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20 Best Math Puzzles to Engage and Challenge Your Students

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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

Math crossword puzzles

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!



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

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 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

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

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.

maths is fun problem solving

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!

Assorted Math Puzzles and Quizzes

  • Make a Letter T ... Letter T Solved
  • Make a Square ... Make a Square Solved
  • Apple Puzzle  
  • Pencils and Jars ... Pencils and Jars Solution
  • Test your addition, subtraction, multiplication and division  
  • Tower of Hanoi ... Tower of Hanoi Solver
  • Make a Letter F  ... Letter F Solved
  • Make a Letter H  ... Letter H Solved
  • Average Arms Question  ... Average Arms Solution
  • Who Owns The Fish?
  • Is This True?  
  • Letters To Numbers  
  • Marquis De L'Hopital  
  • Flipping Discs  
  • Pentomino Challenge
  • Logic Puzzle - Who owns the crocodile  
  • Connect 4  
  • Guess my number  
  • Lazy Gardener Puzzle ... Lazy Gardener Solution
  • Chicken Crossing ... Chicken Crossing Solution
  • Census Puzzle ... Census Solution
  • Bank Puzzle ... Bank Solution
  • Cellar Puzzle   ... Cellar Solution
  • Pool Balls Puzzle ... Pool Balls Solution
  • Calculator Puzzles

Not all puzzles have a solution here. But if you need help you can ask at the Math is Fun Forum

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[FREE] Themed Math Problems: Winter Term (Jan – Mar)

Grade 4 and 5 themed math problems and answers that link to key dates in the Winter Term!

25 Fun Math Problems For Elementary And Middle School (From Easy To Very Hard!)

Fun math problems and brainteasers are loved by mathematicians; they provide an opportunity to apply mathematical knowledge, logic, and problem-solving skills all at once.

In this article, we’ve compiled 25 fun math problems and brainteasers covering various topics and question types. They’re aimed at students in upper elementary (3rd-5th grade) and middle school (6th grade, 7th grade, and 8th grade). We’ve categorized them as:

Math word problems

Math puzzles, fraction problems, multiplication and division problems, geometry problems, problem-solving questions, math puzzles are everywhere, how should teachers use these math problems.

Teachers could make use of these math problem solving questions in a number of ways. They can:

  • incorporate the questions into a relevant math lesson.
  • set tasks at the beginning of lessons.
  • break up or extend a math worksheet.
  • keep students thinking mathematically after the main lesson has finished.

Some are based on real life or historical math problems, and some include ‘bonus’ math questions to help extend the problem-solving fun! As you read through these problems, think about how you could adjust them to be relevant to your students and their grade level or to practice different math skills. 

These math problems can also be used as introductory puzzles for math games such as those introduced at the following links:

  • Math games for grade 4
  • Middle school math games

25 Math Problems

Want the fun and challenging questions from this blog wrapped up in a downloadable question and answer format? Then you're in luck!

1. Home on time – easy 

Type: elapsed time, number, addition.

A movie theater screening starts at 2:35 pm. The movie lasts for 2 hours, 32 minutes after 23 minutes of previews. It took 20 minutes to get to the movie theater. What time should you tell your family that you’ll be home?

Answer: 6:10 pm

2. A nugget of truth – mixed

Type: Multiplication Facts, Multiplication, Multiples, Factors, Problem-Solving 

Chicken nuggets come in boxes of 6, 9 or 20, so you can’t order 7 chicken nuggets. How many other impossible quantities can you find (not including fractions or decimals)?

Answer: 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, or 43

There is actually a theorem which can be used to prove that every integer quantity greater than 43 can be ordered.

3. A pet problem – mixed

Type: Number, Problem Solving, Forming and Solving Equations, Simultaneous Equations, Algebra

Eight of my pets aren’t dogs, five aren’t rabbits, and seven aren’t cats. How many pets do I have?

Answer: 10 pets (5 rabbits, 3 cats, 2 dogs)

Looking for more word problems, solutions and explanations? Read our article on word problems for elementary school.

4. the price of things – mixed.

Type: lateral thinking problem

A mouse costs $10, a bee costs $15, and a spider costs $20. Based on this, how much does a duck cost?

Answer: $5 ($2.50 per leg)

5. A dicey math challenge – easy

Type: Place value, number, addition, problem-solving

Roll three 6 sided dice to generate three place value digits. What’s the biggest number you can make out of these digits? What’s the smallest number you can make?

Add these two numbers together. What do you get?

Answer: If the digits are the same, the maximum is 666 and the minimum is 111. Then, if you add the numbers together, 666 + 111 = 777. If the digits are different, the maximum is 654 and the minimum is 456. Then, if you add the numbers together, 654 + 456 = 1,110.

Bonus: Who got a different result? Why?

6. PIN problem solving – mixed

Type: Logic, problem solving, reasoning

I’ve forgotten my PIN. Six incorrect attempts locks my account: I’ve used five! Two digits are displayed after each unsuccessful attempt: “2, 0” means 2 digits from that guess are in the PIN, but 0 are in the right place. No two digits in my PIN are the same.

What should my sixth attempt be?

pin math problem

Answer: 6347

7. So many birds – mixed

Type: Triangular Numbers, Sequences, Number, Problem Solving

On the first day of Christmas my true love gave me one gift. On the second day they gave me another pair of gifts plus a copy of what they gave me on day one. On day 3, they gave me three new gifts, plus another copy of everything they’d already given me. If they keep this up, how many gifts will I have after twelve days?

Answer: 364

Bonus: This could be calculated as 1 + (1 + 2) + (1 + 2 + 3) + … but is there an easier way? What percentage of my gifts do I receive on each day?

8. I 8 sum math questions – mixed

Type: Number, Place Value, Addition, Problem Solving, Reasoning

Using only addition and the digit 8, can you make 1,000? You can put 8s together to make 88, for example.

Answer: 888 + 88 + 8 + 8 + 8 = 1,000 Bonus: Which other digits allow you to get 1,000 in this way?

4 friends entered a math quiz. One answered \frac {1}{5} of the math questions, one answered \frac {1}{10} , one answered \frac{1}{4} , and the other answered \frac{4}{25} . What percentage of the questions did they answer altogether?

Answer: 71%

10. Ancient problem solving – easy

Type: Fractions, Reasoning, Problem Solving

Ancient Egyptians only used unit fractions (like \frac {1}{2} , \frac{1}{3} or \frac{1}{4} . For \frac {2}{3} they’d write \frac{1}{3} + \frac{1}{3} . How might they write \frac{5}{8} ?


\frac {1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} is correct. So is \frac {1}{2} + \frac{1}{8} (They are both still unit fractions even though they have different denominators.)

Bonus: Which solution is better? Why? Can you find any more? What if subtractions are allowed?

Learn more about unit fractions here .

11. everybody wants a pizza the action – hard.

An infinite number of mathematicians buy pizza. The first wants \frac{1}{2} . The second wants \frac{1}{4} pizza. The third & fourth want \frac{1}{8} and \frac{1}{16} each, and so on. How many pizzas should they order?

