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20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

Math strategies for problem-solving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.  

Problem-solving skills are essential to math in the general classroom and real-life. They require logical reasoning and critical thinking skills. Students must be equipped with strategies to help them find solutions to problems.

This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in real-life situations. 

What are problem-solving strategies?

Problem-solving strategies in math are methods students can use to figure out solutions to math problems. Some problem-solving strategies: 

• Draw a model
• Use different approaches
• Check the inverse to make sure the answer is correct

Students need to have a toolkit of math problem-solving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better. 

Strategies can help guide students to the solution when it is difficult ot know when to start.

The ultimate guide to problem solving techniques

Download these ready-to-go problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.

20 Math Strategies For Problem-Solving

Different problem-solving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem. 

Here are 20 strategies to help students develop their problem-solving skills. 

Strategies to understand the problem

Strategies that help students understand the problem before solving it helps ensure they understand: 

• The context
• What the key information is
• How to form a plan to solve it

Following these steps leads students to the correct solution and makes the math word problem easier .

Here are five strategies to help students understand the content of the problem and identify key information. 

1. Read the problem aloud

Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.

2. Highlight keywords 

When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed.  For example, if the word problem asks how many are left, the problem likely requires subtraction.  Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.

3. Summarize the information

Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary.  Summaries should include only the important information and be in simple terms that help contextualize the problem.

4. Determine the unknown

A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer.  Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.

5. Make a plan

Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it.  The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer.  Encourage students to make a list of each step they need to take to solve the problem before getting started.

Strategies for solving the problem 

1. draw a model or diagram.

Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process.  It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.

Similarly, you could draw a model to represent the objects in the problem:

2. Act it out

This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives .  When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts.  The examples below show how 1st-grade students could “act out” an addition and subtraction problem:

3. Work backwards

Working backwards is a popular problem-solving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution.  This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.

For example,

To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71.  Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.

4. Write a number sentence

When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved.  It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.

5. Use a formula

Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve.   Make sure students know the important mathematical formulas they will need in tests and real-life. It can help to display these around the classroom or, for those who need more support, on students’ desks.

Strategies for checking the solution 

Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense. 

There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work. 

2. Estimate to check for reasonableness

Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable.  Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten.  For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10.  When the estimate is clear the two numbers are close. This means your answer is reasonable. 

3. Plug-In Method

This method is particularly useful for algebraic equations. Specifically when working with variables.  To use the plug-in method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.

If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓

4. Peer Review

Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly.   Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixed-ability partners or similar-ability partners. In mixed-ability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills.  If partners have comparable ability levels and problem-solving skills, they may find that they approach problems differently or have unique insights to offer each other about the problem-solving process.

5. Use a Calculator

A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to real-world situations.  

Step-by-step problem-solving processes for your classroom

In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4-step process to solve problems. 

Polya’s 4 steps include:

• Understand the problem
• Devise a plan
• Carry out the plan

Today, in the style of George Polya, many problem-solving strategies use various acronyms and steps to help students recall. 

Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom. 

Here are 5 problem-solving strategies to introduce to students and use in the classroom.

How Third Space Learning improves problem-solving 

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Third Space Learning offers a free resource library is filled with hundreds of high-quality resources. A team of experienced math experts carefully created each resource to develop students mental arithmetic, problem solving and critical thinking. 

Explore the range of problem solving resources for 2nd to 8th grade students. 

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Third Space Learning offers one-on-one math tutoring to help students improve their math skills. Highly qualified tutors deliver high-quality lessons aligned to state standards. 

Former teachers and math experts write all of Third Space Learning’s tutoring lessons. Expertly designed lessons follow a “my turn, follow me, your turn” pedagogy to help students move from guided instruction and problem-solving to independent practice. 

Throughout each lesson, tutors ask higher-level thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problem-solving skills.

Problem-solving

Educators can use many different strategies to teach problem-solving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra. 

Teaching students how to choose and implement problem-solving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to real-life problem-solving.

READ MORE : 8 Common Core math examples

There are many different strategies for problem-solving; Here are 5 problem-solving strategies: • draw a model  • act it out  • work backwards  • write a number sentence • use a formula

Here are 10 strategies of problem-solving: • Read the problem aloud • Highlight keywords • Summarize the information • Determine the unknown • Make a plan • Draw a model  • Act it out  • Work backwards  • Write a number sentence • Use a formula

1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back

Some strategies you can use to solve challenging math problems are: breaking the problem into smaller parts, using diagrams or models, applying logical reasoning, and trying different approaches.

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Differentiated Instruction: 9 Differentiated Curriculum And Instruction Strategies For Teachers 

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• 6th Grade Percent Word Problems
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Math Problems By Topic

Addition problems, subtraction problems, multiplication problems, division problems.

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Percentage problems, ratio problems, finding all possibilities problems.

Here you will find a range of addtion word problems to help your child apply their addition facts.

The worksheets cover addition problems from 1st to 3rd grade.

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Here you will find a range of subtraction word problems to help your child apply their subtraction facts.

The worksheets cover subtraction problems from 1st to 3rd grade.

These sheets involve solving a range of subtraction word problems up to 100.

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Addition & Subtraction Problems

Here you will find a range of addition and subtraction word problems to help your child apply their knowledge.

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The worksheets cover multiplication problems from 2nd to 5th grade.

