles one might come across and the problem-solving techniques used to solve them.

Elimination Grids

Truth tellers and liars, cryptograms, arithmetic puzzles, river crossing puzzle, tour puzzles, battleship puzzles, chess puzzles, k-level thinking, other puzzles.

Main Article: Propositional Logic See Also: Predicate Logic

One of the simplest types of logical puzzles is a syllogism . In this type of puzzle, you are given a set of statements, and you are required to determine some truth from those statements. These types of puzzles can often be solved by applying principles from propositional logic and predicate logic . The following syllogism is from Charles Lutwidge Dodgson, better known under his pen name, Lewis Carroll.

I have a dish of potatoes. The following statements are true: No potatoes of mine, that are new, have been boiled. All my potatoes in this dish are fit to eat. No unboiled potatoes of mine are fit to eat. Are there any new potatoes in this dish? The first and third statements can be connected by a transitive argument. All of the new potatoes are unboiled, and unboiled potatoes aren't fit to eat, so no new potatoes are fit to eat. The second statement can be expressed as the equivalent contrapositive. All of the potatoes in the dish are fit to eat; if there is a potato that is not fit to eat, it isn't in the dish. Then, once again, a transitive argument is applied. New potatoes aren't fit to eat, and inedible potatoes aren't in the dish. Thus, there are no new potatoes in the dish. \(_\square\)

Given below are three statements followed by three conclusions. Take the three statements to be true even if they vary from commonly known facts. Read the statements and decide which conclusions follow logically from the statements.

Statements: 1. All actors are musicians. 2. No musician is a singer. 3. Some singers are dancers.

Conclusions: 1. Some actors are singers. 2. Some dancers are actors. 3. No actor is a singer.

Answer Choices: a) Only conclusion 1 follows. b) Only conclusion 2 follows. c) Only conclusion 3 follows. d) At least 2 of the conclusions follows.

Main Article: Elimination Grids

Some logical puzzles require you to determine the correct pairings for sets of objects. These puzzles can often be solved with the process of elimination, and an elimination grid is an effective tool to apply this process.

An example of an elimination grid

Elimination grids are aligned such that each row represents an object within a set, and each column represents an object to be paired with an object from that set. Check marks and X marks are used to show which objects pair, and which objects do not pair.

Mr. and Mrs. Tan have four children--three boys and a girl-- who each like one of the colors--blue, green, red, yellow-- and one of the letters--P, Q, R, S.

• The oldest child likes the letter Q.
• The youngest child likes green.
• Alfred likes the letter S.
• Brenda has an older brother who likes R.
• The one who likes blue isn't the oldest.
• The one who likes red likes the letter P.
• Charles likes yellow.

Based on the above facts, Darius is the \(\text{__________}.\)

Main Article: Truth-Tellers and Liars

A variation on elimination puzzles is a truth-teller and liar puzzle , also known as a knights and knaves puzzle . In this type of puzzle, you are given a set of people and their respective statements, and you are also told that some of the people always tell the truth and some always lie. The goal of the puzzle is to deduce the truth from the given statements.

20\(^\text{th}\) century mathematician Raymond Smullyan popularized these types of puzzles.

You are in a room with three chests. You know at least one has treasure, and if a chest has no treasure, it contains deadly poison.

Each chest has a message on it, but all the messages are lying .

• Left chest: "The middle chest has treasure."
• Middle chest: "All these chests have treasure."
• Right chest: "Only one of these chests has treasure."

Which chests have treasure?

There are two people, A and B , each of whom is either a knight or a knave.

A says, "At least one of us is a knave."

What are A and B ?

\(\) Details and Assumptions:

• A knight always tells the truth.
• A knave always lies.

Main Article: Cryptograms

A cryptogram is a puzzle in which numerical digits in a number sentence are replaced with characters, and the goal of the puzzle is to determine the values of these characters.

\[ \begin{array} { l l l l l } & &P & P & Q \\ & &P & Q & Q \\ + && Q & Q & Q \\ \hline & & 8 & 7 & 6 \\ \end{array} \]

In the sum shown above, \(P\) and \(Q\) each represent a digit. What is the value of \(P+Q\)?

\[ \overline{EVE} \div \overline{DID} = 0. \overline{TALKTALKTALKTALK\ldots} \]

Given that \(E,V,D,I,T,A,L\) and \(K \) are distinct single digits, let \(\overline{EVE} \) and \( \overline{DID} \) be two coprime 3-digit positive integers and \(\overline{TALK} \) be a 4-digit integer, such that the equation above holds true, where the right hand side is a repeating decimal number.

Find the value of the sum \( \overline{EVE} + \overline{DID} + \overline{TALK} \).

Main Articles: Fill in the Blanks and Operator Search

Arithmetic puzzles contain a series of numbers, operations, and blanks in order, and the object of the puzzle is to fill in the blanks to obtain a desired result.

\[\huge{\Box \times \Box \Box = \Box \Box \Box}\]

Fill the boxes above with the digits \(1,2,3,4,5,6\), with no digit repeated, such that the equation is true.