Answer: 1 Each successive mathematician wants a slice that is exactly half of what is left:

fractions math problem

12. Shade it black – hard

Type: Fractions, Reasoning, Problem Solving What fraction of this image is shaded black?

another math problem on fractions


Look at the L-shaped part made up of two white and one black squares: 

\frac{1}{3} of this part is shaded. Zoom in on the top-right quarter of the image, which looks exactly the same as the whole image, and use the same reasoning to find what fraction of its L-shaped portion is shaded. Imagine zooming in to do the same thing again and again…

13. Giving is receiving – easy

Type: Number, Reasoning, Problem Solving

5 people give each other a present. How many presents are given altogether?

14. Sharing is caring – mixed

I have 20 candies. If I share them equally with my friends, there are 2 left over. If one more person joins us, there are 6 candies left. How many friends am I with?

Answer: 6 people altogether (so 5 friends!)

15. Multiplication facts secrets – mixed

Type: Area, 2D Shape, Rectangles

Here are 77 letters:


How many different rectangular grids could you arrange all 77 letters into?

Can you reveal the secret message?

Answer: Four: 1 × 77, 77 × 1, 11 × 7 and 7 × 11. If the letters are arranged into one of these, a message appears, reading down each column starting from the top left.

Another math problem on multiplication

Bonus: Can you find any more integers with the same number of factors as 77? What do you notice about these factors (think about prime numbers)? Can you use this system to hide your own messages?

16. Laugh it up – hard

Type: Multiples, Least Common Multiple, Multiplication Facts, Division, Time

One friend jumps every \frac{1}{3} of a minute. Another jumps every 31 seconds. When will they jump together? Answer: After 620 seconds

US lesson slide

17. Pictures of matchstick triangles – easy

Type: 2D Shapes, Equilateral Triangles, Problem Solving, Reasoning

Look at the matchsticks arranged below. How many equilateral triangles are there?

triangles math problem

Answer: 13 (9 small, 3 medium, 1 large)

Bonus: What if the biggest triangle only had two matchsticks on each side? What if it had four?

18. Dissecting squares – mixed

Type: Reasoning, Problem Solving

What’s the smallest number of straight lines you could draw on this grid such that each square has a line going through it?


19. Make it right – mixed

Type: Pythagorean theorem

This triangle does not agree with Pythagorean theorem. 

Adding, subtracting, multiplying or dividing each of the side lengths by the same whole number can fix it. What is the number?

pythagorean theorem math problem

Answer: 3 

The new side lengths are 3, 4 and 5 and  32 + 42 = 52.

20. A most regular math question – hard

Type: Polygons, 2D Shapes, tessellation, reasoning, problem-solving, patterns

What is the regular polygon with the largest number of sides that will self-tessellate?

Answer: Hexagon.

Regular polygons tessellate if one interior angle is a factor of 360 ° . The interior angle of a hexagon is 120°. This is the largest factor less than 180°.

21. Pleased to meet you – easy

Type: Number Problem, Reasoning, Problem Solving

5 people meet; each shakes everyone else’s hand once. How many handshakes take place?

Person A shakes 4 people’s hands. Person B has already shaken Person A’s hand, so only needs to shake 3 more, and so on.

Bonus: How many handshakes would there be if you did this with your class?

22. All relative – easy

Type: Number, Reasoning, Problem-Solving

When I was twelve my brother was half my age. I’m 40 now, so how old is he?

23. It’s about time – mixed

Type: Time, Reasoning, Problem-Solving

When is “8 + 10 = 6” true?

Answer: When you’re telling the time (8am + 10 hours = 6pm)

24. More than a match – mixed

Type: Reasoning, Problem-Solving, Roman Numerals, Numerical Notation

Here are three matches:

matches math problem

How can you add two more matches, but get eight? Answer: Put the extra two matches in a V shape to make 8 in Roman Numerals:

roman numerals math problem

25. Leonhard’s graph – hard

Type: Reasoning, Problem-Solving, Logic

Leonhard’s town has seven bridges as shown below. Can you find a route around the town that crosses every bridge exactly once?


Answer: No!

This is a classic real life historical math problem solved by mathematician Leonhard Euler (rhymes with “boiler”). The city was Konigsberg in Prussia (now Kaliningrad, Russia). Not being able to find a solution is different from proving that there aren’t any! Euler managed to do this in 1736, practically inventing graph theory in the process.

Many of these 25 math problems are rooted in real life, from everyday occurrences to historical events. Others are just questions that might arise if you say “what if…?”. The point is that although there are many lists of such problem-solving math questions that you can make use of, with a little bit of experience and inspiration you could create your own on almost any topic – and so could your students. 

For a kick-starter on creating your own math problems, read our article on middle school math problem solving .

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

The content in this article was originally written by primary school teacher Tom Briggs and has since been revised and adapted for US schools by math curriculum specialist and former elementary math teacher Katie Keeton.

12 Math Club Games & Activities [FREE]

Make math even more enjoyable for your students with this easy-to-use collection of games and activities to try during your math club.

From multiplication skills to reasoning, the minimal resources needed for each activity means you won’t spend half the time setting up and packing away!

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Teaching and Learning

Elevating math education through problem-based learning, by lisa matthews     feb 14, 2024.

Elevating Math Education Through Problem-Based Learning

Image Credit: rudall30 / Shutterstock

Imagine you are a mountaineer. Nothing excites you more than testing your skill, strength and resilience against some of the most extreme environments on the planet, and now you've decided to take on the greatest challenge of all: Everest, the tallest mountain in the world. You’ll be training for at least a year, slowly building up your endurance. Climbing Everest involves hiking for many hours per day, every day, for several weeks. How do you prepare for that?

The answer, as in many situations, lies in math. Climbers maximize their training by measuring their heart rate. When they train, they aim for a heart rate between 60 and 80 percent of their maximum. More than that, and they risk burning out. A heart rate below 60 percent means the training is too easy — they’ve got to push themselves harder. By combining this strategy with other types of training, overall fitness will increase over time, and eventually, climbers will be ready, in theory, for Everest.

maths is fun problem solving

Knowledge Through Experience

The influence of constructivist theories has been instrumental in shaping PBL, from Jean Piaget's theory of cognitive development, which argues that knowledge is constructed through experiences and interactions , to Leslie P. Steffe’s work on the importance of students constructing their own mathematical understanding rather than passively receiving information .

You don't become a skilled mountain climber by just reading or watching others climb. You become proficient by hitting the mountains, climbing, facing challenges and getting right back up when you stumble. And that's how people learn math.

maths is fun problem solving

The Traditional Approach

Problem-based learning has a rich history in American education, with John Dewey laying the theoretical groundwork in 1916 and McMaster University pioneering the PBL program for medical education in 1969. More recently, the National Council of Teachers of Mathematics published Principles and Standards for School Mathematics in 2000, setting forth a vision that emphasized problem-solving, reasoning and communication, aligning closely with the principles of PBL. It encouraged teachers to design learning experiences that engaged students in meaningful mathematical thinking and problem-solving.