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The worksheets cover division problems from 3rd to 5th grade.

Inequalities Word Problems

These sheets involve changing a word problem into an inequality.

Here you will find a range of fraction word problems to help your child apply their fraction learning.

The worksheets cover a range of fraction objectives, from adding and subtracting fractions to working out fractions of numbers. The sheets support fraction learning from 2nd grade to 5th grade.

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Here you will find a range of ratio word problems to help your child understand what a ratio is and how ratios work.

The sheets support ratio learning at a 5th grade level.

This is our finding all possibilities area where all the worksheets involve finding many different answers to the problem posed.

The sheets here encourage systematic working and logical thinking.

The problems are different in that, there is typically only one problem per sheet, but the problem may take quite a while to solve!

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Number Line

• \mathrm{Lauren's\:age\:is\:half\:of\:Joe's\:age.\:Emma\:is\:four\:years\:older\:than\:Joe.\:The\:sum\:of\:Lauren,\:Emma,\:and\:Joe's\:age\:is\:54.\:How\:old\:is\:Joe?}
• \mathrm{Kira\:went\:for\:a\:drive\:in\:her\:new\:car.\:She\:drove\:for\:142.5\:miles\:at\:a\:speed\:of\:57\:mph.\:For\:how\:many\:hours\:did\:she\:drive?}
• \mathrm{The\:sum\:of\:two\:numbers\:is\:249\:.\:Twice\:the\:larger\:number\:plus\:three\:times\:the\:smaller\:number\:is\:591\:.\:Find\:the\:numbers.}
• \mathrm{If\:2\:tacos\:and\:3\:drinks\:cost\:12\:and\:3\:tacos\:and\:2\:drinks\:cost\:13\:how\:much\:does\:a\:taco\:cost?}
• \mathrm{You\:deposit\:3000\:in\:an\:account\:earning\:2\%\:interest\:compounded\:monthly.\:How\:much\:will\:you\:have\:in\:the\:account\:in\:15\:years?}
• How do you solve word problems?
• To solve word problems start by reading the problem carefully and understanding what it's asking. Try underlining or highlighting key information, such as numbers and key words that indicate what operation is needed to perform. Translate the problem into mathematical expressions or equations, and use the information and equations generated to solve for the answer.
• How do you identify word problems in math?
• Word problems in math can be identified by the use of language that describes a situation or scenario. Word problems often use words and phrases which indicate that performing calculations is needed to find a solution. Additionally, word problems will often include specific information such as numbers, measurements, and units that needed to be used to solve the problem.
• Is there a calculator that can solve word problems?
• Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems.
• What is an age problem?
• An age problem is a type of word problem in math that involves calculating the age of one or more people at a specific point in time. These problems often use phrases such as 'x years ago,' 'in y years,' or 'y years later,' which indicate that the problem is related to time and age.

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Mathematicians Are Edging Close to Solving One of the World's 7 Hardest Math Problems

And there’s $1 million at stake.

• In new research, mathematicians have narrowed down one of the biggest outstanding problems in math.
• Huge breakthroughs in math and science are usually the work of many people over many years.
• Seven math problems were given a $1 million bounty each in 2000, and just one has been solved so far.

The “Millennium Problems” are seven infamously intractable math problems laid out in the year 2000 by the prestigious Clay Institute, each with $1 million attached as payment for a solution. They span all areas of math , as the Clay Institute was founded in 1998 to push the entire field forward with financial support for researchers and important breakthroughs.

But the only solved Millennium Problem so far, the Poincare conjecture, illustrates one of the funny pitfalls inherent to offering a large cash prize for math. The winner, Grigori Perelman, refused the Clay prize as well as the prestigious Fields Medal. He withdrew from mathematics and public life in 2006, and even in 2010, he still insisted his contribution was the same as the mathematician whose work laid the foundation on which he built his proof, Richard Hamilton.

Math, all sciences, and arguably all human inquiries are filled with pairs or groups that circle the same finding at the same time until one officially makes the breakthrough. Think about Sir Isaac Newton and Gottfried Leibniz, whose back-and-forth about calculus led to the combined version of the field we still study today. Rosalind Franklin is now mentioned in the same breath as her fellow discoverers of DNA, James Watson and Francis Crick. Even the Bechdel Test for women in media is sometimes called the Bechdel-Wallace Test, because humans are almost always in collaboration.

That’s what makes this new paper so important. Two mathematicians—Larry Guth of the Massachusetts Institute of Technology (MIT) and James Maynard of the University of Oxford—collaborated on the new finding about how certain polynomials are formed and how they reach out into the number line. Maynard is just 37, and won the Fields Medal himself in 2022. Guth, a decade older, has won a number of important prizes with a little less name recognition.

The Riemann hypothesis is not directly related to prime numbers , but it has implications that ripple through number theory in different ways (including with prime numbers). Basically, it deals with where and how the graph of a certain function of complex numbers crosses back and forth across axes. The points where the function crosses an axis is called a “zero,” and the frequency with which those zeroes appear is called the zero density.

In the far reaches of the number line, prime numbers become less and less predictable (in the proverbial sense). They are not, so far, predictable in the literal sense—a fact that is an underpinning of modern encryption , where data is protected by enormous strings of integers made by multiplying enormous prime numbers together. The idea of a periodic table of primes, of any kind of template that could help mathematicians better understand where and how large primes cluster together or not, is a holy grail.

