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How to Solve Math Problems

Last Updated: May 16, 2023 Fact Checked

This article was co-authored by Daron Cam . Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 587,586 times.

Although math problems may be solved in different ways, there is a general method of visualizing, approaching and solving math problems that may help you to solve even the most difficult problem. Using these strategies can also help you to improve your math skills overall. Keep reading to learn about some of these math problem solving strategies.

Understanding the Problem

• Draw a Venn diagram. A Venn diagram shows the relationships among the numbers in your problem. Venn diagrams can be especially helpful with word problems.
• Draw a graph or chart.
• Arrange the components of the problem on a line.
• Draw simple shapes to represent more complex features of the problem.

Developing a Plan

Solving the Problem

Expert Q&A

• Seek help from your teacher or a math tutor if you get stuck or if you have tried multiple strategies without success. Your teacher or a math tutor may be able to easily identify what is wrong and help you to understand how to correct it. Thanks Helpful 1 Not Helpful 1
• Keep practicing sums and diagrams. Go through the concept your class notes regularly. Write down your understanding of the methods and utilize it. Thanks Helpful 2 Not Helpful 0

You Might Also Like

• ↑ Daron Cam. Math Tutor. Expert Interview. 29 May 2020.
• ↑ http://www.interventioncentral.org/academic-interventions/math/math-problem-solving-combining-cognitive-metacognitive-strategies
• ↑ http://tutorial.math.lamar.edu/Extras/StudyMath/ProblemSolving.aspx
• ↑ https://math.berkeley.edu/~gmelvin/polya.pdf

About This Article

To solve a math problem, try rewriting the problem in your own words so it's easier to solve. You can also make a drawing of the problem to help you figure out what it's asking you to do. If you're still completely stuck, try solving a different problem that's similar but easier and then use the same steps to solve the harder problem. Even if you can't figure out how to solve it, try to make an educated guess instead of leaving the question blank. To learn how to come up with a solid plan to use to help you solve a math problem, scroll down! Did this summary help you? Yes No

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Solving Equations

What is an equation.

An equation says that two things are equal. It will have an equals sign "=" like this:

That equations says:

what is on the left (x − 2)  equals  what is on the right (4)

So an equation is like a statement " this equals that "

What is a Solution?

A Solution is a value we can put in place of a variable (such as x ) that makes the equation true .

Example: x − 2 = 4

When we put 6 in place of x we get:

which is true

So x = 6 is a solution.

How about other values for x ?

• For x=5 we get "5−2=4" which is not true , so x=5 is not a solution .
• For x=9 we get "9−2=4" which is not true , so x=9 is not a solution .

In this case x = 6 is the only solution.

You might like to practice solving some animated equations .

More Than One Solution

There can be more than one solution.

Example: (x−3)(x−2) = 0

When x is 3 we get:

(3−3)(3−2) = 0 × 1 = 0

And when x is 2 we get:

(2−3)(2−2) = (−1) × 0 = 0

which is also true

So the solutions are:

x = 3 , or x = 2

When we gather all solutions together it is called a Solution Set

The above solution set is: {2, 3}

Solutions Everywhere!

Some equations are true for all allowed values and are then called Identities

Example: sin(−θ) = −sin(θ) is one of the Trigonometric Identities

Let's try θ = 30°:

sin(−30°) = −0.5 and

−sin(30°) = −0.5

So it is true for θ = 30°

Let's try θ = 90°:

sin(−90°) = −1 and

−sin(90°) = −1

So it is also true for θ = 90°

Is it true for all values of θ ? Try some values for yourself!

How to Solve an Equation

There is no "one perfect way" to solve all equations.

A Useful Goal

But we often get success when our goal is to end up with:

x = something

In other words, we want to move everything except "x" (or whatever name the variable has) over to the right hand side.

Example: Solve 3x−6 = 9

Now we have x = something ,

and a short calculation reveals that x = 5

Like a Puzzle

In fact, solving an equation is just like solving a puzzle. And like puzzles, there are things we can (and cannot) do.

Here are some things we can do:

• Add or Subtract the same value from both sides
• Clear out any fractions by Multiplying every term by the bottom parts
• Divide every term by the same nonzero value
• Combine Like Terms
• Expanding (the opposite of factoring) may also help
• Recognizing a pattern, such as the difference of squares
• Sometimes we can apply a function to both sides (e.g. square both sides)

Example: Solve √(x/2) = 3

And the more "tricks" and techniques you learn the better you will get.

Special Equations

There are special ways of solving some types of equations. Learn how to ...

• solve Quadratic Equations
• solve Radical Equations
• solve Equations with Sine, Cosine and Tangent

Check Your Solutions

You should always check that your "solution" really is a solution.

How To Check

Take the solution(s) and put them in the original equation to see if they really work.

Example: solve for x:

2x x − 3 + 3 = 6 x − 3     (x≠3)

We have said x≠3 to avoid a division by zero.