Yet, PBL as a pedagogy has seen varied levels of adoption in math classrooms across the United States, with many math teachers falling back on traditional teachings of formulas and procedures . In 1998, critical research that examined data from the Third International Mathematics and Science Study (TIMSS) revealed teaching methods varied significantly across cultures , resulting in a “ teaching gap .” According to video documentation in the study, U.S. math teaching predominantly focused on procedural skills, where students spent most of their time acquiring isolated skills through repetition. In contrast, Japanese teaching was measurably more focused on deeper understanding, where students engaged more with solving challenging problems collaboratively.

Not “Math People”

The procedural approach can be effective for students who thrive in structured environments and excel at memorizing formulas and algorithms. However, for many students, this approach leads to a sense of disengagement and a belief that they are not "math people," a negative self-perception connected with math anxiety. Math anxiety can increase cognitive load, which is the mental effort required to process information. When students are anxious about math, their working memory resources may be diverted to managing the anxiety rather than focusing on solving mathematical problems , resulting in decreased performance in procedural tasks .

So what makes PBL different? The key to making it work is introducing the right level of problem. Remember Vygotsky’s Zone of Proximal Development? It is essentially the space where learning and development occur most effectively – where the task is not so easy that it is boring but not so hard that it is discouraging. As with a mountaineer in training, that zone where the level of challenge is just right is where engagement really happens.

I’ve seen PBL build the confidence of students who thought they weren’t math people. It makes them feel capable and that their insights are valuable. They develop the most creative strategies; kids have said things that just blow my mind. All of a sudden, they are math people.

maths is fun problem solving

Skills and Understanding

Despite the challenges, the trend toward PBL in math education has been growing , driven by evidence of its benefits in developing critical thinking, problem-solving skills and a deeper understanding of mathematical concepts, as well as building more positive math identities. The incorporation of PBL aligns well with the contemporary broader shift toward more student-centered, interactive and meaningful learning experiences. It has become an increasingly important component of effective math education, equipping students with the skills and understanding necessary for success in the 21st century.

At the heart of Imagine IM lies a commitment to providing students with opportunities for deep, active mathematics practice through problem-based learning. Imagine IM builds upon the problem-based pedagogy and instructional design of the renowned Illustrative Mathematics curriculum, adding a number of exclusive videos, digital interactives, design-enhanced print and hands-on tools.

The value of imagine im's enhancements is evident in the beautifully produced inspire math videos, from which the mountaineer scenario stems. inspire math videos showcase the math for each imagine im unit in a relevant and often unexpected real-world context to help spark curiosity. the videos use contexts from all around the world to make cross-curricular connections and increase engagement..

This article was sponsored by Imagine Learning and produced by the Solutions Studio team.

Imagine Learning

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  • Problem-solving skills
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  • Develops child’s reasoning skills and makes them a logical thinker
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Let's Roam Explorer

30 Math Riddles (With Answers)

maths is fun problem solving

Math may not be everyone’s cup of tea, but when you think about it, it’s everywhere. Most of us likely do some type of math just about every day. So how can we make learning math and improving our problem-solving abilities more enjoyable? The answer is easy—riddles! And these math riddles are a great place to start.

These riddles span various age levels and abilities. Try some out yourself or gather the kids and make it a family or school activity. Math is fundamental, so there’s no time like the present to fine-tune those skills!

Calling All Math Masters!

If you enjoy math riddles, you’re bound to love the Math Masters category of our in-home scavenger hunts ! All you’ll need is your family, your internet-enabled device, and your competitive spirit. Our intuitive interface will have you connected, solving clues, and completing challenges in no time.

Developed in conjunction with 2 elementary school teachers, this hunt makes problem-solving a blast! When you’re done with that category, there are several more from which you can choose. The one thing they all have in common is FUN!

1. Riddle: A merchant can place 8 large boxes or 10 small boxes into each carton for shipping. In one shipment, he sent a total of 96 boxes. If there are more large boxes than small boxes, how many cartons did he ship? Answer: 11 cartons total: 7 cartons of large boxes (7 * 8 = 56 boxes), 4 cartons of small boxes (4 * 10 = 40 boxes).

2. Riddle: I have a calculator that can display ten digits. How many different ten-digit numbers can I type using just the 0-9 keys once each, and moving from one keypress to the next using the knight’s move in chess? (In chess, the knight move in an L-shape—one square up and two across, two squares down and one across, two squares up and one across, and other like combinations.) Answer: You can form the numbers 5034927618 and 5038167294. You can also form their reverses: 8167294305 and 4927618305. Hence four different numbers can be made.

3. Riddle: When my dad was 31, I was just 8 years old. Now his age is twice as old as my age. What is my present age? Answer: When you calculate the difference between the ages, you can see that it is 23 years. So you must be 23 years old now.

4. Riddle: An insurance salesman walks up to a house and knocks on the door. A woman answers, and he asks her how many children she has and how old they are. She says I will give you a hint. If you multiply the 3 children’s ages, you get 36. He says this is not enough information. So she gives a him 2nd hint. If you add up the children’s ages, the sum is the number on the house next door. He goes next door and looks at the house number (13) and says this is still not enough information. So she says she’ll give him one last hint which is that her oldest of the 3 plays piano. Why would he need to go back to get the last hint after seeing the number on the house next door? Answer: After the first hint, there are several different options. When he realizes the sum is 13, the answer is one of two possibilities. The third hint indicates there is an “oldest” daughter. Since one of the two options includes older twins (1 + 6 + 6 = 13) and the other includes young twins (2 + 2 + 9 = 13), the daughters must be 2, 2, and 9.

5. Riddle: Scott has $28.75. He purchased three cookies that cost $1.50 each, five newspapers that each cost $0.50, five flowers for $1.25 each, and used the remainder of the cash on a pair of sunglasses. How much were the sunglasses? Answer: $15.50.

6. Riddle: What is the smallest whole number that is equal to seven times the sum of its digits? Answer: 21. The two-digit number ab stands for 10a + b since the first digit represents 10s and the second represents units. If 10a + b = 7(a + b), then 10a + b = 7a + 7b, and so 3a = 6b, or, more simply, a = 2b. That is, the second digit must be twice the first. The smallest possible number is 21.

7. Riddle: A monkey is trying to climb a coconut tree. He takes 3 steps forward and slips back 2 steps downward. Each forward step is 30 cm and each backward step is 40 cm. How many steps are required to climb a 100 cm tree? Answer: 50 steps.

8. Riddle: In a mythical land 1/2 of 5 = 3. If the same proportion holds, what is the value of 1/3 of 10? Answer: 4.

9. Riddle: It is 9 am now. Rita studies for 2 hours, takes a bath for 1 hour, and then has lunch for 1 hour. How many hours are left before 9 am tomorrow? Answer: 20 hours.

10. Riddle: There is an empty basket that is one foot in diameter. What is the total number of eggs that you can put in this empty basket? Answer: Only one egg! Once you put an egg into the basket, it’s no longer an empty basket!

Need a break from all that math? Our list of funny riddles is giggle-worthy!