In the new paper, Maynard and Guth focus on a new limitation of Dirichlet polynomials. These are special series of complex numbers that many believe are of the same type as the function involved in the Riemann hypothesis involves. In the paper, they claim they’ve proven that these polynomials have a certain number of large values, or solutions , within a tighter range than before.

In other words, if we knew there might be an estimated three Dirichlet values between 50 and 100 before, now we may know that range to be between 60 and 90 instead. The eye exam just switched a blurry plate for a slightly less blurry one, but we still haven’t found the perfect prescription. “If one knows some more structure about the set of large values of a Dirichlet polynomial, then one can hope to have improved bound,” Maynard and Guth conclude.

No, this is not a final proof of the Riemann hypothesis. But no one is suggesting it is. In advanced math, narrowing things down is also vital. Indeed, even finding out that a promising idea turns out to be wrong can have a lot of value—as it has a number of times in the related Twin Primes Conjecture that still eludes mathematicians.

In a collaboration that has lasted 160 years and counting, mathematicians continue to take each step together and then, hopefully, compare notes.

Caroline Delbert is a writer, avid reader, and contributing editor at Pop Mech. She's also an enthusiast of just about everything. Her favorite topics include nuclear energy, cosmology, math of everyday things, and the philosophy of it all. 

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How to Score More in Math

Mathematics often evokes mixed feelings among students. While some find joy in solving problems, others struggle with anxiety and frustration. As a parent, you play a pivotal role in shaping your child’s attitude towards math and helping them achieve academic success. This guide provides practical strategies you can implement to support your child’s math learning journey, with a focus on application-based learning, problem reframing, and breaking down complex problems into manageable steps. By encouraging a positive mindset and leveraging innovative educational tools, you can make math a rewarding experience for your child.

Encourage Application-Based Learning

Application-based learning is an effective approach to make math more relatable and engaging for students. Here’s how you can foster this type of learning at home:

• Show how math is used in everyday life, such as budgeting, cooking, or planning trips.
• Discuss how professionals like engineers, architects, and scientists use math in their work.
• Encourage your child to participate in projects that involve mathematical concepts, such as building a model, creating a budget for a family event, or designing a simple game.
• Use household items to demonstrate mathematical principles, like measuring ingredients for a recipe to teach fractions.

Reframe Math Problems

Reframing math problems can help reduce anxiety and make challenging concepts more approachable. Here are some strategies to help your child view math problems from a different perspective:

• Encourage your child to approach problems in various ways. If they are stuck, suggest they try a different method or look at the problem from a new angle.
• Use visual aids, such as graphs, diagrams, or physical objects, to represent abstract concepts.

Break Down Problems into Manageable Steps

Complex problems can be intimidating, but breaking them down into smaller steps can simplify the process and enhance understanding. Here’s how to guide your child in this approach:

• Help your child clearly define what the problem is asking. Encourage them to write down what they know and what they need to find out.
• Discuss the problem and ask guiding questions to ensure they understand the requirements.
• Work with your child to brainstorm different strategies for solving the problem. Discuss which method might be the most effective.
• Encourage them to choose a strategy and outline the steps they need to take.
• Have your child follow the steps to solve the problem, providing guidance and support as needed.
• Once they have an answer, encourage them to review their solution and check their work for accuracy.

Practice Makes Perfect:

Math is a subject that requires consistent practice to solidify understanding and build problem-solving fluency. Here are some ways to encourage your child to practice regularly:

• Set aside dedicated practice time: Schedule regular practice sessions into your child’s routine. Short, daily practice sessions can be more effective than cramming right before a test.
• Find a variety of practice problems: Don’t just rely on the problems assigned in class. Look for worksheets, online resources, or math workbooks that offer a range of difficulty levels and question types.
• Focus on understanding over speed: Encourage your child to take their time and focus on understanding the concepts behind the problems rather than rushing to get the answer.
• Review mistakes: Analyze mistakes together and identify areas where your child needs more practice.

Leveraging BYJU’S Offerings for Math Success

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BYJU’S offers many such resources to help your child prepare for their exams. When every chapter is covered in such detail, it helps your child to learn and not worry about their exams.

BYJU’S has many such offerings for different grades. Here are the three main ways in which we deliver content:

BYJU’S The Learning App

BYJU’S The Learning App is designed to foster a love for learning through innovative methods:

• Dynamic Visuals: The app uses high-quality videos and interactive animations to simplify complex mathematical theories and concepts.
• Adaptive Learning Pathways: It tailors the learning experience to match your child’s pace, ensuring they grasp fundamental concepts before progressing to advanced topics.
• Extensive Practice Modules: With a wide range of practice questions and quizzes, the app helps reinforce learning and test knowledge effectively.

BYJU’S Tuition Centres (Hybrid)

If your child prefers a mix of offline and online learning, BYJU’S Tuition Centres offer a hybrid model that combines the best of both worlds:

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• Hybrid Learning Model: Flexibility to attend classes either in person or online allows for convenience and comfort.
• Interactive Learning Environment: Engaging classroom activities and practical experiments enhance understanding and retention of mathematical concepts.

