## Free Math Worksheets — Over 100k free practice problems on Khan Academy

You’ve found something even better!

That’s because Khan Academy has over 100,000 free practice questions. And they’re even better than traditional math worksheets – more instantaneous, more interactive, and more fun!

Kindergarten, basic geometry, pre-algebra, algebra basics, high school geometry.

• Trigonometry

## Statistics and probability

High school statistics, ap®︎/college statistics, precalculus, differential calculus, integral calculus, ap®︎/college calculus ab, ap®︎/college calculus bc, multivariable calculus, differential equations, linear algebra.

• Place value (tens and hundreds)
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• Prepare for the 2020 AP®︎ Statistics Exam
• AP®︎ Statistics Standards mappings
• Polynomials
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• AP Calculus AB solved free response questions from past exams
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Khan Academy’s 100,000+ free practice questions give instant feedback, don’t need to be graded, and don’t require a printer.

## What do Khan Academy’s interactive math worksheets look like?

Here’s an example:

“My students love Khan Academy because they can immediately learn from their mistakes, unlike traditional worksheets.”

## What do Khan Academy’s interactive math worksheets cover?

Our 100,000+ practice questions cover every math topic from arithmetic to calculus, as well as ELA, Science, Social Studies, and more.

## Is Khan Academy a company?

Khan Academy is a nonprofit with a mission to provide a free, world-class education to anyone, anywhere.

## Want to get even more out of Khan Academy?

Then be sure to check out our teacher tools . They’ll help you assign the perfect practice for each student from our full math curriculum and track your students’ progress across the year. Plus, they’re also 100% free — with no subscriptions and no ads.

## Generating PDF...

• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode Scientific Notation Arithmetics
• Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Logical Sets Word Problems
• Pre Calculus Equations Inequalities Simultaneous Equations System of Inequalities Polynomials Rationales Complex Numbers Polar/Cartesian Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
• Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
• Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
• Linear Algebra Matrices Vectors
• Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
• Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
• Physics Mechanics
• Chemistry Chemical Reactions Chemical Properties
• Finance Simple Interest Compound Interest Present Value Future Value
• Economics Point of Diminishing Return
• Conversions Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time
• Pre Algebra
• Pre Calculus
• Linear Algebra
• Trigonometry
• Conversions

## Most Used Actions

Number line.

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• line\:(1,\:2),\:(3,\:1)
• prove\:\tan^2(x)-\sin^2(x)=\tan^2(x)\sin^2(x)
• \frac{d}{dx}(\frac{3x+9}{2-x})
• (\sin^2(\theta))'
• \lim _{x\to 0}(x\ln (x))
• \int e^x\cos (x)dx
• \int_{0}^{\pi}\sin(x)dx
• \sum_{n=0}^{\infty}\frac{3}{2^n}
• Is there a step by step calculator for math?
• Symbolab is the best step by step calculator for a wide range of math problems, from basic arithmetic to advanced calculus and linear algebra. It shows you the solution, graph, detailed steps and explanations for each problem.
• Is there a step by step calculator for physics?
• Symbolab is the best step by step calculator for a wide range of physics problems, including mechanics, electricity and magnetism, and thermodynamics. It shows you the steps and explanations for each problem, so you can learn as you go.
• How to solve math problems step-by-step?
• To solve math problems step-by-step start by reading the problem carefully and understand what you are being asked to find. Next, identify the relevant information, define the variables, and plan a strategy for solving the problem.
• Practice, practice, practice Math can be an intimidating subject. Each new topic we learn has symbols and problems we have never seen. The unknowing... Read More

## Solve any math problem with GeoGebra Math Solver

• Solve equations and inequalities
• Simplify expressions
• Factor polynomials
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• All solvers
• Arithmetics
• Determinant
• Percentages
• Scientific Notation
• Inequalities

## Math Topics

More solvers.

• Simplify Fractions

Examples for

## Mathematics

Wolfram|Alpha has broad knowledge and deep computational power when it comes to math. Whether it be arithmetic, algebra, calculus, differential equations or anything in between, Wolfram|Alpha is up to the challenge. Get help with math homework, solve specific math problems or find information on mathematical subjects and topics.

Do basic arithmetic. Work with fractions, percentages and similar fundamentals. Solve place value and word problems.

## Do basic arithmetic:

Do exact arithmetic with fractions:.

Compute the properties of geometric objects of various kinds in 2, 3 or higher dimensions. Explore and apply ideas from many subfields of geometry.

## Compute properties of a geometric figure:

Plot a conic section and identify its type:, compute properties of a polyhedron:.

Work with various kinds of numbers. Check for membership in larger sets, such as the rationals or the transcendental numbers. Convert between bases.

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Convert a decimal number to another base:.