Enter your answer by concatenating all digits in the order they appear. For example, if the answer is \(1 \times 23 = 456\), enter \(123456\) as your final answer.

Also try its sister problem.

\[ \LARGE{\begin{eqnarray} \boxed{\phantom0} \; + \; \boxed{\phantom0} \; &=& \; \boxed{\phantom0} \\ \boxed{\phantom0} \; - \; \boxed{\phantom0} \; &=& \; \boxed{\phantom0} \\ \boxed{\phantom0} \; \times \; \boxed{\phantom0}\; &=& \; \boxed{\phantom0} \\ \boxed{\phantom0} \; \div \; \boxed{\phantom0} \; &=& \; \boxed{\phantom0} \\ \end{eqnarray}} \]

Put one of the integers \(1, 2, \ldots , 13\) into each of the boxes, such that twelve of these numbers are used once for each (and one number is not used at all) and all four equations are true.

What is the sum of all possible values of the missing (not used) number?

Main Article: River Crossing Puzzles

In a river crossing puzzle , the goal is to find a way to move a group of people or objects across a river (or some other kind of obstacle), and to do it in the fewest amount of steps or least amount of time.

A famous river crossing problem is Richard Hovasse's bridge and torch problem , written below.

Four people come to a river in the night. There is a narrow bridge, but it can only hold two people at a time. They have one torch and, because it's night, the torch has to be used when crossing the bridge. Person A can cross the bridge in one minute, B in two minutes, C in five minutes, and D in eight minutes. When two people cross the bridge together, they must move at the slower person's pace. The question is, can they all get across the bridge in 15 minutes or less? Assume that a solution minimizes the total number of crosses. This gives a total of five crosses--three pair crosses and two solo crosses. Also, assume we always choose the fastest for the solo cross. First, we show that if the two slowest persons (C and D) cross separately, they accumulate a total crossing time of 15. This is done by taking persons A, C, D: D+A+C+A = 8+1+5+1=15. (Here we use A because we know that using A to cross both C and D separately is the most efficient.) But, the time has elapsed and persons A and B are still on the starting side of the bridge and must cross. So it is not possible for the two slowest (C and D) to cross separately. Second, we show that in order for C and D to cross together that they need to cross on the second pair cross: i.e. not C or D, so A and B, must cross together first. Remember our assumption at the beginning states that we should minimize crosses, so we have five crosses--3 pair crossings and 2 single crossings. Assume that C and D cross first. But then C or D must cross back to bring the torch to the other side, so whoever solo-crossed must cross again. Hence, they will cross separately. Also, it is impossible for them to cross together last, since this implies that one of them must have crossed previously, otherwise there would be three persons total on the start side. So, since there are only three choices for the pair crossings and C and D cannot cross first or last, they must cross together on the second, or middle, pair crossing. Putting all this together, A and B must cross first, since we know C and D cannot and we are minimizing crossings. Then, A must cross next, since we assume we should choose the fastest to make the solo cross. Then we are at the second, or middle, pair crossing, so C and D must go. Then we choose to send the fastest back, which is B. A and B are now on the start side and must cross for the last pair crossing. This gives us, B+A+D+B+B = 2+1+8+2+2 = 15. It is possible for all four people to cross in 15 minutes. \(_\square\)

Main Article: Tour Puzzles See Also: Eulerian Path

In a tour puzzle , the goal is to determine the correct path for an object to traverse a graph. These kinds of puzzles can take several forms: chess tours, maze traversals, eulerian paths , and others.

Find the path that leads from the star in the center back to the star in the center. Paths can only go in the direction of an arrow. Image Credit: Eric Fisk Show Solution The solution path is outlined in red below.

Determine a path through the below graph such that each edge is traversed exactly once . Show Solution There are several possible solutions. One possible solution is shown below, with the edges marked in the order they are traversed.

A chess tour is an interesting type of puzzle in its own right, and is explained in detail further down the page.

Main Article: Nonograms

A nonogram is a grid-based puzzle in which a series of numerical clues are given beside a rectangular grid. When the puzzle is completed, a picture is formed in the grid.

The puzzle begins with a series of numbers on the left and above the grid. Each of these numbers represents a consecutive run of shaded spaces in the corresponding row or column. Each consecutive run is separated from other runs by at least one empty space. The puzzle is complete when all of the numbers have been satisfied. The primary technique to solve these puzzles is the process of elimination. If the puzzle is designed correctly, there should be no guesswork required.

Complete the nonogram: Show Solution

One of the many logical puzzles is the Battleship puzzle (sometimes called Bimaru, Yubotu, Solitaire Battleships or Battleship Solitaire). The puzzle is based on the Battleship game.

Solitaire Battleships was invented by Jaime Poniachik in Argentina and was first featured in the magazine Humor & Juegos.