11. Riddle: You know 2 + 2 gives you the same total as 2 x 2. Find a set of three different whole numbers whose sum is equal to their total when multiplied. Answer: The three different whole numbers whose sum is equal to their total when multiplied are 1, 2, and 3.

12. Riddle: If you multiply this number by any other number, the answer will always be the same. What is the number? Answer: Zero.

13. Riddle: A grandfather, two fathers, and two sons went to a movie theater together and everyone bought one movie ticket each. Movie tickets cost $7.00 and their total was $21.00. How? Answer: They purchased 3 tickets. The grandfather is also a father and the father is also a son.

14. Riddle: A small number of cards has been lost from a complete deck. If I deal cards to four people, three cards remain. If I deal to three people, two remain. If I deal to five people, two cards remain. How many cards are there? Answer: There are 47 cards. A complete deck is 52 cards. 5 cards have been lost.

15. Riddle: Tom was on the way to the park. He met a man with seven wives and each of them came with seven sacks. All these sacks contain seven cats and each of these seven cats had seven kittens. So in total, how many were going to the park? Answer: 1. Only Tom was going to the park.

16. Riddle: I am an odd number; take away a letter and I become even. What number am I? Answer: Seven. (SEVEN-S=EVEN.)

17. Riddle: How can you add eight 8’s to get the number 1,000? Answer: 888 + 88 + 8 + 8 + 8 = 1,000.

18. Riddle: If there are four apples and you take away three, how many do you have? Answer: Three apples.

19. Riddle: I am one with a couple of friends. Quarter a dozen, and you’ll find me again. What number am I? Answer: 3.

20. Riddle: Tom weighs half as much as Peter and Jerry weigh three times the weight of Tom. Their total weight is 720 pounds. Can you figure out the individual weights of each man? Answer: Tom weighs 120, Peter weighs 240, and Jerry weighs 360.

Find these math riddles too hard? How about some easy riddles ?

21. Riddle: If it is two hours later, then it will take half as much time till it’s midnight as it would be if it were an hour later. What time is it? Answer: 9:00 PM.

22. Riddle: In an alien land far away, half of 10 is 6. If the same proportion holds true, then what is 1/6th of 30 in this alien land? Answer: 6.

23. Riddle: In two years, Tom will be twice as old as he was five years ago. How old is Tom? Answer: 12.

24. Riddle: A little boy goes shopping and purchases 12 tomatoes. On the way home, all but 9 get mushed and ruined. How many tomatoes are left in good condition? Answer: 9 tomatoes

25. Riddle: A carton contains apples that were divided into two equal parts and sold to two traders Tarun and Tanmay. Tarun had two fruit shops and decided to sell an equal number of apples in both shops, A and B respectively. A mother visited shop A and bought all the apples in the shop for her kids. But one apple was left after dividing all the apples among her children. If each child got one apple, what is the minimum number of apples in the carton? Answer: 12. That is the minimum possible number of apples for each child to have at least one.

26. Riddle: A duck was given $9, a spider was given $36, and a bee was given $27. Based on this information, how much money would be given to a cat? Answer: $18 ($4.50 per leg).

27. Riddle: The speed of a train is 3 meters/second and it takes 10 seconds to cross a lamp post. What is the length of the train? Answer: 30 meters. The time taken by the train to pass the stationary object is equal to the length of the train divided by the speed of the train.

28. Riddle: It takes 12 men 12 hours to construct a wall. Then how long will it take for 6 men to complete the same wall? Answer: It would probably take 24 hours, but there is no need to make it again. The job is already done!

29. Riddle: What are four consecutive prime numbers that, when added, equal 220? Answer: 220= 47+ 53+ 59+ 61

30. Riddle: An athlete can jump FOREVER. However, every time she jumps, she goes half as far as her prior jump. On her very first jump, she goes half a foot. On her second jump, she goes a quarter of a foot, and so. How long will it take her to get a foot away from her starting point? Answer: She will NEVER travel a full foot because the distance keeps being reduced by half.

Say it with us—math is cool! Which of our math questions got you thinking? Share your thoughts in the comments section.

And don’t forget to try a Let’s Roam Scavenger Hunt or any of our other activities that will let you roam from home . When you’re ready to get out and about, we have city art walks and scavenger hunts all over the world. Try one today!

Frequently Asked Questions

Use puzzles, riddles, and games based on mathematics. This list of math riddles should help! Or consider a math-themed scavenger hunt —you’ll have a little math master on your hands!

Math riddles make math more fun and, when learning is fun, it becomes easier. Before you know it, you (or your children) will be thinking like a true mathematician!

888 + 88 + 8 + 8 + 8 = 1,000. Find this and 30 math riddles at LetsRoam.com, along with other family-friendly activities , including a math-themed scavenger hunt .

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Maths and Logic Puzzles

I absolutely adore a good maths puzzle or a logic puzzle. This page is a collection of some of my favourites that I have complied over the years. Some you will know, but hopefully some you will not. Many come from the outstanding Numberplay Blog from the New York Times.

The puzzles are not arranged by topic (as that might give away some of the secrets to solving them) or by age or difficulty, because that depends very much on the puzzle solver themselves.

Please do not email me for any of the answers, as  am afraid I do not have time to reply. Anyway, that would spoil the fun. But I guess for relatively large sums of money, I may be tempted to divulge some information. And if you have a favorite puzzle that you think would fit well into this collection, please let me know via Twitter

1. The Circle of Death keyboard_arrow_up Back to Top

There are one hundred people standing in a circle. They count off beginning at one and ending at one hundred. Since they are in a circle, ONE is next to TWO and ONE HUNDRED. ONE has a sword and kills TWO. He passes the sword to THREE who kills FOUR. And so forth. NINETY-NINE kills ONE HUNDRED and passes the sword to ONE. Then ONE kills THREE and passes the sword to FIVE. This goes on until only one person is left standing. Which number is he?

2. The 4 Coins Problem keyboard_arrow_up Back to Top

You’re creating a new coin system for your country. You must use only four coin values and you must be able to create the values 1 through 10 using one coin at a minimum and two coins maximum. What 4 coins do you choose, and can you think of a second set of 4 coins that achieves the same goal?

3. The Wedding Ring keyboard_arrow_up Back to Top

In certain parts of rural Russia, a would-be bride would gather six long pieces of straw or grass and grasp them in her hand. She then would randomly tie pairs of knots on the top and the bottom. Since there are six blades of grass sticking out above and below the hand, she will tie three knots on the top and three knots on the bottom. The story goes, that if she formed one big ring, she would get married soon. What is the probability that she will get married?

4. The Averages Problem keyboard_arrow_up Back to Top

Consider a list of all the ways you could take four distinct digits from 1 to 9 and arrange them to make the sum of two 2-digit numbers. Some numbers might appear several times: 134, for example, is 93 + 41, and also 91 + 43. What are the mean, median and mode of all the numbers on this list? (Mean is the sum of all the numbers divided by the number of numbers, median is the middle value in a sorted list of the numbers, and the mode is the number that appears most often.)