BYJU’S Tuitions (Online)

For those who prefer online learning, BYJU’S Tuitions provide comprehensive support from the comfort of home:

• Live Interactive Sessions: Real-time classes enable your child to ask questions and clarify doubts instantly.
• Recorded Lectures: Access to 24/7 recorded lectures allows your child to revisit and review concepts at their own pace.
• Assessments: Regular assessments help gauge your child’s understanding and provide opportunities to reinforce their learning.

Helping your child succeed in math requires a combination of encouragement, practical strategies, and the right resources. By promoting application-based learning, reframing math problems, and breaking down complex problems into manageable steps, you can create a positive and effective learning environment at home. Additionally, leveraging BYJU’S offerings can provide your child with the tools and support they need to excel in math and develop a lifelong understanding of the subject. Embrace these strategies and watch your child’s confidence and competence in math soar.

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July 1, 2024

The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved

Number theorists have been trying to prove a conjecture about the distribution of prime numbers for more than 160 years

By Manon Bischoff

Weiquan Lin/Getty Images

The Riemann hypothesis is the most important open question in number theory—if not all of mathematics. It has occupied experts for more than 160 years. And the problem appeared both in mathematician David Hilbert’s groundbreaking speech from 1900 and among the “Millennium Problems” formulated a century later. The person who solves it will win a million-dollar prize.

But the Riemann hypothesis is a tough nut to crack. Despite decades of effort, the interest of many experts and the cash reward, there has been little progress. Now mathematicians Larry Guth of the Massachusetts Institute of Technology and James Maynard of the University of Oxford have posted a sensational new finding on the preprint server arXiv.org. In the paper, “the authors improve a result that seemed insurmountable for more than 50 years,” says number theorist Valentin Blomer of the University of Bonn in Germany.

Other experts agree. The work is “a remarkable breakthrough,” mathematician and Fields Medalist Terence Tao wrote on Mastodon , “though still very far from fully resolving this conjecture.”

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The Riemann hypothesis concerns the basic building blocks of natural numbers: prime numbers, values greater than 1 that are only divisible by 1 and themselves. Examples include 2, 3, 5, 7, 11, 13, and so on.*

Every other number, such as 15, can be clearly broken down into a product of prime numbers: 15 = 3 x 5. The problem is that the prime numbers do not seem to follow a simple pattern and instead appear randomly among the natural numbers. Nineteenth-century German mathematician Bernhard Riemann proposed a way to deal with this peculiarity that explains how prime numbers are distributed on the number line—at least from a statistical point of view.

A Periodic Table for Numbers

Proving this conjecture would provide mathematicians with nothing less than a kind of “periodic table of numbers.” Just as the basic building blocks of matter (such as quarks, electrons and photons) help us to understand the universe and our world, prime numbers also play an important role, not just in number theory but in almost all areas of mathematics.

There are now numerous theorems based on the Riemann conjecture. Proof of this conjecture would prove many other theorems as well—yet another incentive to tackle this stubborn problem.

Interest in prime numbers goes back thousands of years. Euclid proved as early as 300 B.C.E. that there are an infinite number of prime numbers. And although interest in prime numbers persisted, it was not until the 18th century that any further significant findings were made about these basic building blocks.

As a 15-year-old, physicist Carl Friedrich Gauss realized that the number of prime numbers decreases along the number line. His so-called prime number theorem (not proven until 100 years later) states that approximately n / ln( n ) prime numbers appear in the interval from 0 to n . In other words, the prime number theorem offers mathematicians a way of estimating the typical distribution of primes along a chunk of the number line.

The exact number of prime numbers may differ from the estimate given by the theorem, however. For example: According to the prime number theorem, there are approximately 100 / ln(100) ≈ 22 prime numbers in the interval between 1 and 100. But in reality there are 25. There is therefore a deviation of 3. This is where the Riemann hypothesis comes in. This hypothesis gives mathematicians a way to estimate the deviation. More specifically, it states that this deviation cannot become arbitrarily large but instead must scale at most with the square root of n , the length of the interval under consideration.

The Riemann hypothesis therefore does not predict exactly where prime numbers are located but posits that their appearance on the number line follows certain rules. According to the Riemann hypothesis, the density of primes decreases according to the prime number theorem, and the primes are evenly distributed according to this density. This means that there are no large areas in which there are no prime numbers at all, while others are full of them.

You can also imagine this idea by thinking about the distribution of molecules in the air of a room: the overall density on the floor is somewhat higher than on the ceiling, but the particles—following this density distribution—are nonetheless evenly scattered, and there is no vacuum anywhere.

A Strange Connection

Riemann formulated the conjecture named after him in 1859, in a slim, six-page publication (his only contribution to the field of number theory). At first glance, however, his work has little to do with prime numbers.

He dealt with a specific function, the so-called zeta function ζ( s ), an infinitely long sum that adds the reciprocal values of natural numbers that are raised to the power of s :

Even before Riemann’s work, experts knew that such zeta functions are related to prime numbers. Thus, the zeta function can also be expressed as a function of all prime numbers p as follows:

Riemann recognized the full significance of this connection with prime numbers when he used not only real values for s but also complex numbers. These numbers contain both a real part and roots from negative numbers, the so-called imaginary part.