This is an example of a solved Battleship puzzle. The puzzle consists of a 10 × 10 small squares, which contain the following:

• 1 battleship 4 squares long
• 2 cruisers 3 squares long each
• 3 destroyers 2 squares long each
• 4 submarines 1 square long each.

They can be put horizontally or vertically, but never diagonally. The boats are placed so that no boats touch each other, not even vertically. The numbers beside the row/column indicate the numbers of squares occupied in the row/column, respectively. ⬤ indicates a submarine and ⬛ indicates the body of a ship, while the half circles indicate the beginning/end of a ship.

The goal of the game is to fill in the grid with water or ships.

Main Article: Sudoku

A sudoku is a puzzle on a \(9\times 9\) grid in which each row, column, and smaller square portion contains each of the digits 1 through 9, each no more than once. Each puzzle begins with some of the spaces on the grid filled in. The goal is to fill in the remaining spaces on the puzzle. The puzzle is solved primarily through the process of elimination. No guesswork should be required to solve, and there should be only one solution for any given puzzle.

Solve the sudoku puzzle: Puzzle generated by Open Sky Sudoku Generator Each row should contain the each of the digits 1 through 9 exactly once. The same is true for columns and the smaller \(3\times 3\) squares. Show Solution

Main Article: Chess Puzzles See Also: Reduced Games , Opening Strategies , and Rook Strategies

Chess puzzles take the rules of chess and challenge you to perform certain actions or deduce board states.

One kind of chess puzzle is a chess tour , related to the tour puzzles mentioned in the section above. This kind of puzzle challenges you to develop a tour of a chess piece around the board, applying the rules of how that piece moves.

Dan and Sam play a game on a \(5\times3\) board. Dan places a White Knight on a corner and Sam places a Black Knight on the nearest corner. Each one moves his Knight in his turn to squares that have not been already visited by any of the Knights at any moment of the match.

For example, Dan moves, then Sam, and Dan wants to go to Black Knight's initial square, but he can't, because this square has been occupied earlier.

When someone can't move, he loses. If Dan begins, who will win, assuming both players play optimally?

This is the seventeenth problem of the set Winning Strategies.

Due to its well-defined ruleset, the game of chess affords many different types of puzzles. The problem below shows that you can even deduce whose turn it is from a certain boardstate (or perhaps you cannot).

Whose move is it now?

Main Article: K-Level Thinking See Also: Induction - Introduction

K-level thinking is the name of a kind of assumption in certain logic puzzles. In these types of puzzles, there are a number of actors in a situation, and each of them is perfectly logical in their decision-making. Furthermore, each of these actors is aware that all other actors in the situation are perfectly logical in their decision-making.

Calvin, Zandra, and Eli are students in Mr. Silverman's math class. Mr. Silverman hands each of them a sealed envelope with a number written inside.

He tells them that they each have a positive integer and the sum of the three numbers is 14. They each open their envelope and inspect their own number without seeing the other numbers.

Calvin says,"I know that Zandra and Eli each have a different number." Zandra replies, "I already knew that all three of our numbers were different." After a brief pause Eli finally says, "Ah, now I know what number everyone has!"

What number did each student get?

Format your answer by writing Calvin's number first, then Zandra's number, and finally Eli's number. For example, if Calvin has 8, Zandra has 12, and Eli has 8, the answer would be 8128.

Two logicians must find two distinct integers \(A\) and \(B\) such that they are both between 2 and 100 inclusive, and \(A\) divides \(B\). The first logician knows the sum \( A + B \) and the second logician knows the difference \(B-A\).

Then the following discussion takes place:

Logician 1: I don't know them. Logician 2: I already knew that.

Logician 1: I already know that you are supposed to know that. Logician 2: I think that... I know... that you were about to say that!

Logician 1: I still can't figure out what the two numbers are. Logician 2: Oops! My bad... my previous conclusion was unwarranted. I didn't know that yet!

What are the two numbers?

Enter your answer as a decimal number \(A.B\). \((\)For example, if \(A=23\) and \(B=92\), write \(23.92.)\)

Note: In this problem, the participants are not in a contest on who finds numbers first. If one of them has sufficient information to determine the numbers, he may keep this quiet. Therefore nothing may be inferred from silence. The only information to be used are the explicit declarations in the dialogue.

Of course, the puzzles outlined above aren't the only types of puzzles one might encounter. Below are a few more logical puzzles that are unrelated to the types outlined above.

You are asked to guess an integer between \(1\) and \(N\) inclusive.

Each time you make a guess, you are told either

(a) you are too high, (b) you are too low, or (c) you got it!

You are allowed to guess too high twice and too low twice, but if you have a \(3^\text{rd}\) guess that is too high or a \(3^\text{rd}\) guess that is too low, you are out.

What is the maximum \(N\) for which you are guaranteed to accomplish this?

\(\) Clarification : For example, if you were allowed to guess too high once and too low once, you could guarantee to guess the right answer if \(N=5\), but not for \(N>5\). So, in this case, the answer would be \(5\).

You play a game with a pile of \(N\) gold coins.