Let's multiply through by (x − 3) :

2x + 3(x−3) = 6

Bring the 6 to the left:

2x + 3(x−3) − 6 = 0

Expand and solve:

2x + 3x − 9 − 6 = 0

5x − 15 = 0

5(x − 3) = 0

Which can be solved by having x=3

Let us check x=3 using the original question:

2 × 3 3 − 3 + 3  =   6 3 − 3

Hang On: 3 − 3 = 0 That means dividing by Zero!

And anyway, we said at the top that x≠3 , so ...

x = 3 does not actually work, and so:

There is No Solution!

That was interesting ... we thought we had found a solution, but when we looked back at the question we found it wasn't allowed!

This gives us a moral lesson:

"Solving" only gives us possible solutions, they need to be checked!

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How to Solve Math Problems Faster: 15 Techniques to Show Students

Written by Marcus Guido

• Teaching Strategies

“Test time. No calculators.”

You’ll intimidate many students by saying this, but teaching techniques to solve math problems with ease and speed can make it less daunting.

This can also  make math more rewarding . Instead of relying on calculators, students learn strategies that can improve their concentration and estimation skills while building number sense. And, while there are educators who  oppose math “tricks”  for valid reasons, proponents point to benefits such as increased confidence to handle difficult problems.

Here are 15 techniques to show students,  helping them solve math problems faster:

Addition and Subtraction

1. two-step addition.

Many students struggle when learning to add integers of three digits or higher together, but changing the process’s steps can make it easier.

The first step is to  add what’s easy.  The second step is to  add the rest.

Let’s say students must find the sum of 393 and 89. They should quickly see that adding 7 onto 393 will equal 400 — an easier number to work with. To balance the equation, they can then subtract 7 from 89.

Broken down, the process is:

• (393 + 7) + (89 – 7)

With this fast technique, big numbers won’t look as scary now.

2. Two-Step Subtraction

There’s a similar method for subtraction.

Remove what’s easy. Then remove what’s left.

Suppose students must find the difference of 567 and 153. Most will feel that 500 is a simpler number than 567. So, they just have to take away 67 from the minuend — 567 — and the subtrahend — 153 — before solving the equation.

Here’s the process:

• (567 – 67) – (153 – 67)

Instead of two complex numbers, students will only have to tackle one.

3. Subtracting from 1,000

You can  give students confidence  to handle four-digit integers with this fast technique.

To subtract a number from 1,000, subtract that number’s first two digits from 9. Then, subtract the final digit from 10.

Let’s say students must solve 1,000 – 438.  Here are the steps:

This also applies to 10,000, 100,000 and other integers that follow this pattern.

Multiplication and Division

4. doubling and halving.

When students have to multiply two integers, they can speed up the process when one is an even number. They just need to  halve the even number and double the other number.

Students can stop the process when they can no longer halve the even integer, or when the equation becomes manageable.

Using 33 x 48 as an example,  here’s the process:

The only prerequisite is understanding the 2 times table.

5. Multiplying by Powers of 2

This tactic is a speedy variation of doubling and halving.

It simplifies multiplication if a number in the equation is a power of 2, meaning it works for 2, 4, 8, 16 and so on.

Here’s what to do:  For each power of 2 that makes up that number, double the other number.

For example, 9 x 16 is the same thing as 9 x (2 x 2 x 2 x 2) or 9 x 24. Students can therefore double 9 four times to reach the answer:

Unlike doubling and halving, this technique demands an understanding of exponents along with a strong command of the 2 times table.

6. Multiplying by 9

For most students, multiplying by 9 — or 99, 999 and any number that follows this pattern — is difficult compared with multiplying by a power of 10.

But there’s an easy tactic to solve this issue, and  it has two parts.

First, students round up the 9  to 10. Second, after solving the new equation, they subtract the number they just multiplied by 10 from the answer.

For example, 67 x 9 will lead to the same answer as 67 x 10 – 67. Following the order of operations will give a result of 603. Similarly, 67 x 99 is the same as 67 x 100 – 67.

Despite more steps, altering the equation this way is usually faster.

7. Multiplying by 11

There’s an easier way for multiplying two-digit integers by 11.

Let’s say students must find the product of 11 x 34.

The idea is to put a space between the digits, making it 3_4. Then, add the two digits together and put the sum in the space.

The answer is 374.

What happens if the sum is two digits? Students would put the second digit in the space and add 1 to the digit to the left of the space.  For example:

It’s multiplication without having to multiply.

8. Multiplying Even Numbers by 5

This technique only requires basic division skills.

There are two steps,  and 5 x 6 serves as an example. First, divide the number being multiplied by 5 — which is 6 — in half. Second, add 0 to the right of number.

The result is 30, which is the correct answer.

It’s an ideal, easy technique for students mastering the 5 times table.