Analyze integers; subsets of the integers, including the prime numbers; and related ideas.

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Solve a diophantine equation:.

Perform numerical analysis and optimization of systems and objects, including packing and covering of objects and control systems.

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Numerically integrate functions that cannot be integrated symbolically:.

Make queries about various definitions and descriptions in mathematics.

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Get a brief definition:.

Compute properties of datasets, perform statistical inference or model data. Work with probability distributions and random variables.

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Find the sample size needed to estimate a binomial parameter:.

Find roots of and expand, factor or simplify mathematical expressions—everything from polynomials to fields and groups.

## Solve an equation:

Factor a polynomial:, simplify an expression:.

Solve differential equations of any order. Examine solutions and plots of the solution families. Specify initial conditions to find exact solutions.

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Specify initial values:, solve a nonlinear equation:.

Perform trigonometric calculations and explore properties of trigonometric functions and identities.

## Compute values of trigonometric functions:

Solve a trigonometric equation:.

Explore sequences and recurrences, solve common problems in combinatorics and compute properties of graphs and lattices.

## Compute a possible formula and continuation for a sequence:

Analyze a graph specified by adjacency rules:, solve a recurrence:.

Evaluate Boolean logic expressions and expressions involving sets and set operators. Solve Boolean equations. Compute truth tables. Generate Venn diagrams.

## Compute a truth table:

Generate a venn diagram:.

## Get information about a mathematical conjecture:

Get historical information about a theorem:.

Compute the probabilities of certain events occurring. Compute joint, disjoint or conditional probabilities and apply them to real-world situations.

## Compute the probability of a union of events:

Compute coin‐toss probabilities:.

Compute integrals, derivatives and limits as well as analyze sums, products and series.

## Compute an integral:

Calculate a derivative:, solve an ordinary differential equation:.

Visualize functions, equations and inequalities. Do so in 1, 2 or 3 dimensions. Make polar and parametric plots.

## Plot a function:

Plot a region satisfying multiple inequalities:.

Explore and compute properties of vectors, matrices and vector spaces.

## Compute properties of a vector:

Calculate properties of a matrix:, determine whether a set of vectors is linearly independent:.

Analyze functions and expressions containing imaginary numbers or complex variables.

## Compute properties of a function of a complex variable (use the variable z ):

Compute the residue of a function at a point:.

Examine the properties of mathematical functions, such as continuity, surjectivity and parity. Utilize notable special functions or number theoretic functions.

## Do computations with special functions:

Do computations with number theoretic functions:, find representations for a function:.

Compute; learn about algorithms, definitions and theorems involving; or find properties of continued fractions.

## Find the continued fraction representation of a number:

Find definitions of continued fraction terminology:, find continued fraction papers by author:.

Get information about math Common Core Standards for kindergarten through eighth grade.

## Evaluate an expression (CCSS.Math.Content.6.EE.A.2c):

Perform multiple operations with rational numbers (ccss.math.content.7.ns.a.2c):.

## Mathway: Math Problem Solver 4+

Homework scanner & calculator, chegg, inc..

• #13 in Education
• 4.9 • 414.8K Ratings
• Offers In-App Purchases

## Screenshots

Description.

Version 5.3.3

Powered by Chegg, Mathway is your one-stop shop for homework help. Is your equation too complex to type? Save time with our snap and solve feature! Just take a photo and instantly see the solution. Our interactive calculator also allows you to select your class subject (now including physics!) to generate all the functions you need to solve that equation. Need more than an answer? Upgrade your subscription and get unlimited access to detailed equation breakdowns, helping you learn how to solve it one step at a time.

414.8K Ratings

## Recommend!!!!!!?

Once I was doing my homework I couldn’t figure it out I asked my mom to help me and I showed her and she fell asleep so I kind of started panicking Went downstairs to the table I started struggling and thinking and thinking so I went to the App Store looked and looked and looked and looked left to right for helping your homeworkIs the best I really do recommend it it is the best it will save your life and when I mean save your life I mean really save your life nobody really understands when they say that they saved their lifeThey never understand how much that person means they don’t even know how serious the persons like saying I really do recommend this app messed up in the world five star review amazing helpful save your life get it get it now get it now it’s the best app ever students allowed teacher teachers not allowed can students really need this app so don’t get mad at them if they have this app it’s the best time with them so hope you get it

## It works but the full version hasn’t for me

I have been using this a lot this semester so I decided to pay for the monthly subscription then I could cancel the membership at the end of the semester, but it doesn’t even work. The subscription shows in my subscriptions list in my settings but I still can’t view the steps, and when I attempt to see how the problem is solved it brings me to the menu to subscribe to the paid part of the app. Honestly though, other than that. This app works pretty great, solves problems the right way and you can select the method of how you want the problem solved. You can change what kind of math you are trying to solve, and the answers are pretty spot on. The predictive text is even really good on this, it’ll help solve word problems and you kind of don’t em have to type anything in sometimes. However, the keyboard is very buggy and you can’t type very fast. It’s also kind of hard to navigate as there are like 4 different keyboard menus to choose from, I always forget where to find things and it kind of takes a while to load the text sometimes. But it does solve the problems.