You and a friend take turns removing 1, 3, or 6 coins from the pile. The winner is the one who takes the last coin.

For the person that goes first, how many winning strategies are there for \(N < 1000?\)

\(\) Clarification: For \(1 \leq N \leq 999\), for how many values of \(N\) can the first player develop a winning strategy?

Problem Loading...

Note Loading...

Set Loading...

20 Tough Riddles for Adults That Will Have You Scratching Your Head

Put your logic and math skills to the test. No cheating!

So go grab a pencil and a piece of scratch paper and prepare to rip your hair out (and we really do mean that in the best way possible). When you think you’ve got the right answer, click the link at the bottom of each riddle to find the solution. Got it wrong? No worries, you have 19 other riddles to test out.

Navigate Through Our Riddles:

Puzzmo / The King’s Orders / How Many Eggs? / The Gold Chain / Pickleball / Circuit Breaker / Two Trains, Two Grandmas / Ant Math / Peppermint Patty / Great American Rail Trail / A Cruel SAT Problem / Movie Stars Cross a River / Tribute to a Math Genius / One Belt, One Earth / Elbow Tapping / Whiskey Problem / Doodle Problem / Stumping Scientists / What ’ s On Her Forehead? / Keanu for President / Who Opened the Lockers?

Riddle No. 1: The King’s Orders Make for One Hell of a Brain Teaser

Difficulty: easy.

King Nupe of the kingdom Catan dotes on his two daughters so much that he decides the kingdom would be better off with more girls than boys, and he makes the following decree: All child-bearing couples must continue to bear children until they have a daughter!

But to avoid overpopulation, he makes an additional decree: All child-bearing couples will stop having children once they have a daughter! His subjects immediately begin following his orders.

After many years, what’s the expected ratio of girls to boys in Catan?

The likelihood of each baby born being a girl is, of course, 50 percent.

Ready for the solution? Click here to see if you’re right .

Riddle No. 2: How Many Eggs Does This Hen Lay?

This problem is in honor of my dad, Harold Feiveson. It’s due to him that I love math puzzles, and this is one of the first problems (of many) that he gave me when I was growing up.

A hen and a half lays an egg and a half in a day and a half. How many eggs does one hen lay in one day?

Riddle No. 3: The Gold Chain Math Problem Is Deceptively Simple

Difficulty: moderate.

You’re rummaging around your great grandmother’s attic when you find five short chains each made of four gold links. It occurs to you that if you combined them all into one big loop of 20 links, you’d have an incredible necklace. So you bring it into a jeweler, who tells you the cost of making the necklace will be $10 for each gold link that she has to break and then reseal.

How much will it cost?

Riddle No. 4: Try to Solve This Pickleball Puzzle

Difficulty: 🚨hard🚨.

Kenny, Abby, and Ned got together for a round-robin pickleball tournament, where, as usual, the winner stays on after each game to play the person who sat out that game. At the end of their pickleball afternoon, Abby is exhausted, having played the last seven straight games. Kenny, who is less winded, tallies up the games played:

Kenny played eight games

Abby played 12 games

Ned played 14 games

Who won the fourth game against whom?

How many total games were played?

Riddle No. 5: Our Circuit Breaker Riddle Is Pure Evil. Sorry.

The circuit breaker box in your new house is in an inconvenient corner of your basement. To your chagrin, you discover none of the 100 circuit breakers is labeled, and you face the daunting prospect of matching each circuit breaker to its respective light. (Suppose each circuit breaker maps to only one light.)

To start with, you switch all 100 lights in the house to “on,” and then you head down to your basement to begin the onerous mapping process. On every trip to your basement, you can switch any number of circuit breakers on or off. You can then roam the hallways of your house to discover which lights are on and which are off.

What is the minimum number of trips you need to make to the basement to map every circuit breaker to every light?

The solution does not involve either switching on or off the light switches in your house or feeling how hot the lightbulbs are. You might want to try solving for the case of 10 unlabeled circuit breakers first.

Riddle No. 6: Two Trains. Two Grandmas. Can You Solve This Tricky Math Riddle?

Jesse’s two grandmothers want to see him every weekend, but they live on opposite sides of town. As a compromise, he tells them that every Sunday, he’ll head to the subway station nearest to his apartment at a random time of the day and will hop on the next train that arrives.

If it happens to be the train traveling north, he’ll visit his Grandma Erica uptown, and if it happens to be the train traveling south, he’ll visit his Grandma Cara downtown. Both of his grandmothers are okay with this plan, since they know both the northbound and southbound trains run every 20 minutes.

But after a few months of doing this, Grandma Cara complains that she sees him only one out of five Sundays. Jesse promises he’s indeed heading to the station at a random time each day. How can this be?

The trains always arrive at their scheduled times.

Riddle No. 7: Here’s a Really F@*#ing Hard Math Problem About Ants

Max and Rose are ant siblings. They love to race each other, but always tie, since they actually crawl at the exact same speed. So they decide to create a race where one of them (hopefully) will win.