You can imagine complex numbers as a two-dimensional construct. Rather than mark a point on the number line, they instead lie on the plane. The x coordinate corresponds to the real part and the y coordinate to the imaginary part:

Никита Воробьев/Wikimedia

The complex zeta function that Riemann investigated can be visualized as a landscape above the plane. As it turns out, there are certain points amid the mountains and valleys that play an important role in relation to prime numbers. These are the points at which the zeta function becomes zero (so-called zeros), where the landscape sinks to sea level, so to speak.

The colors represent the values of the complex zeta function, with the white dots indicating its zeros.

Jan Homann/Wikimedia

Riemann quickly found that the zeta function has no zeros if the real part is greater than 1. This means that the area of the landscape to the right of the straight line x = 1 never sinks to sea level. The zeros of the zeta function are also known for negative values of the real part. They lie on the real axis at x = –2, –4, –6, and so on. But what really interested Riemann—and all mathematicians since—were the zeros of the zeta function in the “critical strip” between 0 ≤ x ≤ 1.

In the critical strip (dark blue), the Riemann zeta function can have “nontrivial” zeros. The Riemann conjecture states that these are located exclusively on the line x = 1/2 (dashed line).

LoStrangolatore/Wikimedia ( CC BY-SA 3.0 )

Riemann knew that the zeta function has an infinite number of zeros within the critical strip. But interestingly, all appear to lie on the straight line x = 1 / 2 . Thus Riemann hypothesized that all zeros of the zeta function within the critical strip have a real part of x = 1 / 2 . That statement is actually at the crux of understanding the distribution of prime numbers. If correct, then the placement of prime numbers along the number line never deviates too much from the prime number set.

On the Hunt for Zeros

To date, billions and billions of zeta function zeros have now been examined— more than 10 13 of them —and all lie on the straight line x = 1 / 2 .

But that alone is not a valid proof. You would only have to find a single zero that deviates from this scheme to disprove the Riemann hypothesis. Therefore we are looking for a proof that clearly demonstrates that there are no zeros outside x = 1 / 2 in the critical strip.

Thus far, such a proof has been out of reach, so researchers took a different approach. They tried to show that there is, at most, a certain number N of zeros outside this straight line x = 1 / 2 . The hope is to reduce N until N = 0 at some point, thereby proving the Riemann conjecture. Unfortunately, this path also turns out to be extremely difficult. In 1940 mathematician Albert Ingham was able to show that between 0.75 ≤ x ≤ 1 there are at most y 3/5+ c zeros with an imaginary part of at most y , where c is a constant between 0 and 9.

In the following 80 years, this estimation barely improved. The last notable progress came from mathematician Martin Huxley in 1972 . “This has limited us from doing many things in analytic number theory,” Tao wrote in his social media post . For example, if you wanted to apply the prime number theorem to short intervals of the type [ x , x + x θ ], you were limited by Ingham’s estimate to θ > 1 / 6 .

Yet if Riemann’s conjecture is true, then the prime number theorem applies to any interval (or θ = 0), no matter how small (because [ x , x + x θ ] = [ x , x + 1] applies to θ = 0).

Now Maynard, who was awarded the prestigious Fields Medal in 2022 , and Guth have succeeded in significantly improving Ingham’s estimate for the first time. According to their work, the zeta function in the range 0.75 ≤ x ≤ 1 has at most y (13/25)+ c zeros with an imaginary part of at most y . What does that mean exactly? Blomer explains: “The authors show in a quantitative sense that zeros of the Riemann zeta function become rarer the further away they are from the critical straight line. In other words, the worse the possible violations of the Riemann conjecture are, the more rarely they would occur.”

“This propagates to many corresponding improvements in analytic number theory,” Tao wrote . It makes it possible to reduce the size of the intervals for which the prime number theorem applies. The theorem is valid for [ x , x + x 2/15 ], so θ > 1 / 6 = 0.166... becomes θ > 2 ⁄ 15 = 0.133...

For this advance, Maynard and Guth initially used well-known methods from Fourier analysis for their result. These are similar techniques to what is used to break down a sound into its overtones. “The first few steps are standard, and many analytic number theorists, including myself, who have attempted to break the Ingham bound, will recognize them,” Tao explained . From there, however, Maynard and Guth “do a number of clever and unexpected maneuvers,” Tao wrote.

Blomer agrees. “The work provides a whole new set of ideas that—as the authors rightly say—can probably be applied to other problems. From a research point of view, that’s the most decisive contribution of the work,” he says.

So even if Maynard and Guth have not solved Riemann’s conjecture, they have at least provided new food for thought to tackle the 160-year-old puzzle. And who knows—perhaps their efforts hold the key to finally cracking the conjecture.

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

*Editor’s Note (7/9/24): This sentence was edited after posting to better clarify that prime numbers exclude 1.

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The Supreme Court Is Not Done Remaking America

Some of the rulings that came before the justices’ decision on presidential immunity could prove to have just as big an impact..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From “The New York Times,” I’m Michael Barbaro. This is “The Daily.”

When the Supreme Court wrapped up its term last week, much of the focus was on the ruling that gave Donald Trump sweeping immunity from criminal prosecution. But as my colleague Adam Liptak explains, a set of rulings that generated far less attention could have just as big an impact on American government and society.

It’s Monday, July 8.