5. Avoiding the Troll keyboard_arrow_up Back to Top

Every day you make a trip from A to B, 2 miles away. But your trip is unpredictable. On 50% of the days, you can walk between A and B without any obstruction. The other 50% of the time, a troll appears at the halfway point, blocking your trip. The troll also activates an invisible barrier that blocks 1 mile perpendicularly in both directions, forcing you to walk around it to complete your trip. You cannot see the troll or the barrier until you are halfway from A to B, so you cannot plan in advance whether you need to walk around the barrier. You do know the troll appears randomly 50% of the time. If you can walk in straight lines to any points between A and B, what is your best strategy so you walk the least distance on average? What is that distance?

6. Ali Baba's Cave keyboard_arrow_up Back to Top

Ali Baba found a cave full of gold and diamonds. A bag full of gold weighs 200 kilograms, a bag full of diamonds weighs 40 kilograms. Ali Baba can carry only 100 kilograms at a time. A kilogram of gold costs $20, and a kilogram of diamonds costs $60. What is the greatest amount of money Ali Baba can earn for the gold and diamonds he can carry out at once (in one attempt in one bag)?

7. Drinking Game Puzzle keyboard_arrow_up Back to Top

The game starts with 6 empty glasses in a row numbered 1 to 6. You roll a standard die. If the number for the glass is empty, then the glass is filled up. If the number for the glass is full, then you drink that glass so it becomes empty. There is a special rule when 5 glasses are full. If you roll the number for the lone empty glass, then the final glass gets filled and you have to drink all 6 glasses. At this point the game ends. From the start of the game, what is the average number of rolls until the game ends?

8. The Raja's Pearls keyboard_arrow_up Back to Top

A rajah, in his will, left his daughters a certain number of pearls with the instructions to divide them as follows: The oldest daughter was to receive 1 pearl plus 1/7 of what was left. The next eldest daughter was to receive 2 pearls plus 1/7 of those left. The next eldest daughter was to receive 3 pearls plus 1/7 of what was left, and so on in the same manner. The youngest daughter received what was left after all the other divisions. At first, the youngest daughter thought the distribution was unfair, but after careful calculations she concluded that all daughters would receive the same number of pearls. How many pearls and how many daughters were there in all?

9. The St Petersburg Lottery keyboard_arrow_up Back to Top

Suppose we play the following game. I toss a coin repeatedly until it comes up heads. If heads appears on the first throw, I pay you $2. If it appears on the second throw, I give you $4; if on the third, I pay $8 and so on, doubling each time. How much would you be willing to pay me to play this game?

10. Cheryl's Birthday keyboard_arrow_up Back to Top

Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates. May 15 May 16 May 19 June 17 June 18 July 14. July 16 August 14. August 15. August 17 Cheryl then tells Bernard and Albert separately the month and the day of her birthday respectively. Albert: I don't know when Cheryl's birthday is, but I know that Bernard does not know too. Bernard: At first I don't know when Cheryl's birthday is, but I know now. Albert: Then I also know when Cheryl's birthday is. When is Cheryl's birthday?

11. Ages of Children keyboard_arrow_up Back to Top

2 mathematicians meet on the street and one says to the other, "I bet you can't guess the ages of my 3 children" (all integer ages). The 2nd mathematician says, "I'll take your bet but only if you give me enough information to solve it." The 1st agrees, and says, "The product of my 3 children's age is 36." "That is not enough information." "Right you are!" he looks around, spies an address on a building and says, "The sum of my children's age is the same as the address over there." Thinking a little bit; "That still isn't enough information." "Right you are again!  My eldest son has red hair." After some thinking the 2nd mathematician won the bet.

12. Band around the earth keyboard_arrow_up Back to Top

Imagine there is a steel band around the equator of the Earth, nice and snug (~25,000 mi). If you then add one foot of steel to this band, how much is the band raised off the ground? High enough to slip a hair under it? Your hand? High enough to walk under it?

13. The School Lockers keyboard_arrow_up Back to Top

There is a school with 1,000 students and 1,000 lockers. On the first day of term the headteacher asks the first student to go along and open every single locker, he asks the second to go to every second locker and close it, the third to go to every third locker and close it if it is open or open it if it is closed, the fourth to go to the fourth locker and so on. The process is completed with the thousandth student. How many lockers are open at the end?

14. The Devil's Wager keyboard_arrow_up Back to Top

You die and the devil says he’ll let you go to heaven if you beat him in a game. He sits you down at a round table. He divides a large pile of quarters in two so you both have the same amount of quarters in your own pile. He says, “O.K., we’ll take turns putting quarters down, no overlapping allowed, and the quarters must rest on the table surface. The first guy who can’t put a quarter down loses. You cannot shift or try to squeeze any quarter into a space that moves another quarter.” The devil says he wants to go first. You realize if the devil goes first, he probably has a strategy to win. You haven’t had time to think this through yet, so you ask for some time to consider your options. He grants you 15 minutes. At that end of that time you know how to beat him … but you also know you must go first for your strategy to win. You convince him to let you go first, saying you really didn’t have enough time to consider a strategy so you should at least go first. What is your winning strategy?

15. Half an Egg keyboard_arrow_up Back to Top

A peasant woman came to a market to sell some eggs. A first buyer took half her eggs plus 1/2 an egg. The same happened with the remaining eggs: a second buyer buyer took half her eggs plus 1/2 an egg. A third only bought what was left over: 1 egg. How many eggs were there initially?

16. Make 6 keyboard_arrow_up Back to Top

You can only use two maths symbols (including factorials and square roots) to make 6 0     0     0     =     6 1     1     1     =     6 2     2     2     =     6 3     3     3     =     6 4     4     4     =     6 5     5     5     =     6 6     6     6     =     6 7     7     7     =     6 8     8     8     =     6 9     9     9     =     6

17. Three Liars Tossing a Coin keyboard_arrow_up Back to Top

Three people each have a tendency to lie 1/3 of the time. There is a coin flip that they all see. They all say it's Heads. What's the probability it is actually Heads?

18. Free the Prisoners keyboard_arrow_up Back to Top

The warden meets with 23 new prisoners when they arrive. He tells them, “You may meet today and plan a strategy. But after today, you will be in isolated cells and will have no communication with one another.

“In the prison is a switch room, which contains two light switches labeled 1 and 2, each of which can be in either up or the down position. I am not telling you their present positions. The switches are not connected to anything.

“After today, from time to time whenever I feel so inclined, I will select one prisoner at random and escort him to the switch room. This prisoner will select one of the two switches and reverse its position. He must flip one switch when he visits the switch room, and may only flip one of the switches. Then he’ll be led back to his cell.

“No one else will be allowed to alter the switches until I lead the next prisoner into the switch room. I’m going to choose prisoners at random. I may choose the same guy three times in a row, or I may jump around and come back. I will not touch the switches, if I wanted you dead you would already be dead.

“Given enough time, everyone will eventually visit the switch room the same number of times as everyone else. At any time, anyone may declare to me, ‘We have all visited the switch room.’ “If it is true, then you will all be set free. If it is false, and somebody has not yet visited the switch room, you will all die horribly. You will be carefully monitored, and any attempt to break any of these rules will result in instant death to all of you”

What is the strategy they come up with so that they can be free?