For this race, each of them will start at the bottom corner of a cuboid, and then crawl as fast as they can to reach a crumb at the opposite corner. The measurements of their cuboids are as pictured:

If they both take the shortest possible route to reach their crumb, who will reach their crumb first? (Don’t forget they’re ants, so of course they can climb anywhere on the edges or surface of the cuboid.)

Remember: Think outside the box.

Riddle No. 8: This Peppermint Patty Riddle Is Practically Impossible

You’re facing your friend, Caryn, in a “candy-off,” which works as follows: There’s a pile of 100 caramels and one peppermint patty. You and Caryn will go back and forth taking at least one and no more than five caramels from the candy pile in each turn. The person who removes the last caramel will also get the peppermint patty. And you love peppermint patties.

Suppose Caryn lets you decide who goes first. Who should you choose in order to make sure you win the peppermint patty?

First, solve for a pile of 10 caramels.

Riddle No. 9: Can You Solve the Great American Rail-Trail Riddle?

This problem was suggested by the physicist P. Jeffrey Ungar.

Finally, the Great American Rail-Trail across the whole country is complete! Go ahead, pat yourself on the back—you’ve just installed the longest handrail in the history of the world, with 4,000 miles from beginning to end. But just after the opening ceremony, your assistant reminds you that the metal you used for the handrail expands slightly in summer, so that its length will increase by one inch in total.

“Ha!” you say, “One inch in a 4,000 mile handrail? That’s nothing!” But … are you right?

Let’s suppose when the handrail expands, it buckles upward at its weakest point, which is in the center. How much higher will pedestrians in the middle of the country have to reach in summer to grab the handrail? That is, in the figure below, what is h ? (For the purposes of this question, ignore the curvature of the Earth and assume the trail is a straight line.)

Pythagoras is a fascinating historical figure.

Riddle No. 10: This Riddle Is Like an Especially Cruel SAT Problem. Can You Find the Answer?

Amanda lives with her teenage son, Matt, in the countryside—a car ride away from Matt’s school. Every afternoon, Amanda leaves the house at the same time, drives to the school at a constant speed, picks Matt up exactly when his chess club ends at 5 p.m., and then they immediately return home together at the same constant speed. But one day, Matt isn’t feeling well, so he leaves chess practice early and starts to head home on his portable scooter.

After Matt has been scooting for an hour, Amanda comes across him in her car (on her usual route to pick him up), and they return together, arriving home 40 minutes earlier than they usually do. How much chess practice did Matt miss?

Consider the case where Amanda meets Matt exactly as she’s leaving their house.

Riddle No. 11: Can You Get These 3 Movie Stars Across the River?

Three movie stars, Chloe, Lexa, and Jon, are filming a movie in the Amazon. They’re very famous and very high-maintenance, so their agents are always with them. One day, after filming a scene deep in the rainforest, the three actors and their agents decide to head back to home base by foot. Suddenly, they come to a large river.

On the riverbank, they find a small rowboat, but it’s only big enough to hold two of them at one time. The catch? None of the agents are comfortable leaving their movie star with any other agents if they’re not there as well. They don’t trust that the other agents won’t try to poach their star.

For example, Chloe’s agent is okay if Chloe and Lexa are alone in the boat or on one of the riverbanks, but definitely not okay if Lexa’s agent is also with them. So how can they all get across the river?

There isn’t just one way to solve this problem.

Riddle No. 12: This Ludicrously Hard Riddle Is Our Tribute to a Late Math Genius. Can You Figure It Out?

On April 11, John Horton Conway , a brilliant mathematician who had an intense and playful love of puzzles and games, died of complications from COVID-19. Conway is the inventor of one of my favorite legendary problems (not for the faint of heart) and, famously, the Game of Life . I created this problem in his honor.

Carol was creating a family tree, but had trouble tracking down her mother’s birthdate. The only clue she found was a letter written from her grandfather to her grandmother on the day her mother was born. Unfortunately, some of the characters were smudged out, represented here with a “___” . (The length of the line does not reflect the number of smudged characters.)

“Dear Virginia,

Little did I know when I headed to work this Monday morning, that by evening we would have a beautiful baby girl. And on our wedding anniversary, no less! It makes me think back to that incredible weekend day, J___ 27th, 19___ , when we first shared our vow to create a family together, and, well, here we are! Happy eighth anniversary, my love.

Love, Edwin”

The question: When was Carol’s mother born?

This problem is inspired by Conway’s Doomsday Rule .

Riddle No. 13: To Solve This Twisty Math Riddle, You Just Need One Belt and One Earth

Imagine you have a very long belt. Well, extremely long, really … in fact, it’s just long enough that it can wrap snugly around the circumference of our entire planet. (For the sake of simplicity, let’s suppose Earth is perfectly round, with no mountains, oceans, or other barriers in the way of the belt.)

Naturally, you’re very proud of your belt. But then your brother, Peter, shows up—and to your disgruntlement, he produces a belt that’s just a bit longer than yours. He brags his belt is longer by exactly his height: 6 feet.