9. Multiplying Odd Numbers by 5

This is another time-saving tactic that works well when teaching students the 5 times table.

This one has three steps,  which 5 x 7 exemplifies.

First, subtract 1 from the number being multiplied by 5, making it an even number. Second, cut that number in half — from 6 to 3 in this instance. Third, add 5 to the right of the number.

The answer is 35.

Who needs a calculator?

10. Squaring a Two-Digit Number that Ends with 1

Squaring a high two-digit number can be tedious, but there’s a shortcut if 1 is the second digit.

There are four steps to this shortcut,  which 812 exemplifies:

• Subtract 1 from the integer: 81 – 1 = 80
• Square the integer, which is now an easier number: 80 x 80 = 6,400
• Add the integer with the resulting square twice: 6,400 + 80 + 80 = 6,560
• Add 1: 6,560 + 1 = 6,561

This work-around eliminates the difficulty surrounding the second digit, allowing students to work with multiples of 10.

11. Squaring a Two-Digit Numbers that Ends with 5

Squaring numbers ending in 5 is easier, as there are  only two parts of the process.

First, students will always make 25 the product’s last digits.

Second, to determine the product’s first digits, students must multiply the number’s first digit — 9, for example — by the integer that’s one higher — 10, in this case.

So, students would solve 952 by designating 25 as the last two digits. They would then multiply 9 x 10 to receive 90. Putting these numbers together, the  result is 9,025.

Just like that, a hard problem becomes easy multiplication for many students.

12. Calculating Percentages

Cross-multiplication is an  important skill  to develop, but there’s an easier way to calculate percentages.

For example, if students want to know what 65% of 175 is, they can multiply the numbers together and move the decimal place two digits to the left.

The result is 113.75, which is indeed the correct answer.

This shortcut is a useful timesaver on tests and quizzes.

13. Balancing Averages

To determine the average among a set of numbers, students can balance them instead of using a complex formula.

Suppose a student wants to volunteer for an average of 10 hours a week over a period of four weeks. In the first three weeks, the student worked for 10, 12 and 14 hours.

To determine the number of hours required in the fourth week, the student must  add how much he or she surpassed or missed the target average  in the other weeks:

• 14 hours – 10 hours = 4 hours
• 12 – 10 = 2
• 10 – 10 = 0
• 4 hours + 2 hours + 0 hours = 6 hours

To learn the number of hours for the final week, the student must  subtract the sum from the target average:

• 10 hours – 6 hours = 4 hours

With practice, this method may not even require pencil and paper. That’s how easy it is. 

Word Problems

14. identifying buzzwords.

Students who struggle to translate  word problems  into equations will benefit from learning how to spot buzzwords — phrases that indicate specific actions.

This isn’t a trick. It’s a tactic.

Teach students to look for these buzzwords,  and what skill they align with in most contexts:

Be sure to include buzzwords that typically appear in their textbooks (or other classroom  math books ), as well as ones you use on tests and assignments.

As a result, they should have an  easier time processing word problems .

15. Creating Sub-Questions

For complex word problems, show students how to dissect the question by answering three specific sub-questions.

Each student should ask him or herself:

• What am I looking for?  — Students should read the question over and over, looking for buzzwords and identifying important details.
• What information do I need?  — Students should determine which facts, figures and variables they need to solve the question. For example, if they determine the question is rooted in subtraction, they need the minuend and subtrahend.
• What information do I have?  — Students should be able to create the core equation using the information in the word problem, after determining which details are important.

These sub-questions help students avoid overload.

Instead of writing and analyzing each detail of the question, they’ll be able to identify key information. If you identify students who are struggling with these, you can use  peer learning  as needed.  

For more fresh approaches to teaching math in your classroom, consider treating your students to a range of  fun math activities .

Final Thoughts About these Ways to Solve Math Problems Faster

Showing these 15 techniques to students can give them the  confidence to tackle tough questions .

They’re also  mental math  exercises, helping them build skills related to focus, logic and critical thinking.

A rewarding class equals an  engaging class . That’s an easy equation to remember.

> Create or log into your teacher account on Prodigy  — a free, adaptive math game that adjusts content to accommodate player trouble spots and learning speeds. Aligned to US and Canadian curricula, it’s loved by more than 500,000 teachers and 15 million students.

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Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b)

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Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left?

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March 12, 2024

The Simplest Math Problem Could Be Unsolvable

The Collatz conjecture has plagued mathematicians for decades—so much so that professors warn their students away from it

By Manon Bischoff

Mathematicians have been hoping for a flash of insight to solve the Collatz conjecture.

James Brey/Getty Images

At first glance, the problem seems ridiculously simple. And yet experts have been searching for a solution in vain for decades. According to mathematician Jeffrey Lagarias, number theorist Shizuo Kakutani told him that during the cold war, “for about a month everybody at Yale [University] worked on it, with no result. A similar phenomenon happened when I mentioned it at the University of Chicago. A joke was made that this problem was part of a conspiracy to slow down mathematical research in the U.S.”