19. Burning Ropes to Measure Time keyboard_arrow_up Back to Top

Warm-up:  You are given a box of matches and a piece of rope.  The rope burns at the rate of one rope per hour, but it may not burn uniformly.  For example, if you light the rope at one end, it will take exactly 60 minutes before the entire rope has burnt up, but it may be that the first 1/10 of the rope takes 50 minutes to burn and that the remaining 9/10 of the rope takes only 10 minutes to burn.  How can you measure a period of exactly 30 minutes?  You can choose the starting time.  More precisely, given the matches and the rope, you are to say the words “start” and “done” exactly 30 minutes apart.

The actual problem:  Given a box of matches and two such ropes, not necessarily identical, measure a period of 15 minutes.

20. Minimum Number of Aircraft keyboard_arrow_up Back to Top

On Bagshot Island, there is an airport. The airport is the homebase of an unlimited number of identical airplanes. Each airplane has a fuel capacity to allow it to fly exactly 1/2 way around the world, along a great circle. The planes have the ability to refuel in flight without loss of speed or spillage of fuel. Though the fuel is unlimited, the island is the only source of fuel. What is the fewest number of aircraft necessary to get one plane all the way around the world assuming that all of the aircraft must return safely to the airport? How did you get to your answer?

Notes: (a) Each airplane must depart and return to the same airport, and that is the only airport they can land and refuel on ground. (b) Each airplane must have enough fuel to return to airport. (c) The time and fuel consumption of refueling can be ignored. (so we can also assume that one airplane can refuel more than one airplanes in air at the same time.) (d) The amount of fuel airplanes carrying can be zero as long as the other airplane is refueling these airplanes. What is the fewest number of airplanes and number of tanks of fuel needed to accomplish this work? (we only need airplane to go around the world)

21. Minimum Number of Weights keyboard_arrow_up Back to Top

A wholesale merchant came to me one day and posed this problem. Every day in his business he has to weigh amounts from one pound to one hundred and twenty-one pounds, to the nearest pound. To do this, what is the minimum number of weights he needs and how heavy should each weight be?

22. Relatively Prime Bet keyboard_arrow_up Back to Top

There is a box in which distinct numbered balls have been kept. You have to pick two balls randomly from the lot.

If someone is offering you a 2 to 1 odds that the numbers will be relatively prime, for example If the balls you picked had the numbers 6 and 13, you lose $1. If the balls you picked had the numbers 5 and 25, you win $2. Will you accept that bet?

23. Crossing the Bridge keyboard_arrow_up Back to Top

Four people need to cross a rickety bridge at night. Unfortunately, they have only one torch and the bridge is too dangerous to cross without one. The bridge is only strong enough to support two people at a time. Not all people take the same time to cross the bridge. Times for each person: 1 min, 2 mins, 7 mins and 10 mins. What is the shortest time needed for all four of them to cross the bridge?

24. 25 Horses, 5 Tracks keyboard_arrow_up Back to Top

Mr John has 25 horses, and he wants to pick the fastest 3 horses out of those 25. He has only 5 tracks, which means only 5 horses can run at a time. He doesn't even have a stop watch . What is the minimum number of races required to find the 3 fastest horses?

25. Flipping Coins Puzzle keyboard_arrow_up Back to Top

There are are twenty coins sitting on the table, ten are currently heads and tens are currently tails. You are sitting at the table with a blindfold and gloves on. You are able to feel where the coins are, but are unable to see or feel if they heads or tails. You must create two sets of coins. Each set must have the same number of heads and tails as the other group. You can only move or flip the coins, you are unable to determine their current state.

How do you create two even groups of coins with the same number of heads and tails in each group?

26. Random Airplane Seats keyboard_arrow_up Back to Top

People are waiting in line to board a 100-seat airplane. Steve is the first person in the line. He gets on the plane but suddenly can’t remember what his seat number is, so he picks a seat at random. After that, each person who gets on the plane sits in their assigned seat if it’s available, otherwise they will choose an open seat at random to sit in.

The flight is full and you are last in line. What is the probability that you get to sit in your assigned seat?

27. Proportion of Girls and Boys keyboard_arrow_up Back to Top

In a country where everyone wants a boy, each family continues having babies till they have a boy. After some time, what is the proportion of boys to girls in the country? (Assuming probability of having a boy or a girl is the same)

28. The Hats and the Monster keyboard_arrow_up Back to Top

There is an island with 10 inhabitants. One day a monster comes and says that he intends to eat every one of them but will give them a chance to survive in the following way: In the morning, the monster will line up all the people - single file so that the last person sees the remaining 9, the next person sees the remaining 8, and so on until the first person that obviously sees no one in front of himself. The monster will then place black or white hats on their heads randomly (they can be all white, all black or any combination thereof). The monster will offer each person starting with the last one (who sees everyone else's hats) to guess the color of his/her own hat. The answer can only be one word: "white" or "black". The monster will eat all ten hostages as soon as a second captive answers incorrectly. All the remaining people will hear the guess but not the outcome of the guess. The monster will then go on to the next to last person (who only sees 8 people), and so on until the end. The monster gives them the whole night to think. The Task: Devise a strategy that guarentees all captive's survival. Assumptions: 1) All the 10 people can easily understand your strategy, and will execute it with perfect precision. 2) If the monster suspects that any of the people are giving away information to any of the remaining team members by intonation of words when answering, or any other signs, or by touch, he will eat everyone. 3) The only allowed response is a short, unemotional "white" or "black".

29. Eggs from a Building keyboard_arrow_up Back to Top

Suppose that there is a building with 100 floors. You are given 2 identical eggs. The most interesting property of the eggs is that every egg has it’s own “threshold” floor. Let’s call that floor N. What this means is that the egg will not break when dropped from any floor below floor N, but the egg will definitely break from any floor above floor N, including floor N itself. For example, if the property of the eggs is that N equals 15, those eggs will always break on any floor higher than or equal to the 15th floor, but those eggs will never break on any floor below floor 15. The same holds true for the other egg since they are identical. These are very strong eggs, because they can be dropped multiple times without breaking as long as they are dropped from floors below their “threshold” floor, floor N. But once an egg is dropped from a floor above it’s threshold floor Here is the puzzle: What strategy should be taken in order to minimize the number of egg drops used to find floor N (the threshold floor) for the egg? Also, what is the minimum number of drops for the worst case using this strategy? Remember that you are given 2 identical eggs which both have the same exact threshold floor.

30. 5 Pirates, 100 Gold Coins keyboard_arrow_up Back to Top

Five ship’s pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all extremely intelligent, treacherous and selfish (especially the captain).

The captain always proposes a distribution of the loot. All pirates vote on the proposal, and if half the crew or more go “Aye”, the loot is divided as proposed, as no pirate would be willing to take on the captain without superior force on their side.

If the captain fails to obtain support of at least half his crew (which includes himself), he faces a mutiny, and all pirates will turn against him and make him walk the plank. The pirates start over again with the next senior pirate as captain.

What is the maximum number of coins the captain can keep without risking his life?