## Developer Response ,

Sorry to hear of the issues you're having - please select 'restore subscription' from the settings of your Mathway account. This should turn the steps "ON". For further assistance you can reach Mathway customer support by emailing: [email protected].

## App Privacy

The developer, Chegg, Inc. , indicated that the app’s privacy practices may include handling of data as described below. For more information, see the developer’s privacy policy .

## Data Used to Track You

The following data may be used to track you across apps and websites owned by other companies:

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## Data Not Linked to You

The following data may be collected but it is not linked to your identity:

Privacy practices may vary, for example, based on the features you use or your age. Learn More

## Information

English, Arabic, French, German, Hindi, Indonesian, Italian, Japanese, Korean, Portuguese, Russian, Simplified Chinese, Spanish, Turkish, Vietnamese

• Steps (Monthly) $9.99 • Steps (Monthly)$19.99
• Steps (Annual) $39.99 • Steps (Annual)$79.99
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Chegg Study - Homework Help

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## Upload a screenshot and solve any math problem instantly with MathGPT!

Drag & drop an image file here, or click to select an image.

✅ Unlimited Solutions

➕ Free Bonuses ($1085 value!) Chapter Contents ⊗ • Search IntMath • Math interactives • About (site info) • Uses of Trignometry • ASCIIMath input, KaTeX output • ASCIIMath input, LaTeX and KaTeX output • Send Math in emails • Syntax for ASCIIMathML • Math Display Experiments • Scientific Notebook ## Math Problem Solver Math Tutoring Need help? Chat with a tutor anytime, 24/7 . Chat Now » ## 🤖 Math Solver & Calculator This tool combines the power of mathematical computation engine that excels at solving mathematical formulas with the power of artificial intelligence large language models to parse and generate natural language answers. This creates a math problem solver that's more accurate than ChatGPT, more flexible than a math calculator, and provides answers faster than a human tutor. Learn more here . ## Problem Solver Subjects Our math problem solver that lets you input a wide variety of math math problems and it will provide a step by step answer. This math solver excels at math word problems as well as a wide range of math subjects. • Math Word Problems • Pre-Algebra • Geometry Graphing • Trigonometry • Precalculus • Finite Math • Linear Algebra Here are example math problems within each subject that can be input into the calculator and solved. This list is constanstly growing as functionality is added to the calculator. ## Basic Math Solutions Below are examples of basic math problems that can be solved. • Long Arithmetic • Rational Numbers • Operations with Fractions • Ratios, Proportions, Percents • Measurement, Area, and Volume • Factors, Fractions, and Exponents • Unit Conversions • Data Measurement and Statistics • Points and Line Segments ## Math Word Problem Solutions Math word problems require interpreting what is being asked and simplifying that into a basic math equation. Once you have the equation you can then enter that into the problem solver as a basic math or algebra question to be correctly solved. Below are math word problem examples and their simplified forms. Word Problem: Rachel has 17 apples. She gives some to Sarah. Sarah now has 8 apples. How many apples did Rachel give her? Simplified Equation: 17 - x = 8 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) Simplified Equation: {r = d + 12, d = b + 6, r = 2 × b} Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left? Simplified: 40 - 10 - 5 ## Pre-Algebra Solutions Below are examples of Pre-Algebra math problems that can be solved. • Variables, Expressions, and Integers • Simplifying and Evaluating Expressions • Solving Equations • Multi-Step Equations and Inequalities • Ratios, Proportions, and Percents • Linear Equations and Inequalities ## Algebra Solutions Below are examples of Algebra math problems that can be solved. • Algebra Concepts and Expressions • Points, Lines, and Line Segments • Simplifying Polynomials • Factoring Polynomials • Linear Equations • Absolute Value Expressions and Equations • Radical Expressions and Equations • Systems of Equations • Quadratic Equations • Inequalities • Complex Numbers and Vector Analysis • Logarithmic Expressions and Equations • Exponential Expressions and Equations • Conic Sections • Vector Spaces • 3d Coordinate System • Eigenvalues and Eigenvectors • Linear Transformations • Number Sets • Analytic Geometry ## Trigonometry Solutions Below are examples of Trigonometry math problems that can be solved. • Algebra Concepts and Expressions Review • Right Triangle Trigonometry • Radian Measure and Circular Functions • Graphing Trigonometric Functions • Simplifying Trigonometric Expressions • Verifying Trigonometric Identities • Solving Trigonometric Equations • Complex Numbers • Analytic Geometry in Polar Coordinates • Exponential and Logarithmic Functions • Vector Arithmetic ## Precalculus Solutions Below are examples of Precalculus math problems that can be solved. • Operations on Functions • Rational Expressions and Equations • Polynomial and Rational Functions • Analytic Trigonometry • Sequences and Series • Analytic Geometry in Rectangular Coordinates • Limits and an Introduction to Calculus ## Calculus Solutions Below are examples of Calculus math problems that can be solved. • Evaluating Limits • Derivatives • Applications of Differentiation • Applications of Integration • Techniques of Integration • Parametric Equations and Polar Coordinates • Differential Equations ## Statistics Solutions Below are examples of Statistics problems that can be solved. • Algebra Review • Average Descriptive Statistics • Dispersion Statistics • Probability • Probability Distributions • Frequency Distribution • Normal Distributions • t-Distributions • Hypothesis Testing • Estimation and Sample Size • Correlation and Regression ## Finite Math Solutions Below are examples of Finite Math problems that can be solved. • Polynomials and Expressions • Equations and Inequalities • Linear Functions and Points • Systems of Linear Equations • Mathematics of Finance • Statistical Distributions ## Linear Algebra Solutions Below are examples of Linear Algebra math problems that can be solved. • Introduction to Matrices • Linear Independence and Combinations ## Chemistry Solutions Below are examples of Chemistry problems that can be solved. • Unit Conversion • Atomic Structure • Molecules and Compounds • Chemical Equations and Reactions • Behavior of Gases • Solutions and Concentrations ## Physics Solutions Below are examples of Physics math problems that can be solved. • Static Equilibrium • Dynamic Equilibrium • Kinematics Equations • Electricity • Thermodymanics ## Geometry Graphing Solutions Below are examples of Geometry and graphing math problems that can be solved. • Step By Step Graphing • Linear Equations and Functions • Polar Equations Looking for the old Mathway Calculator? We've moved it to here . This tool combines the power of mathematical computation engine that excels at solving mathematical formulas with the power of GPT large language models to parse and generate natural language. This creates math problem solver thats more accurate than ChatGPT, more flexible than a calculator, and faster answers than a human tutor. Learn More. ## Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class. Email Address Sign Up ## Equation Solver Enter the Equation you want to solve into the editor. Please ensure that your password is at least 8 characters and contains each of the following: • a special character: @$#!%*?&