If Peter were also to wrap his belt around the circumference of Earth, how far above the surface could he suspend the belt if he pulled it tautly and uniformly?

Earth’s circumference is about 25,000 miles, or 130 million feet … but you don’t need to know that to solve this problem.

Riddle No. 14: This Elbow Tapping Riddle Is Diabolical. Good Luck Solving It.

In some future time, when the shelter-in-place bans are lifted, a married couple, Florian and Julia, head over to a bar to celebrate their newfound freedom.

They find four other couples there who had the same idea.

Eager for social contact, every person in the five couples enthusiastically taps elbows (the new handshake) with each person they haven’t yet met .

It actually turns out many of the people had known each other prior, so when Julia asks everyone how many elbows they each tapped, she remarkably gets nine different answers!

The question: How many elbows did Florian tap?

What nine answers did Julia hear?

Riddle No. 15: You’ll Need a Drink After Trying to Solve This Whisky Riddle

Alan and Claire live by the old Scottish saying, “Never have whisky without water, nor water without whisky!” So one day, when Alan has in front of him a glass of whisky, and Claire has in front of her a same-sized glass of water, Alan takes a spoonful of his whisky and puts it in Claire’s water. Claire stirs her whisky-tinted water, and then puts a spoonful of this mixture back into Alan’s whisky to make sure they have exactly the same amount to drink.

So: Is there more water in Alan’s whisky, or more whisky in Claire’s water? And does it matter how well Claire stirred?

The size of the spoon does not matter.

Riddle No. 16: The Doodle Problem Is a Lot Harder Than It Looks. Can You Solve It?

This week’s riddle is relatively simple—but sinister all the same.

The question: Can you make 100 by interspersing any number of pluses and minuses within the string of digits 9 8 7 6 5 4 3 2 1? You can’t change the order of the digits! So what’s the least number of pluses and minuses needed to make 100?

For instance, 98 - 7 - 6 + 54 - 32 shows one way of interspersing pluses and minuses, but since it equals 107, it’s not a solution.

I call this a “doodle problem”: one that’s best worked on during meetings where you might be doodling otherwise.

You might want to start looking for solutions that use a total of seven pluses and minuses (although there are ways to use fewer).

Ready for the solution? Click here to see if you’re right.

Riddle No. 17: This Math Puzzle Stumped Every Scientist but One. Think You Can Crack It?

Difficulty: hard.

In honor of Freeman Dyson, the renowned physicist who died last month , here’s a legendary tale demonstrating his quick wit and incredible brain power.

One day, in a gathering of top scientists, one of them wondered out loud whether there exists an integer that you could exactly double by moving its last digit to its front. For instance, 265 would satisfy this if 526 were its exact double—which it isn’t.

After apparently just five seconds , Dyson responded, “Of course there is, but the smallest such number has 18 digits.”

This left some of the smartest scientists in the world puzzling over how he could have figured this out so quickly.

So given Dyson’s hint, what is the smallest such number?

My second grader has recently learned how to add a 3-digit number to itself using the classic vertical method:

18-digit numbers, of course, can be added in the same way.

Riddle No. 18: Figure Out What’s on Her Forehead

Cecilia loves testing the logic of her very logical friends Jaya, Julian, and Levi, so she announces:

“I’ll write a positive number on each of your foreheads. None of the numbers are the same, and two of the numbers add up to the third.”

She scribbles the numbers on their heads, then turns to Jaya and asks her what her number is. Jaya sees Julian has 20 on his forehead, and Levi has 30 on his. She thinks for a moment and then says, “I don’t know what my number is.” Julian pipes in, “I also don’t know my number,” and then Levi exclaims, “Me neither!” Cecilia gleefully says, “I’ve finally stumped you guys!”

“Not so fast!” Jaya says. “Now I know my number!”

What is Jaya’s number?

Jaya could be one of two numbers, but only one of those numbers would lead to Julian and Levi both not knowing their numbers. Why?

Riddle No. 19: Can You Get Keanu Reeves Elected As President?

It’s 2024, and there are five candidates running in the democratic primary: Taylor Swift, Oprah Winfrey, Mark Cuban, Keanu Reeves, and Dwayne Johnson. (Hey, it could happen.) As usual, the first primary is in Iowa.

In an effort to overcome its embarrassment after the 2020 caucus debacle , the Iowa Democratic Party has just announced a new, foolproof way of finding the best candidate: there will be four consecutive elections.

First, candidate 1 will run against candidate 2. Next, the winner of that will run against candidate 3, then that winner will run against candidate 4, and finally the winner of that election will run against the final candidate. By the transitive property, the winner of this last election must be the best candidate ... so says the Iowa Democratic Party.

Candidate Keanu has been feeling pretty low, as he knows he is ranked near the bottom by most voters, and at the top by none. In fact, he knows the Iowa population is divided into five equal groups, and that their preferences are as follows:

Keanu is childhood friends with Bill S. Preston, Esq., the new head of the Iowa Democratic Party. Preston, confident that the order of the candidates doesn’t matter for the outcome, tells Keanu he can choose the voting order of the candidates.