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Parents are stumped over first grader’s math question - can you solve it?

‘the question sets them up to fail,’ says facebook user, article bookmarked.

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Math curriculums have been changing in the school system over the last few years, with many parents claiming that “new” math is too difficult.

A mother named Tiesha Sanders recently took to Facebook to share her first-grader’s homework problem that she had no idea how to solve. “The new Math is NOT IT!” her post’s caption began, next to a photo of the problem and a note that she wrote to the teacher .

“Disclaimer: I am not upset with the teacher, she’s just teaching what she’s supposed to. And #2, don’t come here like we’re the dumb ones, I taught elementary for the last six years, this question ain’t it! Also, this is 1st grade math,” the caption read.

The problem required the student to split the number 27 into tens and ones and then just into ones. Her daughter, Summer, wrote that there were two tens and seven ones, so she had assumed when she was asked for the ones again that it was still seven. After getting the question marked wrong, Sanders left a note for the teacher asking for the right answer to help her child next time.

“Hello!” the note began. “I just wanted to ask how Summer got #3 wrong? Her father and I were going over her mistakes and wanted to be sure we were on the right track.”

Her daughter’s teacher responded to the note, writing: “Hello! This is the new math they have us teaching.”

“It is 27 ones. It wants her to know that having two tens and seven ones is the same as 27 ones. If you have any other questions you can call or text me.”

After posting, many people took to the comments where they agreed with Sanders that the question was confusing.

“If they wanted her to decompose the number and show the ways in which the number can be logically made... why NOT = signs? The arrows make sense to who?” one comment read.

Another commenter agreed, writing: “I wonder why they wouldn’t word it differently? Like maybe what does the tens place + the ones place = that threw me for a loop because it’s really unclear.”

Others called out the flawed structure of the question, saying: “But if they have the box that labels ‘tens’ and ‘ones’ then only ask for the ‘ones’, how in the entire world is this math, mathing?”

Someone agreed, adding: “The question sets them up to fail.”

This isn’t the first time a parent has publicly voiced their confusion over an elementary school homework question. Back in December, one mother in Buckinghamshire, England, had become so confused helping her six-year-old with a worksheet that she posted the question in a private Facebook group . 

“At first I thought I was losing my mind. I was like, ‘What am I missing here?’” So I posted in a group with loads of moms hoping they would have the answer,” Laura Rathbone said in an interview with Today .

Rathbone’s daughter, Lilly-Mo, was asked on the worksheet to pick the odd item out based on the five items she was given. The items listed were: friend, toothbrush, desk, silver, and egg.

“So… my six-year-old daughter who’s in year one got this homework question,” her Facebook post read. “It’s confusing in my opinion, to say the least, especially considering the age it’s aimed at… but I’d love to hear your answers!”

She added: “I think it’s something you’d find in a Puzzler magazine personally but let me know your thoughts.”

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February 5, 2024

How String Theory Solved Math’s Monstrous Moonshine Problem

A concept from theoretical physics helped confirm the strange connection between two completely different areas of mathematics

By Manon Bischoff

Night sky with full moon and old tree. On dark background

VladGans/Getty Images

After the star-studded mystery thriller The Number 23 debuted in cinemas in 2007, many people became convinced that they were seeing the eponymous number everywhere. I was in school at that time, and some of my classmates would shudder whenever the number 23 appeared in any context. Other people became fascinated by this form of numerology because as soon as you pay more attention to a certain thing—including a number— you get the feeling that you see it too often to be purely coincidence.

For a long time, people assumed that the late mathematician John McKay might have fallen victim to this same phenomenon, known as the “frequency illusion,” or the Baader-Meinhof phenomenon . In McKay’s case, the number that captured his imagination was 196,884.

It doesn’t seem too surprising that a two-digit number such as 23 might come up repeatedly. But would a six-digit figure do so? McKay came across this number by chance in 1978 when he was looking through a paper in a mathematical field that was not his specialty. He was working in geometry and was studying the symmetry of figures. That day, however, he was looking at results from number theory , which deals with the properties of integers such as prime numbers . He came across a sequence of numbers that started with the value 196,884.

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This figure sounded familiar to McKay. He had previously worked on a mathematical structure—still hypothetical at the time— known as the monster . This strange algebraic structure was intended to describe the symmetries of a geometric object that lives in 196,883 dimensions (only one fewer than the number 196,884). And because a one-dimensional point fulfills every symmetry anyway, the monster can also describe its symmetrical properties. So McKay found the number 196,884 again in an extraordinary way. He added the first two dimensions in which mathematicians believed the monster’s symmetry applied: 196,883 + 1 = 196,884.

Does that sound far-fetched? Others thought so, too. Experts paid little attention to McKay’s result. After all, a structure such as the monster contains a number of numbers, as does the consequence from number theory that McKay had associated with it. “If you have a whole lot of numbers, then a few of them are going to be roughly the same as each other just by coincidence,” said mathematician Richard Borcherds, who has made major contributions to the field, in an explanatory YouTube video .

But McKay couldn’t shake the feeling that the two extremely different mathematical fields of geometry and number theory could be connected. He even reportedly wore T-shirts with the inscription “196,883 + 1 = 196,884” at conferences.

Complete Madness or a Stroke of Genius?

A short time later, mathematician John Thompson realized that there might be something to McKay’s suspicions after all. He succeeded in linking the next higher dimension, in which an object follows the symmetries of the monster, with the next member of the mysterious sequence of numbers from number theory. The dimension is 21,296,876. The values differ—but if you add up all the monster dimensions as before (1 + 196,883 + 21,296,876), the result is 21,493,760.

That was surprising because, as you may recall, when McKay first spotted 196,884, he was looking at a special sequence in number theory. The second number in that sequence is 21,493,760—Thompson’s result. In other words, it began to seem that there really could be a link between two seemingly unrelated areas of mathematics.

At this point the math community began to get curious. Maybe McKay was right after all—even if that sounded totally absurd. What could this strange structure, which described symmetries of unimaginable objects and had not even been fully constructed, have to do with number theory?

By 1979 evidence was mounting that other numbers and dimensions seemed to follow this unexpected pattern. Mathematicians John Conway and Simon Norton finally published a paper entitled “Monstrous Moonshine,” in which they set out the conjecture of a connection between geometry and number theory. “They called it moonshine because it appeared so far-fetched,” said number theorist Don Zagier of the Max Planck Institute for Mathematics in Bonn, Germany, to Quanta Magazine in 2015.

And indeed, there was likely very little hope of ever proving this moonshine conjecture. Quite apart from the fact that there was no indication that the two distant mathematical areas were connected, it was not even completely clear whether the monster really existed.

The Monster in the Moonlight

The monster was a theoretical prediction of group theory, an area of geometry that deals with the symmetrical properties of objects. In the 1970s mathematicians began to create a kind of periodic table of groups: they wanted to find the “atoms” of finite symmetries. According to this way of thinking, every finite group can be represented by a combination of these atoms. After decades of research, the geometers finally seemed to have reached their goal. Unlike the chemical elements, there are an infinite number of “finite simple groups,” but almost all can be divided into 18 categories, the arrangement of which is reminiscent of the periodic table. In addition, the experts came across a total of 26 outsiders that do not fit into these 18 classes.