The Collatz conjecture—the vexing puzzle Kakutani described—is one of those supposedly simple problems that people tend to get lost in. For this reason, experienced professors often warn their ambitious students not to get bogged down in it and lose sight of their actual research.

The conjecture itself can be formulated so simply that even primary school students understand it. Take a natural number. If it is odd, multiply it by 3 and add 1; if it is even, divide it by 2. Proceed in the same way with the result x : if x is odd, you calculate 3 x + 1; otherwise calculate x / 2. Repeat these instructions as many times as possible, and, according to the conjecture, you will always end up with the number 1.

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For example: If you start with 5, you have to calculate 5 x 3 + 1, which results in 16. Because 16 is an even number, you have to halve it, which gives you 8. Then 8 / 2 = 4, which, when divided by 2, is 2—and 2 / 2 = 1. The process of iterative calculation brings you to the end after five steps.

Of course, you can also continue calculating with 1, which gives you 4, then 2 and then 1 again. The calculation rule leads you into an inescapable loop. Therefore 1 is seen as the end point of the procedure.

Following iterative calculations, you can begin with any of the numbers above and will ultimately reach 1.

Credit: Keenan Pepper/Public domain via Wikimedia Commons

It’s really fun to go through the iterative calculation rule for different numbers and look at the resulting sequences. If you start with 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Or 42: 42 → 21 → 64 → 32 → 16 → 8 → 4 → 2 → 1. No matter which number you start with, you always seem to end up with 1. There are some numbers, such as 27, where it takes quite a long time (27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → ...), but so far the result has always been 1. (Admittedly, you have to be patient with the starting number 27, which requires 111 steps.)

But strangely there is still no mathematical proof that the Collatz conjecture is true. And that absence has mystified mathematicians for years.

The origin of the Collatz conjecture is uncertain, which is why this hypothesis is known by many different names. Experts speak of the Syracuse problem, the Ulam problem, the 3 n + 1 conjecture, the Hasse algorithm or the Kakutani problem.

German mathematician Lothar Collatz became interested in iterative functions during his mathematics studies and investigated them. In the early 1930s he also published specialist articles on the subject , but the explicit calculation rule for the problem named after him was not among them. In the 1950s and 1960s the Collatz conjecture finally gained notoriety when mathematicians Helmut Hasse and Shizuo Kakutani, among others, disseminated it to various universities, including Syracuse University.

Like a siren song, this seemingly simple conjecture captivated the experts. For decades they have been looking for proof that after repeating the Collatz procedure a finite number of times, you end up with 1. The reason for this persistence is not just the simplicity of the problem: the Collatz conjecture is related to other important questions in mathematics. For example, such iterative functions appear in dynamic systems, such as models that describe the orbits of planets. The conjecture is also related to the Riemann conjecture, one of the oldest problems in number theory.

Empirical Evidence for the Collatz Conjecture

In 2019 and 2020 researchers checked all numbers below 2 68 , or about 3 x 10 20 numbers in the sequence, in a collaborative computer science project . All numbers in that set fulfill the Collatz conjecture as initial values. But that doesn’t mean that there isn’t an outlier somewhere. There could be a starting value that, after repeated Collatz procedures, yields ever larger values that eventually rise to infinity. This scenario seems unlikely, however, if the problem is examined statistically.

An odd number n is increased to 3 n + 1 after the first step of the iteration, but the result is inevitably even and is therefore halved in the following step. In half of all cases, the halving produces an odd number, which must therefore be increased to 3 n + 1 again, whereupon an even result is obtained again. If the result of the second step is even again, however, you have to divide the new number by 2 twice in every fourth case. In every eighth case, you must divide it by 2 three times, and so on.

In order to evaluate the long-term behavior of this sequence of numbers , Lagarias calculated the geometric mean from these considerations in 1985 and obtained the following result: ( 3 / 2 ) 1/2 x ( 3 ⁄ 4 ) 1/4 x ( 3 ⁄ 8 ) 1/8 · ... = 3 ⁄ 4 . This shows that the sequence elements shrink by an average factor of 3 ⁄ 4 at each step of the iterative calculation rule. It is therefore extremely unlikely that there is a starting value that grows to infinity as a result of the procedure.

There could be a starting value, however, that ends in a loop that is not 4 → 2 → 1. That loop could include significantly more numbers, such that 1 would never be reached.

Such “nontrivial” loops can be found, for example, if you also allow negative integers for the Collatz conjecture: in this case, the iterative calculation rule can end not only at –2 → –1 → –2 → ... but also at –5 → –14 → –7 → –20 → –10 → –5 → ... or –17 → –50 → ... → –17 →.... If we restrict ourselves to natural numbers, no nontrivial loops are known to date—which does not mean that they do not exist. Experts have now been able to show that such a loop in the Collatz problem, however, would have to consist of at least 186 billion numbers .