So what order should Keanu choose?

How would Keanu fare in one-to-one races against each candidate?

Riddle No. 20: Who Opened All These Damn Lockers?

There are 100 lockers that line the main hallway of Chelm High School. Every night, the school principal makes sure all the lockers are closed so that there will be an orderly start to the next day. One day, 100 mischievous students decide that they will play a prank.

The students all meet before school starts and line up. The first student then walks down the hallway, and opens every locker. The next student follows by closing every other locker (starting at the second locker). Student 3 then goes to every third locker (starting with the third) and opens it if it’s closed, and closes it if it’s open. Student 4 follows by opening every fourth locker if it’s closed and closing it if it’s open. This goes on and on until Student 100 finally goes to the hundredth locker. When the principal arrives later in the morning, which lockers does she find open?

Make sure you pay attention to all of the factors.

Laura Feiveson is an economist for the government, a storyteller, and a lifelong enthusiast of math puzzles.  She lives in Washington, DC with her husband and two daughters. 

.css-cuqpxl:before{padding-right:0.3125rem;content:'//';display:inline;} Pop Mech Pro .css-xtujxj:before{padding-left:0.3125rem;content:'//';display:inline;}

Ukraine Is Using an Ancient Weapon on Russia

Inside the CIA’s Quest to Steal a Soviet Sub

Jumpstart Your Car With a Cordless Tool Battery

Lost Villa of Rome’s Augustus Potentially Found

Could Freezing Your Brain Help You Live Forever?

Air Force’s Combat Drone Saga Has Taken a Turn

China’s Building a Stealth Bomber. U.S. Says ‘Meh’

A Supersonic Bomber's Mission Ended in Flames

Who Wants to Buy the A-10 Warthog?

US Army Accepts Delivery of First M10 Assault Gun

Underwater UFO is a Threat, Says Ex-Navy Officer

Please ensure that your password is at least 8 characters and contains each of the following:

• a special character: @$#!%*?&

Game Central

Get step-by-step explanations

Graph your math problems

Practice, practice, practice

Get math help in your language

• Film and TV

To make sure you never miss out on your favourite NEW stories , we're happy to send you some reminders

Click ' OK ' then ' Allow ' to enable notifications

Two high school students solve math puzzle thought to be impossible for 2,000 years

The girls cracked an ancient equation.

Britt Jones

If you thought you were smart, you’re not as smart as these two teenagers.

There has been buzz online and in the math community over an astounding breakthrough from an unlikely source.

I always thought high school math was the hardest it would get and couldn’t fathom anything more difficult than calculus, but these two girls knocked my embarrassing grades out of the park.

Calcea Johnson and Ne’Kiya Jackson are both whiz kids, as they have proven the Pythagoras’ theorem by using trigonometry .

'What's so special about that?' you may ask. Well, it’s literally stumped academics for thousands of years.

The students of St Mary’s Academy, New Orleans, are now being encouraged by a prominent US math research organization to submit their work to a peer-reviewed journal. It’s just that good.

Johnson and Jackson attended the American Mathematical Society south-eastern chapter’s semi-annual meeting in Georgia, where they revealed their findings to the group.

This meeting was attended by math researchers from many US universities such as Alabama, Georgia, Louisiana State, Ohio State and Oklahoma and the girls were the only two high schoolers in attendance.

I’d be embarrassed if I was a top scholar and was outsmarted by kids, honestly.

During their speech, Johnson and Jackson spoke about how they discovered new proof for the Pythagorean theorem.

But what is it?

Let’s dumb it down:

The theorem is 2,000 years old and established the sum of the squares of a right triangle’s two shorter sides equals the exact same of the square of the hypotenuse.

That’s confusing…

It has always been argued that any proof based on trigonometry would need to the circular, but the schoolgirls denied this as truth.

Instead, Johnson and Jackson said: "In our lecture we present a new proof of Pythagoras’s Theorem which is based on a fundamental result in trigonometry—the Law of Sines—and we show that the proof is independent of the Pythagorean trig identity \sin^2x + \cos^2x = 1."

Johnson and Jackson have been speaking to local media in recent days, including local TV news station WWL, where Johnson said it was an 'unparalleled feeling' to present her work alongside Jackson with people that are a lot more experienced than them standing alongside.

She said: "There’s nothing like it – being able to do something that people don’t think that young people can do.

"You don’t see kids like us doing this – it’s usually, like, you have to be an adult to do this."

Jackson went on to shout out her school teachers for challenging them to complete something that has previously been deemed impossible.

Topics:  Education , US News , Good News

Choose your content:

Cybertruck owner tests 'dangerous' feature after new update and gets 'freaked out' by results

The tiktoker tested a software update introduced to address concerns about the trunk on the cybertruck.