Triangles labeled to show varied symmetries.

The first of these outliers was the “monster,” which mathematicians Bernd Fischer and Robert Griess predicted in 1973. The name comes from the sheer size of this group: it contains more than 8 x 10 53 symmetries. For comparison, the symmetry group of a 20-sided “D20” die ( an icosahedron ) contains 60 symmetries, meaning 60 possible transformations (rotations or reflections) can be carried out without changing the orientation of the D20.

Colorful circles connected by lines represent groups of symmetries.

Because of its sheer size, the monster presented mathematicians with massive challenges. “Most people thought it was going to be hopeless to construct it since much, much, much smaller groups required computer constructions at that time,” explained Borcherds in his YouTube video. Meanwhile even powerful computers struggle with a structure consisting of 8 x 10 53 elements.

Yet this pessimistic forecast ultimately proved wrong. In 1980 Griess constructed the monster and thus proved its existence —without the help of computers.

A Sine Function on Steroids

Number theory is mostly about integers, which seems quite simple at first glance. But to investigate the relationships between them, experts resort to complicated concepts, such as so-called modular forms. These are functions f ( z ) that are extremely symmetrical. As with the sine function, you only need to know a specific section of a modular form to know what it looks like everywhere else.

“Modular forms are something like trigonometric functions, but on steroids,” mathematician Ken Ono told Quanta Magazine .

Colorful arcs represent moduli space

Nevertheless, they play an extremely important role in mathematics. Andrew Wiles of the University of Oxford used them, for example, to prove Fermat’s theorem, and Maryna Viazovska of the Swiss Federal Institute of Technology in Lausanne used them to find the densest sphere-packing arrangement in eight spatial dimensions . Because modular forms are so complicated, however, they are often approximated by an infinitely long polynomial, such as:

f (q ) = ( 1 ⁄ q ) + 744 + 19,688 q + 21,493,760 q 2 + 864,299,970 q 3 + …

The prefactors in front of the variable q form a number sequence with interesting properties from a number-theoretical perspective. McKay associated this sequence of numbers with the monster.

A Surprising Link

Borcherds first heard about the moonshine conjecture in the 1980s. “I was just completely blown away by this,” he recalled in an interview with YouTuber Curt Jaimungal . Borcherds was sitting in one of Conway’s lectures at the time and learned that number theory and group theory could be mysteriously connected. The subject never let go of him. He began to search for the suspected connection until he found it. In 1992 he published his groundbreaking result , for which he received a Fields Medal, one of the highest awards in mathematics, six years later. His conclusion: a highly speculative area of physics, string theory, could provide the missing piece of the puzzle between the monster and the sequence of numbers.

String theory attempts to unite the four fundamental forces of physics (electromagnetism, strong and weak nuclear forces and gravity). Instead of relying on particles or waves to make up the basic building blocks of the universe, as in conventional theories, string theory involves one-dimensional structures: tiny threads vibrate like the strings of an instrument and thus generate the familiar particles and interactions that we perceive in the universe.

Borcherds knew that string theory was based on many mathematical principles related to symmetries. As it turns out, moduli also play a role. When the tiny threads are closed and move through spacetime in a wobbly manner, their track forms a two-dimensional tube. This structure has the same symmetry as modular shapes—regardless of how the thread oscillates.

The type of string theory that Borcherds investigated can only be mathematically formulated in 25 spatial dimensions. Because our world consists of only three visible spatial dimensions, however, string theorists assume that the remaining 22 dimensions are rolled up into tiny spheres or doughnut-shaped tori. But the physics depends on their exact shape: a string theory in which the dimensions are rolled up as cylinders provides different predictions than one in which they form a sphere. In order to describe the particles and their interactions in a way that fits our world, physicists have to find the right “compactification” in their calculations.

Borcherds rolled up 24 dimensions into a 24-dimensional doughnut surface and discovered that the associated string theory had the symmetry of the monster. The fact that only one free spatial dimension remained did not bother him. After all, he was interested in the mathematical properties of the model and not in a physical theory that describes our world.

In this constructed world, the threads swing along the 24-dimensional doughnut. The dimensions of the monster count all the ways in which a thread can vibrate at a certain energy. So at the lowest energy, it only vibrates in one way; at the next highest energy, there are already 196,883 different possibilities. And the trace that the thread leaves behind has the symmetry of a modular shape.

Borcherds had thus proven the connection between the monster group and a modular form. And it was not to remain the only such case: in the meantime, mathematicians have been able to connect other finite groups with other modular forms —and there, too, string theory provides the link. So even if it turns out that the speculative theory is not suitable for describing our universe, it can still help us discover completely new mathematical worlds.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

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MISSION UPDATES | February 12, 2024

Sols 4096-4097: fun math and a new butte.

This image was taken by Left Navigation Camera onboard NASA's Mars rover Curiosity on Sol 4094.

This image was taken by Left Navigation Camera onboard NASA's Mars rover Curiosity on Sol 4094. Credits: NASA/JPL-Caltech. Download image ›

Earth planning date: Monday, February 12, 2024

One side effect of enjoying a very long mission is reaching fun number milestones. Usually that is "Sol 3000" or some other nice round number, but today we are planning the mathematically fun sol 2^12, also known as sol 4096.

Today I was on shift as a MAHLI uplink lead, so I was excited to think about contact science targets. Unfortunately, the team determined that Curiosity’s wheels are perched on some small rocks. When this happens, we worry about the rover shifting around if the weight of the rover causes the rocks to move a little. Even though the rover would likely only shift a tiny amount, when we have the arm ~2 cm away from the surface to take close approach MAHLI images, a tiny amount could have major consequences. Safety is always first on Mars, so we decided to focus on remote science today.

Fortunately, there are quite a few interesting targets in the area. The team identified a possible small, eroded crater not too far from Curiosity. Mastcam will target one side of the crater rim in the "Fischer Pass" mosaic, and ChemCam will perform a LIBS observation on a block (also called Fischer Pass) within the potential crater rim that has alternating dark toned rough layers and light toned smooth layers. Mastcam will target the other side of the potential crater rim in the "Hitchcock Lake" mosaic to document apparent distortions in the bedrock.

Mastcam will also take a mosaic of the base of Fascination Turret, which is a part of the Gediz Valley Ridge that we have a spectacular view of right now. ChemCam will take a Long Distance RMI mosaic of the Texoli butte.

Curiosity has been driving through an area surrounded by many buttes (for example, the Texoli butte). In today’s location a new butte came into view (see the image at the top of this blog post)! This butte, called "Wilkerson," is located across the Gediz Valley Ridge, so it has been obscured until now. We finally drove to a high enough location to see over the ridge, so Mastcam will capture the butte in a mosaic. Orbital images show that Wilkerson butte may have a mantling layer of unique rocks on top, so it will be interesting to see what it looks like from the ground.

Finally, Curiosity will continue to drive along Gediz Valley Ridge, and I’m sure we will have a great view at our next stop as well.


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