Google updated its search engine and Lens tool with new features to help you visualize and solve problems in more difficult subjects like geometry, physics, trigonometry and calculus.

The update allows you to type an equation into the Search bar or take a photo with Lens to get the correct answer and a step-by-step explanation. The new feature also provides solutions to word problems. For instance, the solver can explain high school physics concepts like finding the average acceleration using the equation “vf= vi + at.”

You can also access the math solver feature by typing “[math solver]” directly into the search bar. However, this function is only available on desktop. It will launch on mobile by the end of the year, the company told TechCrunch.

Another useful new feature is the geometry solver. Since solving geometry problems requires a lot of visualization, Lens can now explain both the visual and the text components of the problem.

However, Lens can only interpret “certain triangle problems,” the company wrote in its blog post, attributed to Google product lead Robert Wong.

We reached out to Wong to clarify, who told us that the geometry solver can currently do the following:

• Find the area of a triangle using A = 1/2 x height x base or Heron’s formula.
• Find the perimeter of a triangle.
• Apply the “Isosceles Triangle Theorem” to determine an unknown side length or angle.
• Apply the “Pythagorean Theorem” to figure out the side lengths in right triangles.

“For our geometry experience, we wanted to focus on a subset of problems that could have a large impact before eventually expanding to cover a more diverse range of geometry problems,” Wong added.

For the physics and geometry solver, Google leveraged its Multitask Unified Model (MUM), which understands specific search queries across various formats, including text, images and videos.

“Language and query understanding are complex challenges that require sophisticated systems to ensure we’re producing the most helpful results — especially in cases where our systems are looking at information in multiple different formats,” Wong explained. “In this case, these features need to understand information across both text and images to correctly interpret a problem and provide accurate step-by-step responses that include both text and diagrams.”

When asked about the accuracy rate for the new experience, the company declined to share specific numbers.

“Our testing shows the accuracy rate is very high,” Wong said.

In addition to problem-solving, Google also launched new 3D models on Search, letting you examine interactive diagrams related to nearly 1,000 topics, such as biology, chemistry, physics, astronomy and more. Google introduced 3D models in 2021, rolling out diagrams for more than 200 chemistry, biology, physics and anatomy concepts.