Adam, welcome back. It hasn’t been very long, but we want to talk to you about the rest of the Supreme Court’s decisions that happened over the past few weeks, the rest meaning the non-Trump decisions. There were a lot of other cases, many of which we covered on the show over the past year, but we haven’t yet talked about where the justices landed as they issued their rulings on these cases over the past few weeks. So I wonder if you can walk us through some of the bigger decisions and what, taken as a whole, this entire term really means. So where should we start?

Well, this term had so many major cases, Michael, on so many important issues touching all aspects of American politics and society, that it’s a little hard to know where to start. But I think one way to think about the term is to ask, how much is this a 6-3 court? There are six conservatives in the majority, the three liberal justices in dissent. Are we going to get that kind of classic lineup time after time after time?

And one way to start answering that question is to look at two areas which are kind of part of the court’s greatest hits, areas where they’ve done a lot of work in the last few terms — guns and abortion.

OK, let’s start with guns.

The court had two big guns cases. One of them involved the Second Amendment and broke 8 to 1 against Second Amendment rights. Only Justice Clarence Thomas, the most avid supporter of gun rights, was in dissent. So let me tell you just a little bit about this case.

There’s a federal law that says people subject to domestic violence restraining orders, it’s a crime for them to have guns. A guy named Zackey Rahimi was subject to such a domestic violence restraining order, but he goes to court and says, this law violates my Second Amendment rights. The Second Amendment protects me and allows me to have a gun even if I’m in this status.

And that goes to the Supreme Court. And the way the Supreme Court analyzes this question is it looks to a test that it established only a couple of years ago, in 2022, which said you judge the constitutionality of gun control laws using history. You kind of go back in time and you see whether the community and the founding era disarmed people in the same way that the current law disarms people.

And you might think that actually, back in the 1700s, there were no such things as domestic violence restraining orders. So you might think that the answer is, this contemporary law is unconstitutional. But Chief Justice John Roberts, writing for an eight-justice majority, says, no, that’s not quite right. We’re going to kind of roll back the specificity of the test and look at very general principles. Can you disarm dangerous people back then? And if you can do that, then you can disarm Rahimi, even under this law that the founding generation could not have contemplated.

That’s really interesting. So the court, its conservative majority especially, seems to be saying that our last big decision made it too hard to regulate guns. We need to fix that. So we’re going to search really hard for a way to make sure that somebody with a restraining order for domestic abuse can’t legally have a gun.

Right. On the other hand, there was a second guns case, not involving the Second Amendment, but posing an important issue. The question in the case was whether the Trump administration was allowed to enact a gun control regulation in 2017 after the Las Vegas shooting in which, at an outdoor music festival, a gunman killed 58 people, wounded 500 more.

And the Trump administration, prompted by this massacre, they issue a regulation that tries to outlaw bump stocks. What are bump stocks? They’re devices that turn semi-automatic weapons into weapons that can fire at rates approaching a machine gun. And drawing on the authority of a 1934 law which bans, for the most part, civilian ownership of machine guns, it said bump stocks are basically the same thing, and we will, by regulation, outlaw them.

And the question for the court was, did the 1934 law authorize that? And here — and this is a typical split on this kind of stuff — the majority, the conservative majority, takes a textualist approach. It bears down on the particular words of the statute. And Justice Thomas looks at the words that Congress said a machine gun is one where a single function of the trigger causes all of these bullets to fly. And a bump stock, he said, is not precisely that. Therefore, we’re going to strike down this regulation.

So how do you reconcile these two divergent gun rulings, one where the court works really hard to allow for gun restrictions in the case of domestic abusers, and another where they seem to have no compunction about allowing for a bump stock that I think most of us, practically speaking, understand as making a semiautomatic weapon automatic in the real world?

I think the court draws a real distinction between two kinds of cases. One is about interpreting the Constitution, interpreting the Second Amendment. And in that area, it is plowing new ground. It has issued maybe four major Second Amendment cases, and it’s trying to figure out how that works and what the limits are. And the Rahimi case shows you that they’re still finding their way. They’re trying to find the right balance in that constitutional realm where they are the last word.

The bump stocks case doesn’t involve the Constitution. It involves an interpretation of a statute enacted by Congress. And the majority, in those kinds of cases, tends to read statutes narrowly. And they would say that that’s acceptable because unlike in a constitutional case, if it’s about a congressional statute, Congress can go back and fix it. Congress can say whatever it likes.

Justice Samuel Alito said, in the bump stocks case, this massacre was terrible, and it’s a pity Congress didn’t act. But if Congress doesn’t act, a regulator can’t step in and do what Congress didn’t do.

That’s interesting, because it suggests a surprising level of open-mindedness among even the court’s most conservative justices to an interpretation of the Constitution that may allow for a greater level of gun regulation than perhaps we think of them as being interested in.

Yeah. When we’re talking about the Constitution, they do seem more open to regulating guns than you might have thought.

OK. You also mentioned, Adam, abortion. Let’s talk about those decisions from this court.

So the Court, in 2022, as everyone knows, overturned Roe v. Wade, eliminated the constitutional right to abortion. But in two cases this term, they effectively enhanced the availability of abortion.

One of them involves emergency rooms. There’s a federal statute that says that emergency rooms that receive federal money have to treat patients and give them stabilizing care if they arrive in the emergency room. That seems to conflict with a strict Idaho law that prohibits abortions except to save the life of the mother.