The length of the Collatz sequences for all numbers from 1 to 9,999 varies greatly.

Credit: Cirne/Public domain via Wikimedia Commons

Even if that sounds unlikely, it doesn’t have to be. In mathematics there are many examples where certain laws only break down after many iterations are considered. For instance,the prime number theorem overestimates the number of primes for only about 10 316 numbers. After that point, the prime number set underestimates the actual number of primes.

Something similar could occur with the Collatz conjecture: perhaps there is a huge number hidden deep in the number line that breaks the pattern observed so far.

A Proof for Almost All Numbers

Mathematicians have been searching for a conclusive proof for decades. The greatest progress was made in 2019 by Fields Medalist Terence Tao of the University of California, Los Angeles, when he proved that almost all starting values of natural numbers eventually end up at a value close to 1.

“Almost all” has a precise mathematical meaning: if you randomly select a natural number as a starting value, it has a 100 percent probability of ending up at 1. ( A zero-probability event, however, is not necessarily an impossible one .) That’s “about as close as one can get to the Collatz conjecture without actually solving it,” Tao said in a talk he gave in 2020 . Unfortunately, Tao’s method cannot generalize to all figures because it is based on statistical considerations.

All other approaches have led to a dead end as well. Perhaps that means the Collatz conjecture is wrong. “Maybe we should be spending more energy looking for counterexamples than we’re currently spending,” said mathematician Alex Kontorovich of Rutgers University in a video on the Veritasium YouTube channel .

Perhaps the Collatz conjecture will be determined true or false in the coming years. But there is another possibility: perhaps it truly is a problem that cannot be proven with available mathematical tools. In fact, in 1987 the late mathematician John Horton Conway investigated a generalization of the Collatz conjecture and found that iterative functions have properties that are unprovable. Perhaps this also applies to the Collatz conjecture. As simple as it may seem, it could be doomed to remain unsolved forever.

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

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100 Math Riddles for Kids (and Grown-Ups Too!)

So many ways to have fun with math.

Math riddles are tricky problems or questions that students have to use their knowledge of math operations, vocabulary, and general knowledge to solve. They’re also a fun way to engage in creative thinking and collaboration as they work to solve math riddles together.

As a warm-up or challenge activity, math riddles:

• Improve problem-solving skills.
• Build resilience. Students are bound to miss one (or many) riddles. Having to try and try again is a skill they’ll need in math and beyond.
• Increase engagement. Math riddles can be a fun way to learn and practice math, which can increase motivation.
• Support collaboration. If students must solve a problem in groups, they’ll practice working together and relying on each other’s unique approaches and thinking styles.

Whatever the reason, check out these 100 math riddles to challenge every student.  

Math Riddles for Elementary School

How many sides does a circle have.

None. A circle does not have sides.

I’m an even, single-digit number. Once you write me, I have no start or end. I look like a standing pair of glasses. Who am I?

Ashley buys one new toy every 30 days. how many toys does he have in one year, which is longer, 10 centimeters or 10 inches, if you multiply me by any other number, the answer will always be the same. what number am i, what weighs more, a pound of feathers or a pound of bricks.

They weigh the same.

When Pam was 8, her sister was half her age. Now, Pam is 14. How old is her sister?

Pam’s sister is 4 years younger, so her sister is now 10 (14 – 4 = 10).

If a boy blows 18 bubbles, and then he pops 7, eats 6, stomps 5, and blows another 1, how many are left?

If you’re at a theater to watch a movie that starts in one hour, how many minutes do you have to wait to watch it.

60 minutes.

Nina has six siblings who were born two years apart. Kevin, the youngest, is 7 years old, while Nina is the oldest. How old is Nina?

Nina is 19: 7 + 2 + 2 + 2 + 2 + 2 + 2 = 19.

How did the soccer fan know before the game that the score would be 0-0?

The score is always 0-0 before the game.

What is the Roman numeral for 100?

You’re 8 years old, and your uncle is 53. how old will your uncle be when you are 23, at what point do parallel lines meet.

Never. Parallel lines never intersect.

If it took seven people 10 hours to build a doghouse, how many hours will it take 15 people to build the same doghouse?

None. The dog house is already built.

A farmer has 23 goats on his farm. One day after a storm, all but 9 ran away. How many goats does the farmer have now?

Place three matches on the table. how can you add two more to make eight.

Add two matches to make the Roman numeral VIII.

How many times can you take 10 away from 100?

Once. After that, the number you’re subtracting from is 90.

There’s something common between the numbers 11, 88, and 96. What is it?

You can rotate them 180° or upside down and they will still look the same.

What can you put between 8 and 9 to make the result greater than 8, but less than 9?

A decimal point (8.9).

This number is the smallest prime number and an even number. What is it?