Over the years, the tech giant has launched multiple features that help you learn, including its interactive feature, “practice problems,” that tests your knowledge of high school topics like math, chemistry and physics. Google Search recently rolled out an English tutoring tool that provides prompts and asks you to say the answers out loud using a provided vocabulary word.

Google takes aim at Duolingo with new English tutoring tool
• Press corner

## More help with math and science problems in Search

Oct 30, 2023

We’re introducing new updates to Search and Lens to help learners solve trickier problems.

Whether you're delving into a math textbook or turning to Search to get more context on a complicated physics problem, it can sometimes be hard to describe exactly what you're looking for. Take that intricate biology concept or pesky geometry problem, for example. With new features across Search and Lens, you can now visualize STEM-related concepts and figure out which equation to use by browsing for them in a more natural and intuitive way.

## Get help with math, physics and geometry

It can feel frustrating when you are trying to solve a math problem and aren’t quite sure what the next step should be. Now, when you look up the problem on Search, you will be able to get even more help on subjects like trigonometry and calculus. Simply type your equation or integral into the Search bar, or take a picture with Lens, to see a step-by-step explanation and solution. You can also type " math solver " to give the experience a try on desktop and coming soon, on mobile.

With advancements in our large language models, you are now able to also get help with word problems , starting with many fundamental topics from high-school physics. This experience helps you identify the known and unknown values and understand which formulas to use and when.

Beyond calculus and physics problems, seeking geometry help on Search presents a unique challenge, because these problems are often harder to describe with words alone. Say you are tasked with finding the area of a triangle, and all you are given is a diagram of the triangle as well as measurements of two of the sides. Without being able to take a picture of the problem, it’s hard to put into words what you need help with. Now, instead of trying to come up with a written query that describes your visual problem, you can use Lens on certain triangle problems, to interpret both the visual and the text components of the problem, and generate a step-by-step explanation of how to solve it.

## Visually explore STEM concepts on Search

Developing a deep visual understanding of STEM-related topics is a key part of mastering the subject as a whole and may even spark your curiosity to want to learn more about similar topics . For example, you might have a general understanding of mitochondria but aren’t quite able to picture the inner folds that create the cristae or the matrix within the inner membrane. With 3D models on Search you will be able to explore interactive diagrams containing definitions and overviews for almost 1,000 biology, chemistry, physics, astronomy and related topics.

With these new and improved features, we hope that people are able to continue to feed their curiosity and learn a few new things along the way.

## New ways to get inspired with generative AI in Search

Let’s stay in touch. Get the latest news from Google in your inbox.

## Google can now solve trickier math problems for you with these new features

Math is a challenging subject because it requires an understanding of how to perform the operation to reach an answer, which makes it more difficult to Google an equation to find the answer difficult -- until now.

Google added new updates to Search and Lens that make it easier for users to get assistance when solving math problems. All users have to do now is type the equation or integral into the Search bar, or take a picture with Lens to get a step-by-step explanation or solution.

Also:  Chrome on iOS unveils a much-anticipated feature. Here's how to access it

To test out the experience for yourself, on desktop, you can type in an equation or type the term "Math Solver" on Google Search where you will be prompted to enter a math problem or select from the examples to see how it works. The math solver experience will be coming to mobile soon.

Lens can also be leveraged by users to take a photo of geometry triangle problems, solving the challenge of trying to put primarily visual problems into words.

Advancements in Google's large language models also give Search the capability to solve word problems.

All you have to do is type the problem into Search, where you will be met with steps that tell you how to solve the problem by identifying the known and unknown values and providing correct formulas.

Also: The AI I want to see in the world: 5 ways it could manage my Gmail inbox for me

Lastly, Google is also making it easier to explore STEM-related concepts on Search by including 3D models and interactive diagrams for almost 1,000 biology, chemistry, physics, astronomy, and related topics, according to Google.

For example, if you Google "mitochondrion" you will have the opportunity to click on and learn from an interactive diagram that provides an overview, as well as specific details about the individual parts.

## The Math Problem That Took Nearly a Century to Solve

Uc san diego mathematicians unlock the secret to ramsey numbers.

## Published Date

We’ve all been there: staring at a math test with a problem that seems impossible to solve. What if finding the solution to a problem took almost a century? For mathematicians who dabble in Ramsey theory, this is very much the case. In fact, little progress had been made in solving Ramsey problems since the 1930s.

Now, University of California San Diego researchers Jacques Verstraete and Sam Mattheus have found the answer to r(4,t), a longstanding Ramsey problem that has perplexed the math world for decades.