The court agrees to hear the case, it hears arguments, and then it dismisses the case. It dismisses it as improvidently granted, which is judicial speak for “never mind.” But it’s very tentative. The court merely dismissed the case. It said it was too early to hear it. They’re going to look at it later. So it’s a very tentative sliver of a victory for abortion rights.

But nonetheless, the effect of this is to suspend the Idaho law, at least to the extent it conflicts with the federal law. And it lets emergency abortions continue. Women in Idaho have more access to emergency abortions as a consequence of this decision than if the court had gone the other way.

And of course, the other abortion case centered on the abortion pill, mifepristone.

Right. And that pill is used in a majority of abortions. And the availability of that pill is crucial to what remains of abortion rights in the United States. Lower courts had said that the Food and Drug Administration exceeded its authority in approving these abortion pills. And the case comes to the Supreme Court.

And here, again, they rule in favor of abortion rights. They maintain the availability of these pills, but they do so, again, in a kind of technical way that does not assure that the pills will forever remain available. What the court says, merely — and unanimously — is that the particular plaintiffs who challenged the law, doctors and medical groups who oppose abortion, didn’t have standing, hadn’t suffered the sort of direct injury, that gave them the right to sue.

And it got rid of the case on standing grounds. But that’s not a permanent decision. Other people, other groups can sue, have sued. And the court didn’t decide whether the FDA approval was proper or not, only that the lawsuit couldn’t go forward. And here, too, this case is a victory for abortion rights, but maybe an ephemeral one, and may well return to the court, which has not given an indication of how it will turn out if they actually address the merits.

Got it. So this is a court, the one you’re describing in these rulings, acting with some nuance and some restraint?

Yeah, this picture is complicated.

This is not the court that we’re used to thinking about. There are a lot of crosscurrents. There are a lot of surprises. And that was true, in those cases, on big issues, on guns and abortion. But in another set of cases, the court moved aggressively to the right and really took on the very power and structure of the federal government.

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We’ll be right back.

So Adam, tell us about these cases where the court was less nuanced, less, perhaps, judicious, and really tried to move aggressively to the right and take on the power of government?

So it’s been a long-term goal of the conservative legal movement to weaken the power of regulators, of taking on what they call the administrative state. And this term, the court really vindicated that decades-old project primarily by overruling the foundational precedent in this area, a precedent that gave expert agencies the power to interpret federal laws and enact regulations to protect consumers, investors, all manner of people.

And the court overruled that decision called Chevron. It was as important as the court, two years ago, overruling the right to abortion, one year ago, overruling affirmative action in higher education. This decision will reshape the way the federal government does its work.

Right. And Adam, as I recall, because we did a whole episode about this with you, Chevron created a framework whereby if a law has any ambiguity about how it’s supposed to play out, that the experts within the federal government, within the EPA or the FDA, you name the agency, that we collectively defer to them and their wisdom, and that that becomes the basis for how these laws get interpreted and carried out.

That’s right. And if you think about it, Michael, Congress can’t anticipate every circumstance. Congress will, on purpose sometimes, and inevitably at other times, leave gaps in the law. And those gaps need to be filled by someone. And the choice that the Chevron decision made was to say, we’re going to let the expert regulator fill in those gaps. If there are ambiguities in statutes, the reasonable interpretation of the regulator will get deference from courts.

Experts, not judges, will decide this matter, is what Chevron said 40 years ago. And it’s really hard to overstate the consequences of overruling Chevron. It will open countless, countless regulations to judicial challenge. It may actually kind of swamp the courts. The courts have relied very heavily on Chevron to make difficult decisions about complicated stuff, questions about the environment, and food safety, and drugs, and securities, questions that really often require quite technical expertise.

So what was the court’s rationale for changing that Chevron framework that’s been in place for so long?

What the six-justice majority opinion written by Chief Justice Roberts says is that Chevron was a wrong turn from the outset, that unelected bureaucrats should not be empowered to say what the law means, that that’s the job of judges. So it moves from the expert agency to federal judges the determination of all sorts of important issues. And it probably has the effect of deregulating much of American society.

I mean, in the old world, the regulator had a thumb on the scale. The regulator’s interpretation of an ambiguous statute was the one that counted. And now, the judge will have a fresh look at it. That doesn’t mean that, in every case, the challenger wins, and in every case, the regulator loses. But it shifts the balance and it makes challenges more likely to succeed.

Where else did we see this instinct by the court to challenge the government’s authority in this term?

So right after the court overturns Chevron, it issues a second decision that really amplifies the power of that decision, because it says that challenges can be brought not only in the usual six-year statute of limitations from when a regulation is issued, but six years from when it first affects a company.

And bear with me, because that’s a big difference. If I start a company tomorrow, I have six years to sue over a regulation that affects it, even though that regulation may have been in place for 30 years. So it restarts the clock on challenges, and that one-two punch, both of them decided by 6-3 majorities, go even further in reshaping the ability of the federal government to regulate.

I just want to be sure I understand something. So in the past, let’s say the Clean Water Act was passed in the 1970s. Under the old statute of limitations, a company could sue and say that regulation is a problem for six years. But you’re saying a new company formed right now could go back and sue over something in a 30 - or 40-year-old law and how it’s being interpreted. In other words, this ruling means there really isn’t a statute of limitations on challenging these regulations any longer.