Henry is 1,000 mm tall, and hank is 100 cm tall. who’s taller.

They are the same height.

Can you find a way to get the number 23 by using only the number 2?

22 + 2/2 = 23

How many cupcakes are in a baker’s dozen?

I get 11 when i add 5 and 6, but when i take 7 and 6, i get 1. what am i, if 1 = 4, 2 = 8, 3 = 12, then 4 = how much.

Since 1 = 4, then 4 = 1.

There is an odd number that becomes even if you take away one letter from it. What is it?

Seven. Take away the S and it is “even.”

What three numbers have the same result if they are added or multiplied together?

1, 2, and 3.

If days on Mars are as long as two weeks on Earth, how long is an hour?

14 Earth hours.

Math Riddles for Middle Schoolers

If you go to the movies and you’re paying, is it cheaper to take one friend to the movies twice or two friends to the movies at the same time.

It is cheaper to take two friends at the same time, since you’d be paying for yourself plus two others (three people) as opposed to paying for yourself and one friend twice, which equals four people.

If four men can build four tables in four hours, how many can eight men build in eight hours?

I am four times as old as my daughter. in 20 years time, i shall be twice as old as her. how old are we now.

I am 40 and my daughter is 10.

There is a lily pad on a small pond. Each day, the lily pad doubles its size. On day 20 it covers the whole pond. On what day was the lily pad the size of half the pond?

At a clothing store, the owner has devised his own method of pricing items. a vest costs $20, socks cost $25, a tie costs $15, and a blouse costs $30. using this method, how much would a pair of underwear cost.

$45. The pricing method consists of charging $5 for each letter required to spell the item.

One is to three as three is to five and five is to four and four is the magic number. What is the pattern?

The word “one” has three letters. The word “three” has five letters. “Five” has four letters. “Four” has four letters.

The ages of a father and son add up to 66. The father’s age is the son’s age reversed. How old could they be? (There are three possible solutions.)

51 and 15, 42 and 24, or 60 and 06.

Pick a number from 1 to 10, multiply it by 2, add 10, divide it by 2, now subtract the number you picked from your total. What is it?

You want to boil a two-minute egg. if you only have a three-minute timer (hourglass), a four-minute timer, and a five-minute timer, how can you boil the egg for only two minutes.

Once the water is boiling, turn the three-minute timer and five-minute timer over. When the three-minute timer runs out, put the egg in the boiling water. When the five-minute timer runs out, two minutes have elapsed and it is time to take the egg out of the water. You do not need the four-minute timer.

81 x 9 = 801. What must you do to make the this equation true?

Turn it upside down. 108 = 6 x 18.

How much will a 38° angle measure when looked at under a microscope that magnifies 10 times?

It will be 38 degrees.

If 1 = 5, 2 = 15, 3 = 215, and 4 = 3,215, what does 5 equal?

5 = 1, because 1 = 5.

If seven people meet one another and each shakes hands only once with each of the others, how many handshakes happened?

Using only addition, how can you add eight 8s to get the number 1,000.

888 + 88 + 8 + 8 + 8 =1,000.

How much dirt is there in a hole that is 3.45 m by 6.21 m?

None. There is no dirt in a hole.

A phone and case cost $110 in total. The price of the phone is $100 more than the case. How much is the phone?

An empty bus pulls up to a stop, and 10 people get on. at the next stop, 5 people get off, and twice as many people get on as at the first stop. at the third stop, 25 get off. how many people are on the bus.

One—the driver.

During what month do people sleep the least?

February because it has the fewest days.

There are 12 kids in a class. Six kids are wearing socks, four are wearing shoes, and three are wearing both. How many have bare feet?

Mr. kim has five sons. each of his sons has a sister. how many children does mr. kim have.

Six. They all have the same sister.

There are six chocolates in a basket. Six people each take one chocolate. How can it be that one chocolate is left in the basket?

The last person took the basket with the chocolate in it.

Kevin finished half of his report at school. He finishes another quarter after school. How much does he have left to finish?

A quarter (1 – 1/2 – 1/4 = 1/4).

What is the next number? 1, 2, 4, 7, 11, 16, __?

At an ice cream shop, a single scoop costs $5. a double scoop costs $7. a triple scoop costs $8. what’s the best value.

The triple scoop.

Maria made brownies. She cuts the brownies into 12 equal pieces. How many cuts did she make?

Five cuts (two vertical, three horizontal).

Hailey has three library books at home. She finished reading one of them and returned it. Then her mom bought her two books. How many books does she currently have at home?

Four (3 – 1 + 2 = 4).

Which statement is correct: “Eight and six are fifteen” or “six and eight are fifteen”?

Neither of them is correct: 8 + 6 =14.

Math Riddles for High Schoolers

Cara leaves new york at 8 a.m., driving 35 mph. nancy starts driving to new york at 50 mph. when they meet, which one will be farthest away from new york.