## What was Ramsey’s problem, anyway?

In mathematical parlance, a graph is a series of points and the lines in between those points. Ramsey theory suggests that if the graph is large enough, you’re guaranteed to find some kind of order within it — either a set of points with no lines between them or a set of points with all possible lines between them (these sets are called “cliques”). This is written as r(s,t) where s are the points with lines and t are the points without lines.

To those of us who don’t deal in graph theory, the most well-known Ramsey problem, r(3,3), is sometimes called “the theorem on friends and strangers” and is explained by way of a party: in a group of six people, you will find at least three people who all know each other or three people who all don’t know each other. The answer to r(3,3) is six.

“It’s a fact of nature, an absolute truth,” Verstraete states. “ It doesn't matter what the situation is or which six people you pick — you will find three people who all know each other or three people who all don't know each other. You may be able to find more, but you are guaranteed that there will be at least three in one clique or the other.”

{/exp:typographee}

Verstraete first saw r(4,t) in a book written by two UC San Diego professors.

What happened after mathematicians found that r(3,3) = 6? Naturally, they wanted to know r(4,4), r(5,5), and r(4,t) where the number of points that are not connected is variable. The solution to r(4,4) is 18 and is proved using a theorem created by Paul Erdös and George Szekeres in the 1930s.

Currently r(5,5) is still unknown.

## A good problem fights back

Why is something so simple to state so hard to solve? It turns out to be more complicated than it appears. Let’s say you knew the solution to r(5,5) was somewhere between 40-50. If you started with 45 points, there would be more than 10 234 graphs to consider!

“Because these numbers are so notoriously difficult to find, mathematicians look for estimations,” Verstraete explained. “ This is what Sam and I have achieved in our recent work. How do we find not the exact answer, but the best estimates for what these Ramsey numbers might be?”

The problem r(4,t) is a conjecture from Erdös, who offered $250 to the first person who could solve it. Math students learn about Ramsey problems early on, so r(4,t) has been on Verstraete’s radar for most of his professional career. In fact, he first saw the problem in print in Erdös on Graphs: His Legacy of Unsolved Problems, written by two UC San Diego professors, Fan Chung and the late Ron Graham. The problem is a conjecture from Erdös, who offered$250 to the first person who could solve it.

“Many people have thought about r(4,t) — it’s been an open problem for over 90 years,” Verstraete said. “But it wasn’t something that was at the forefront of my research. Everybody knows it's hard and everyone’s tried to figure it out, so unless you have a new idea, you’re not likely to get anywhere.”

Then about four years ago, Verstraete was working on a different Ramsey problem with a mathematician at the University of Illinois-Chicago, Dhruv Mubayi. Together they discovered that pseudorandom graphs could advance the current knowledge on these old problems.

In 1937, Erdös discovered that using random graphs could give good lower bounds on Ramsey problems. What Verstraete and Mubayi discovered was that sampling from pseudo random graphs frequently gives better bounds on Ramsey numbers than random graphs. These bounds — upper and lower limits on the possible answer — tightened the range of estimations they could make. In other words, they were getting closer to the truth.

In 2019, to the delight of the math world, Verstraete and Mubayi used pseudorandom graphs to solve r(3,t). However, Verstraete struggled to build a pseudorandom graph that could help solve r(4,t).

It took a pseudorandom graph from finite geometry, called an O'Nan configuration, to solve a longstanding Ramsey problem.

He began pulling in different areas of math outside of combinatorics, including finite geometry, algebra and probability. Eventually he joined forces with Mattheus, a postdoctoral scholar in his group whose background was in finite geometry.

“It turned out that the pseudorandom graph we needed could be found in finite geometry,” Verstraete stated. “Sam was the perfect person to come along and help build what we needed.”

Once they had the pseudorandom graph in place, they still had to puzzle out several pieces of math. It took almost a year, but eventually they realized they had a solution: r(4,t) is close to a cubic function of t . If you want a party where there will always be four people who all know each other or t people who all don’t know each other, you will need roughly t 3 people present. There is a small asterisk (actually an o) because, remember, this is an estimate, not an exact answer. But t 3 is very close to the exact answer.

The findings are currently under review with the Annals of Mathematics . A preprint can be viewed on arXiv.

“It really did take us years to solve,” Verstraete stated. “And there were many times where we were stuck and wondered if we’d be able to solve it at all. But one should never give up, no matter how long it takes.”

Verstraete emphasizes the importance of perseverance — something he reminds his students of often. “If you find that the problem is hard and you're stuck, that means it's a good problem. Fan Chung said a good problem fights back. You can't expect it just to reveal itself.”