That’s right. And it’s not as though you can’t form a company just for the purpose of litigation. I mean, it completely opens up the ability of industries, trade groups just to set up a trivial nothing company that will then be said to be affected by the regulation and then can sue from now until the end of time.

And the liberal justices sure understood what was happening here, that this one-two punch, as Justice Ketanji Brown Jackson wrote in dissent, was a catastrophe for regulators. She wrote, “At the end of a momentous term, this much is clear — that tsunami of lawsuits against agencies that the court’s holdings in this case have authorized has the potential to devastate the functioning of the federal government.”

It’s a pretty searing warning.

Yeah. I mean, talking about regulations and administrative law might put some people to sleep, but this is a really big deal, Michael. And as if those two cases were not a substantial enough attack on the federal government’s regulatory authority, the court also issues a third 6-3 decision undoing one of the main ways that regulators file enforcement actions against people who they say have violated the law.

They don’t always go to court. Sometimes, they go to administrative tribunals within the agency. The court says, no, that’s no good. Only courts can adjudicate these matters. So it’s just another instance of the court being consistently hostile to the administrative state.

Adam, all three of these decisions might sound pretty dangerous if you have a lot of confidence in the federal government and in the judgments of regulators and bureaucrats to interpret things. But if you’re one of the many Americans who doesn’t have a whole lot of faith in the federal government, I have to imagine all of these rulings might seem pretty constructive.

That’s an excellent point. Lots of people are skeptical of regulators, are skeptical of what they would call the deep state, of unelected bureaucrats, of even the idea of expertise. And so for those people, this is a step in the right direction. It’s taking power away from bureaucrats and handing it to what we would hope are independent, fair-minded judges.

What does seem clear, Adam, is that even though this episode was supposed to be about the rest of the Supreme Court’s rulings this year, the less sexy-sounding decisions than Trump and immunity and how much power and protection all future presidents have, the rulings that you’re describing around the government’s administrative power, they seem like they’re going to have the greatest long-term impact on how our government functions, and in a sense, what our society looks like.

Well, the biggest case of the term is obviously the Trump immunity case. That’s a decision for the ages. But close behind these decisions, reshaping the administrative state and vindicating a long-held goal of the conservative legal movement going back to the Reagan administration, that the Federalist Society, the conservative legal group, has been pushing for decades, and really unraveling a conception of what the federal government does that’s been in place since the administration of Franklin Delano Roosevelt and his New Deal.

So as much as we’ve been talking about other cases where the court was tentative, surprising, nuanced in the biggest cases of the term, all delivered by six to three votes, all controlled by the conservative supermajority, the court was not nuanced. It was straightforward, and it reshaped American government.

In the end, a hard right court is going to, no matter how much it might deviate, operate like a hard right court.

Yes, Michael. It’s possible to look at the balance of the decisions and draw all kinds of complicated conclusions about the court. But when you look at the biggest cases, the picture you see is a conservative court moving the law to the right.

Well, Adam, thank you very much. We appreciate it.

Thank you, Michael.

Here’s what else you need to know today. “The Times” reports that four senior Democratic house members have told colleagues that President Biden must step aside as the party’s nominee over fears that he is no longer capable of winning. They include the top Democrats on the House Judiciary Committee, the Armed Services Committee, and the Veterans Affairs Committee.

Those top Democrats joined five rank and file House Democrats who have publicly called for Biden to step down. The latest of those was representative Angie Craig of Minnesota, who represents a swing district in the state. In a statement, Craig said that after watching Biden in the first debate, quote, “I do not believe that the president can effectively campaign and win against Donald Trump.” Senate Democrats remain largely quiet on the question of Biden’s future.

Now, you probably heard, I had a little debate last week. I can’t say it was my best performance.

In several appearances over the weekend, Biden acknowledged the growing skepticism of his candidacy —

Well, ever since then, there’s been a lot of speculation. What’s Joe going to do?

— but emphatically rejected the calls to step aside.

Well, let me say this clearly as I can. I’m staying in the race.

And in a surprise electoral upset, France’s political left was projected to win the largest number of seats in the National Assembly after the latest round of voting. The anti-immigrant far right had been expected to make history by winning the most seats, but a last-minute scramble by left wing parties averted that result.

Today’s episode was produced by Rikki Novetsky, Shannon Lin, and Rob Szypko. It was edited by Devon Taylor and Lisa Chow. Contains original music by Dan Powell and Sophia Lanman, and was engineered by Chris Wood. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Michael Barbaro. See you tomorrow.

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When the Supreme Court wrapped up its term last week, much of the focus was on the ruling that gave former President Donald J. Trump sweeping immunity from criminal prosecution. But another set of rulings that generated less attention could have just as big an impact on American government and society.

Adam Liptak, who covers the Supreme Court for The Times, explains.

On today’s episode

Adam Liptak , who covers the Supreme Court for The New York Times and writes Sidebar, a column on legal developments.

Background reading

In a volatile term, a fractured Supreme Court remade America .

Here’s a guide to the major Supreme Court decisions in 2024 .

In video: How a fractured Supreme Court ruled this term .

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Maddy Masiello, Isabella Anderson, Nina Lassam and Nick Pitman.

Adam Liptak covers the Supreme Court and writes Sidebar, a column on legal developments. A graduate of Yale Law School, he practiced law for 14 years before joining The Times in 2002. More about Adam Liptak

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