They will be the same distance from New York.

A teacher says that there are fewer than 30 students but more than 20 in a classroom. The children can be divided into groups of two, three, four, six, and eight with no leftover students. How many students are in the class?

Seven brothers were born two years apart. the youngest brother is 7. how old is the oldest brother, charlie has one child, a boy. he’s having another child. what chance does he have of having another boy, what is the next number: 1, 3, 7, 15, 31, __, what is the next number: 2, 4, 6, 10, 16, __, there are 100 houses in the neighborhood where sam and his brother live. sam’s house number is the reverse of his brother’s house number. the difference between their house numbers ends with 2. what are their house numbers, in a three-digit number, the second digit is four times greater than the third digit. the first digit is three lower than the second digit. what is the number, it is raining at midnight and the weather forecast says that tomorrow and the next day will be clear. will it be sunny in 48 hours.

No, in 48 hours, it will be night so it will not be sunny.

A 300-foot train traveling 300 feet per minute must travel through a 300-foot-long tunnel. How long will it take the train to travel through the tunnel?

Two minutes. It takes the front of the train one minute, and the rest of the train will take two minutes to clear the tunnel.

Robert and David played several golf matches against each other in a week. They played for a pizza at each match, but no pizzas were purchased until the end of the week. If Robert and David had the same number of wins at any time, those pizzas were canceled. Robert won four matches (but no pizzas), and David won three pizzas. How many rounds of golf were played?

11. David won seven matches—four to cancel out Robert’s four wins and three more to win the pizzas.

I add 5 to 9 and get 2. The answer is correct, but how?

When it is 9 a.m., add five hours to it, and you will get to 2 p.m.

When Carlos was 6 years old, his little sister, Julia, was half his age. If Carlos is 40 years old today, how old is Julia?

Tom was asked to paint numbers outside 100 apartments, which means he will have to paint numbers 1 through 100. how many times will he have to paint the number 8, there are 100 pairs of dogs in a zoo. a pair of puppies are born for each dog. unfortunately, 23 of the dogs have not survived. how many dogs are left in total.

977 dogs. 100 x 2 = 200; 200 + 800 = 1,000; 1,000 – 23 = 977

If a hen and a half lay an egg and a half in a day and a half, how many eggs will half a dozen hens lay in half a dozen days?

Leon works at the aquarium. when he tries to put each turtle in its own tank, he has one turtle too many. but if he puts two turtles per tank, he has one tank too many. how many turtles and how many tanks does leon have.

He has three tanks and four turtles.

Math Riddles for Adults

You have a perfect cube. can you divide your cube into smaller cubes that all have different volumes.

No. There is no way for cubes of all different volumes to correctly fit inside of a larger cube. There would be empty spaces as the cubes got smaller and smaller and you would have no way to fill these spaces. Eventually, the cubes would be too small to continue.

One morning you need to find some socks, but your light is broken. You know there are 10 purple socks and 10 orange socks in your drawer, but they’re all mixed up. They are completely the same except for their color. How many socks do you have to pick to ensure you have a matching pair?

You only have to pick three socks. Because there are only two colors of socks in the drawer, you will definitely have a matching pair if you take three.

You are cutting up a pie. First you cut it into two pieces. With your next cut, you produce four pieces. The third cut results in seven pieces. How many pieces can you cut with seven slices?

What is special about the number 854,917,632.

It’s the numbers one through ten in alphabetical order.

You are 8 feet away from a door, and with each move you advance half the distance to the door. How many moves will it take to reach the door?

You will never reach it because you will always have another half distance to cross.

Joe has 10 coins totaling $1.19. From these coins, he cannot make exact change for a dollar, half dollar, quarter, dime, or nickel. What are the coins?

A half dollar, a quarter, four dimes, and four pennies.

When shipping something, Tom can place 10 small boxes or 8 large boxes into a carton. A total of 96 boxes were sent in one shipment and the number of small boxes was less than large boxes. How many cartons did he ship?

11 cartons.

A certain number has three digits. The sum of the three digits equals 36 times this number. Seven times the left digit plus nine equals five times the sum of the two other digits. Eight times the second digit minus nine is equal to the sum of the first and third. What is the number?

Can you arrange four nines to make it equal to 100.

99 + 9 / 9 = 100

Easy Math Riddles

I am a number that is 5 more than the number of toes you have on your feet. what am i, i am a number that is two more than the number of days in a week. what am i, i’m less than 15 but more than 10. you can find me by adding 10 to 3. what am i, i am more than 10 but less than 14. i am one more than the number of months in one year. what am i, i’m less than 8 but more than 5. i’m one more than the fingers on your left hand. what am i, i’m a number you can find by adding the number of sides of a triangle. what am i, i’m between 1 and 5. i’m less than 4 and more than 2. what am i, i am a number you get when you add the sides of a rectangle and a square., hard math riddles, a snail is at the bottom of a 20-meter pit. every day the snail climbs 5 meters up, but at night it slides down 4 meters. how many days does it take before the snail reaches the top.