Verstraete knows such dogged determination is well-rewarded: “I got a call from Fan saying she owes me $250.” ## You May Also Like Bacteria-virus arms race provides rare window into rapid and complex evolution, intentional avocado consumption may improve physical activity, model suggests that mammalian sperm cells have two modes of swimming, women produce skin temperature data that are just as predictable as men, stay in the know. Keep up with all the latest from UC San Diego. Subscribe to the newsletter today. You have been successfully subscribed to the UC San Diego Today Newsletter. ## Campus & Community Arts & culture. • Media Resources & Contacts Signup to get the latest UC San Diego newsletters delivered to your inbox. Award-winning publication highlighting the distinction, prestige and global impact of UC San Diego. Popular Searches: Covid-19 Ukraine Campus & Community Arts & Culture Voices ## After 90 Years, Mathematicians Finally Solved the Most Notorious Ramsey Problem “Everyone’s tried to figure it out,” says the scientist who actually did. • For 90 years, mathematicians have been stumped by a problem that attempts to explain how many people would need to be invited to a party so at least 4 people always know one another. • To finally crack the case, researchers from UC San Diego delved into the world of finite geometry and found a pseudograph to estimate an answer to the question . • The researchers hope that this approach could help estimate other mathematical functions that are still left unsolved. Okay ... let’s back up a little. Named after mathematician Frank Ramsey, who first proved his eponymous theorem in the 1920s, Ramsey problems seek to find order in disorder. Ramsey’s original theory says that if a graph is large enough—in mathematics , a graph is a collection of points and lines between those points—you can find order within all that chaos. Chiefly, the theorem states that a set of points will have no lines between them, or a set of points with all possible lines between them (aka cliques). This is expressed as r(s, t ), where s means “points with lines” and t means “points without lines.” ## More From Popular Mechanics While this is enough to make a non-mathematician dizzy, the concept is usually described as “the theorem on friends and strangers,” and is often allegorically explained as the idea that a party of six people will inevitably produce three people who know each other and three people who don’t. This is expressed as r(3,3), with the answer being six. (While I’m sure you have five close friends you can invite to a party, we’re talking about points and lines on a graph here—not actually people.) “It’s a fact of nature , an absolute truth,” Jacques Verstraete, whose groundbreaking finding is currently under review with the journal Annals of Mathematics , says in a press release . “It doesn’t matter what the situation is or which six people you pick—you will find three people who all know each other or three people who all don't know each other. You may be able to find more, but you are guaranteed that there will be at least three in one clique or the other.” Expanding this idea, mathematicians Paul Erdös and George Szekeres discovered that r(4,4) equals 18 in 1935, and the solution to r(5,5) is still unknown. However, Verstraete and UC San Diego researcher Sam Mattheus have found an estimated solution for r(4, t ), where t means that “points without lines” is variable. “Because these numbers are so notoriously difficult to find, mathematicians look for estimations ,” Verstraete said in the press statement. “This is what Sam and I have achieved in our recent work. How do we find not the exact answer, but the best estimates for what these Ramsey numbers might be?” Verstraete first saw the problem stated in a book about Erdös written by two UC San Diego researchers. These problems are usually solved using random graphs, but this Ramsey problem required more out-of-the-box thinking. Using a concept known as pseudorandom graphs, Verstaraete discovered a solution for the r(3, t ) in 2019 . But the technique hit a brick wall when applied to the notoriously tricky r(4, t ) problem, so Verstraete ventured into other areas of mathematics—specifically, finite geometry —to figure out the answer. “Many people have thought about r(4, t )—it’s been an open problem for over 90 years ,” Verstraete said in a press statement. “But it wasn’t something that was at the forefront of my research. Everybody knows it’s hard and everyone’s tried to figure it out, so unless you have a new idea, you’re not likely to get anywhere.” Drawing on Mattheus’s expertise in the field, the researchers discovered a pseudograph in finite geometry called an O’Nan configuration that worked for their problem. After a year of fine-tuning the math, Verstraete found that the answer was very close to a cubic function of t . This means that if you want a party where there will always be four people who know one another then, you need to invite t 3 people who don’t know each other. Reminder: this is just a close estimate, not an exact answer. Now that it has produced a close estimate, Verstraete hopes that the technique could help open up new mathematical frontiers that are still waiting to be explored. Darren lives in Portland, has a cat, and writes/edits about sci-fi and how our world works. You can find his previous stuff at Gizmodo and Paste if you look hard enough. ## .css-8psjmo:before{content:'//';display:inline;padding-right:0.3125rem;} Math .css-v6ym3h:before{content:'//';display:inline;padding-left:0.3125rem;} Why Some People Think 2+2=5 Get a Die to Roll Any Way You Want…With Math! Scientists Create Quantum 'Alice Rings' Scientists