Man admits to killing his wife because he can’t afford her medical bills

Ronnie wiggs has been charged after admitting to attacking his wife in hospital.

Explorer shares his ‘scariest travel experience’ after spending 24 hours in world’s highest town

The town sits in the andes mountains and nearly 17,000ft above sea level.

Panera is getting rid of its Charged Lemonade after lawsuits claim two people died after drinking it

Panera bread has found its 'charged sips' drinks mentioned in at least three lawsuits, two of which are alleged wrongful death lawsuits.

• High school seniors sneak into principal's house and spend the night there to prank her
• People baffled at video of high school students from 1989
• High school teacher suspended after Satan costume causes controversy
• High school graduate speaks out after she’s declined diploma for dancing during ceremony

Watch CBS News

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

More from CBS News

Heineken vows nearly $50 million to transform "tired pubs" in U.K.

Should you use a reverse mortgage to cover long-term care costs?

Afghan diplomat accused of smuggling gold into India resigns

How technology helped a nonspeaking autistic woman find her voice

New Orleans teens solve 2,000-year-old math problem

W hen she started a math contest with a bonus question challenging students to create a new proof for the Pythagorean theorem using trigonometry, teacher Michelle Blouin Williams didn’t expect anyone to complete the task.

“I was just looking for some ingenuity,” she said, per CBS News .

Calcea Johnson and Ne’Kiya Jackson, however, blew Williams’ expectations out of the water by figuring it out in 2023. The teens were seniors at St. Mary’s Academy in New Orleans, a prestigious Catholic school for girls which has maintained a 100% acceptance rate to colleges and 100% graduation rate for 17 years, CBS News reported.

They appeared in an episode of CBS News’ “60 Minutes ” on Sunday to talk about their achievement.

How did the teens find the answer?

While they were motivated initially by the math competition’s $500 prize, an internal drive to finish what they started manifested when they reached the tricky bonus question. For two months, the high school seniors worked tirelessly to finish their proof.

CeCe Johnson, Calcea’s mother, told “60 Minutes,” “It was pages and pages and pages of, like, over 20 or 30 pages for this one problem.”

Her father, Cal Johnson, added, “Yeah, the garbage can was full of papers, which she would, you know, work out the problems and — if that didn’t work she would ball it up, throw it in the trash.”

When they finished, teachers at St. Mary’s recognized the importance of their work and submitted their proof to the American Mathematical Society for recognition at a conference in March 2023, where the students presented their work.

What is the Pythagorean theorem and what’s a proof?

In essence, the mathematical theorem states that knowing the lengths of two sides of a right triangle enables you to figure out the length of the third using this formula: a² + b² = c².

It’s associated with Greek mathematician Pythagoras, but evidence suggests it was known earlier, in Babylon and Iron Age India, per Britannica . Its practical uses include construction and architecture, two-dimensional navigation, and surveying.

A mathematical proof is exactly what it sounds like: reasoning that proves a mathematical theorem is true. American mathematician Daniel Kane explains proofs as being like essays, but using math.

Why is Calcea Johnson and Ne’Kiya Jackson’s work significant?

According to the “ 60 Minutes ” episode, “there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible.”

In 1927, mathematician Elisha Loomis said as much in his book, “ The Pythagorean Proposition .” Loomis argued that there could be no trigonometric proof of the theorem because it would be circular.

Stuart Anderson, a professor emeritus of mathematics at Texas A&M University–Commerce, told Scientific American , “A lot of the basic trig ‘identities’ are nothing more than Pythagoras’ theorem.”

So, because trigonometric functions are based upon the Pythagorean theorem, using them reflexively to prove the theorem would be akin to going in circles and a fundamental mathematical error, Loomis argued.

According to Scientific American , the teens refuted this in their presentation in 2023 and said that “a trigonometric identity called the law of sines didn’t depend on the Pythagorean theorem and that they could use it to prove the theorem.”

Calcea and Ne’Kiya have joined an extremely small group who’ve accomplished the same feat, including mathematician Jason Zimba, who successfully created a new proof in 2009. The two submitted their proof for final peer review this spring and continue to work on creating more proofs.

How did the world respond to their accomplishment?

The teens were given the keys to the city of New Orleans and a commendation from the governor of Louisiana, among other public recognitions.

While their achievement “blew up,” as Ne’Kiya described it, the two students remain humble, and laughed at being called geniuses.

When news of their accomplishment broke, some people seemed to be shocked and dismissed the news as fake, St. Mary’s president Pamela Rogers said in the interview .

“They were saying, ‘Oh, they could not have done it. African Americans don’t have the brains to do it.’ ... People — have a vision of who can be successful. And — to some people, it is not always an African American female. And to us, it’s always an African American female.”

When interviewer Bill Whitaker asked why they thought there’d been such a response, Ne’Kiya said, “Probably because we’re African American, one. And we’re also women. So I think — oh, and our age. Of course our ages probably played a big part.”

“I’d like to actually be celebrated for what it is. Like, it’s a great mathematical achievement,” she continued.