The snail reaches the top of the pit on the 16th day. On the first day, the snail reaches a height of 5 meters and slides down 4 meters at night, and thus ends at a height of 1 meter. On the second day, he reaches 6 meters but slides back to 2 meters. On the third day, he reaches 7 meters but slides back to 3 meters. On the 15th day, he reaches 19 meters but slides back to 15 meters. On the 16th day, he reaches 20 meters, so now he is at the top of the pit.

There are eight balls with eight different numbers in them: 1, 3, 5, 7, 9, 11, 13, 15. Select three balls that equal 30.

Flip the nine and turn it into a 6 and then the three balls are 6, 11, and 13.

Can you make 24 from the numbers 3, 3, 8, and 8? You can only use the operations + – x / (), and you have to use all four numbers.

8 / (3 – 8/3) = 24.

How can you take 2 from 5 and leave 4?

Remove the 2 letters F and E from five and you have IV which is the Roman numeral for four.

Granny left half her money to her granddaughter and half that amount to her grandson. She left a sixth to her brother, and the remainder, $1,000, to the dogs’ home. How much did she leave altogether?

Focus on the fractions. Add one-half, one-quarter, and one-sixth for a total that is a fraction of twelfths (2 + 4 + 6 = 12). You can also think about it as 6/12, 3/12, 2/12, which equals 11/12. If the remainder is $1,000, that must be one-twelfth, so the total is $12,000.

What single digit appears most frequently between and including the numbers 1 and 1,000?

The most common digit is 1. Every number 1 through 9 appears exactly the same number of times in every 10 numbers. But because we included the number 1,000, there’s an extra occurrence of the number 1. In total, the number 1 appears 301 times, and every other number appears 300 times.

A small number of cards has been lost from a complete pack. If I deal among four people, three cards remain. If I deal among three people, two remain, and if I deal among five people, two cards remain. How many cards are there?

What is the smallest whole number that is equal to seven times the sum of its digits, what is the smallest number that increases by 12 when it is flipped and turned upside-down, you have two u.s. coins worth a total value of 30 cents. one of them is not a nickel. what are the two coins.

One quarter and one nickel.

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Can you solve NASA's Pi Day 2024 challenge?

Hungry for Pi? Check out NASA's Pi Day challenge and put your wits to the test solving problems just like NASA scientists and engineers.

Happy Pi Day 2024!

Have you ever wondered what it would be like to solve problems for NASA to help with the exploration of other planets in the solar system ? 

In celebration of Pi Day 2024 , you can do just that and take the annual NASA Pi Day Challenge . This is a fun way to put on your scientist and engineer thinking cap and try your best at a series of questions all surrounding the mathematical constant, pi. 

Related: What is the smallest known planet?

What is pi? If you recall from mathematics class back in grade school, it's approximately 3.14159 and can be used to figure out the circumference of a circle of the volume of a square. 

While there are many uses for it in different STEM jobs and fields, it's also very important for engineers and scientists at NASA to help study not just our planet but others across the solar system and even other galaxies .

This challenge is a tradition that has been on-going for the last decade put on by NASA's Jet Propulsion Laboratory's Education Office and features numerous math problems you have to solve using pi. 

Some of the questions you can answer this year pertain to missions including the Deep Space Optical Communications technology on NASA's Psyche spacecraft , the Double Asteroid Redirection Test (DART) spacecraft, Earth-orbiting satellites, rovers on the Moon, and even the Hubble Space Telescope and James Webb Space Telescope .

—  As scientists find real exoplanets, sci-fi writers change their vision of alien worlds

 — 10 surprising facts about pi

— Pi Day turns 35: The circle of mathematics needs more diversity, advocates say

So, let's get solving! You can find each of the problems with an accompanying worksheet you can do all your work online and the answers will be posted by NASA so you can check your work! 

There are nearly four dozen different questions you can figure out, so try a few or do them all to "cook up" a unique way to get space-y and celebrate Pi Day 2024! 

Join our Space Forums to keep talking space on the latest missions, night sky and more! And if you have a news tip, correction or comment, let us know at: [email protected].

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Meredith is a regional Murrow award-winning Certified Broadcast Meteorologist and science/space correspondent. She most recently was a Freelance Meteorologist for NY 1 in New York City & the 19 First Alert Weather Team in Cleveland. A self-described "Rocket Girl," Meredith's personal and professional work has drawn recognition over the last decade, including the inaugural Valparaiso University Alumni Association First Decade Achievement Award, two special reports in News 12's Climate Special "Saving Our Shores" that won a Regional Edward R. Murrow Award, multiple Fair Media Council Folio & Press Club of Long Island awards for meteorology & reporting, and a Long Island Business News & NYC TV Week "40 Under 40" Award.

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