Find Boat Load of Three-Body Solutions Solution to Riddle of the Week #6 AI Is Helping Build the 'Periodic Table of Shapes" Solution to Riddle of the Week #53 Forget 50/50, Coin Tosses Have a Bias Solution to Riddle of the Week #52 How M.C. Escher Created His Mathematical Artwork Scientists Solve 50-Year-Old Möbius Mystery ## Easy Finger Math Tricks to Help Kids Solve Problems Posted: November 9, 2023 | Last updated: November 9, 2023 While using your fingers isn't the fastest way to recall a multiplication fact while doing a problem, finger math tricks can help kids figure out how to answer the problem at hand — and as they work on their math, they will eventually learn all the facts by repetition. Note that before your child can understand other finger tricks, they must be able to count by 2s, 5s, and 10s and multiply by 2s, 3s, and 4s. ## Quick Finger Math Tricks for Threes and Fours The tricks for multiplying by threes and fours are really a matter of counting out the answer on your fingers. As your children count out the answer repeatedly, they'll memorize it and then be able to move on to larger numbers. ## Multiplying by Three Did you realize that all of your fingers have three segments? Therefore, you can figure out anything from 3 x 1 to 3 x 10 by counting the segments on each finger. To start: • Hold up the number of fingers you're going to multiply by 3. For example, if the problem is 3 x 4 — hold up four fingers. • Count each segment on each finger you're holding up, and you should come up with 12 — which is the correct answer. ## Multiplying by Four Multiplying by four is the same as multiplying by two — twice. To start: • Hold up the number of fingers to correspond with the number you are multiplying by four. For example, if you are multiplying 4 x 6 — hold up six fingers. • Count each finger by two, moving from left to right. Then count each finger again, continuing to count by twos, until you've counted every finger twice. Helpful Hack To keep track of the fingers you've counted twice, sometimes it's easier to put your finger down as you count the first time, and back up as you count the second time. ## Finger Math Tricks for Multiplying by 6, 7, 8, and 9 While numbers one through five are easy for most kids to remember, six and up often pose a problem. This handy trick will make it a little easier to work those problems out. ## Multiplying 6, 7, 8, and 9 by Hand To begin, assign each finger a number. For example, your thumbs represent 6, your index fingers each represent 7, etc. This will remain the same throughout the finger math hack. Your left hand will represent the first number that you are multiplying and your right hand will represent the second number you are multiplying. In this example, we are multiplying 7 x 8. To Determine the Part of Your Answer: • On your left hand, put down the finger that represents the number you are multiplying as well as any fingers whose number value is less than this figure. In this example, you are multiplying 7 x 8, so the left hand will represent 7. You will drop your index finger (number 7) and your thumb (number 6). • Similarly, the right hand will represent eight, so you will drop down your middle finger (number 8), your index finger (number 7), and your thumb (number 6). • Now, just multiply the fingers that are still pointed upwards. In this case, you will have three fingers on your left hand and two on your right, so you will multiply 3 x 2 to get 6. This is the first part of your answer! To Determine the Second Part of Your Answer: • Keeping your fingers in the same positions, count how many fingers are folded down. In the 7 x 8 example, you should have five fingers folded. • You will count each of these in quantities of ten. So, 10, 20, 30, 40, 50. • 50 is your answer. To Determine Your Final Answer: • Add your two numbers together. In this example, you would add 6 + 50, which gives you 56! ## Another Finger Math Trick Just for Nine There is a trick that works separately, just for multiplying by the number nine. • To start, hold up all ten fingers, with your palms facing you. • Assign each finger a number, starting with your left-hand thumb and ending with your right-hand thumb. The left-hand thumb will be one, the left-hand index finger will be two, and so on until you reach the number 10 for your right-hand thumb. • To tackle a problem, put down the corresponding finger of the number you're multiplying by nine. For example, if you are multiplying 9 x 8, you'd put down the eighth finger (which will be on your right hand). • Count all the fingers to the left of the finger you have folded down. This will give you 7. This is the first digit of your answer. • Count all the fingers to the right of the finger you have folded down. This will give you 2. This is the second digit of your answer. • Put the numbers together! Your answer is 72. ## Finger Multiplication Tricks Can Make Math Easy and Fun While the hope is that your kids will eventually memorize their multiplication charts , using some quick hand tricks for multiplication and letting them count things out on their fingers is not a bad way to learn. It keeps frustration at bay since the answer is always a fingertip away, and the repetition of having to figure it out will help cement those facts into their brains. ## More for You James Comer's Biden Subpoenas Backfire A Scientist Says the Singularity Will Happen by 2031 Risky intercepts show China's fighter pilots are 'willing to challenge' the US as their planes increasingly meet in the air, former aviator says Jury in Robert De Niro trial reaches verdict, actor’s company liable for$1.2M in damages

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