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A value is said to be a root of a polynomial if ..

The largest exponent of appearing in is called the degree of . If has degree , then it is well known that there are roots, once one takes into account multiplicity. To understand what is meant by multiplicity, take, for example, . This polynomial is considered to have two roots, both equal to 3.

One learns about the "factor theorem," typically in a second course on algebra, as a way to find all roots that are rational numbers. One also learns how to find roots of all quadratic polynomials, using square roots (arising from the discriminant) when necessary. There are more advanced formulas for expressing roots of cubic and quartic polynomials, and also a number of numeric methods for approximating roots of arbitrary polynomials. These use methods from complex analysis as well as sophisticated numerical algorithms, and indeed, this is an area of ongoing research and development.

Systems of linear equations are often solved using Gaussian elimination or related methods. This too is typically encountered in secondary or college math curricula. More advanced methods are needed to find roots of simultaneous systems of nonlinear equations. Similar remarks hold for working with systems of inequalities: the linear case can be handled using methods covered in linear algebra courses, whereas higher-degree polynomial systems typically require more sophisticated computational tools.

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10 Best AI Math Solvers For Instant Homework Solutions

Math ai tools make solving math problems simple. here well reveal some of the best ai math solvers for solving complex math problems..

If you need extra help with math homework, you're not alone. AI math solvers are powerful tools that can help students at all grade levels develop their math skills.

Students are not the only group that can benefit from using AI for math. Parents and educators can also use these tools to help students succeed. Math AI solvers break down math problems into small components that are easier to understand.

In this article, we'll compare some of the best AI math solver tools available. There were several things we looked for to find the best ones, such as ease of use, features, and price. Read on!

10 Best AI Math Solver Tools

1. ai math – best ai math solver overall, 2. mathful – best ai math solver for boosting your math grade.

3. HIX Tutor – Best AI Math Solver for Step-by-Step Support

4. ExamUp – Best AI Math Solver for Studying for Exams

5. homeworkai – best ai math solver for reviewing textbook material, 6. microsoft math solver – best ai math solver for use on mobile devices, 7. mathway – best ai math solver for solving algebra problems, 8. maths.ai – best ai math solver for a personalized learning experience, 9. snapxam – best ai math solver for access to online math calculators, 10. air math – best ai math solver for solving problems from the web.

For the overall best math AI tool, try AI Math. This tool enables students to master advanced math skills without the burden of scouring textbooks for answers. From deciphering word problems to simplifying algebraic expressions, AI Math covers nearly every branch of math to provide students with comprehensive support.

Use AI Math to solve math problems anytime and anywhere you have an internet connection. Along with 24/7 accessibility, the AI math solver offers near-precision with a 99% accuracy rate. Best of all, it doesn't cost anything to get started.

To begin, simply input your math question by typing the problem. In seconds, you'll receive an accurate solution and a detailed explanation that helps you learn the material.

● Provides step-by-step solutions.

● Designed to handle both simple and complex math questions.

● Gives instant feedback to reinforce math concepts.

● Requires a paid subscription after the trial period.

● May go against school policies that prohibit AI use.

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Mathful is an advanced AI math problem solver that uses machine learning technology and powerful algorithms to rapidly generate accurate solutions to nearly every math question imaginable.

Not all students can successfully learn new math concepts in the classroom, but Mathful is different. Students can learn at their own pace and review challenging math concepts as often as they need.

Mathful covers over 10 branches of math, including but not limited to, algebra, geometry, and calculus. It also helps younger students with elementary math subjects, such as fractions, decimals, and measurement. The platform is available 24/7 to help students study in the comfort of their homes day or night.

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3. HIX Tutor – Best AI Math Solver for Step-by-Step Math Support

HIX Tutor is a revolutionary tool that enhances students' mathematical education. With HIX Tutor, users can get instant step-by-step explanations of homework problems, whether they need assistance with limits and derivatives or want to hone their algebraic skills.

HIX Tutor promotes streamlined learning by giving students the guidance they need each step of the way. Just submit your math query and wait seconds while the tool works its magic. You can then check the final answer against your own and gain a better understanding of how to reach the final solution.

This diverse AI math solver uses the latest in AI math technology to ensure accurate answers to math questions at all grade levels.

● Helps boost math grades and overall school success.

● Is trained in geometry, trigonometry, calculus, and more.

● Allows students to type in math questions or upload an image or file.

● Could have difficulty reading files with messy or illegible handwriting.

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ExamUp is the perfect solution for students who want to prep for exams or have last-minute study sessions. The reliable AI homework solver can handle math topics across all branches, from number theory to applied mathematics.

ExamUp provides detailed explanations of math problems that improve student comprehension and ensure that the knowledge sticks. The AI math problem solver has a 99% accuracy rate, making it a trustworthy study sidekick.

ExamUp offers a free plan that allows students to access certain features at no cost. Gain access to additional math features and capabilities with a paid subscription.

● Supports 30+ languages, including English, Spanish, French, and Italian.

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● Offers homework help for other subjects like chemistry, physics, and biology.

● Cannot replace an in-person classroom education.

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Math textbooks certainly have value but can be difficult to learn from due to the sheer amount of information. HomeworkAI simplifies the learning process by analyzing textbook material with practice questions and providing instant feedback and solutions to problems.

The AI homework problem solver can also help students with regular homework assignments. Type in a math problem or upload an image or text file to start the process. The platform is easy to use and is accessible 24/7 for convenience.

HomeworkAI can help users of all educational levels, from elementary age to older professionals. Students can try HomeworkAI at no cost.

● Supports a variety of file formats, such as PDF, image files, DOC, and DOCX.

● Can handle math problems with multiple solution methods.

● Helps with other school subjects like biology, chemistry, physics, and more.

● The tool does not offer advanced math features.

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It's inevitable that you won't always be home when you need to study or do math homework. That is why having a mobile-friendly AI math solver like Microsoft Math Solver can be a lifesaver.

Available for download in the App Store or on Google Play, Microsoft Math Solver can be used on both Android and iOS devices. After downloading the app, you'll be prompted to type in a math question, such as a quadratic equation or trigonometry problem. The tool will then provide simple steps to solve the problem and show your work.

● Covers pre-algebra, algebra, calculus, and trigonometry.

● Provides video resources for new features, tips, and more.

● Allows students to graph their math problems.

● The tool can only be used on mobile devices.

Mathway is specifically designed to help students with a variety of math subjects, such as calculus, physics, and statistics, but it is especially beneficial for algebra. The tailored AI math solver offers a simple interface with a chatbot and calculator to make completing algebra problems fast and simple.

When you enter an algebra problem, the tool will present a series of options for you to choose from. For example, it may ask if you want to “Simplify,” “Calculate in the Square Root,” or “Convert to Radical Form.” Once you've made your selection, the tool provides an answer and explanation.

● The AI math solver is free to use.

● Has a clean interface that doesn't distract you from learning.

● Can help with other math subjects like basic math, statistics, and physics.

● Requires users to sign up for advanced features.

● Explanations of math problems can sometimes be too brief.

Every student is unique in the way that they learn and comprehend information. With Maths.ai, students can learn at their own level and pace. The cutting-edge math AI solver gradually adapts to the user's learning style for a personalized academic experience.

Maths.ai prides itself on being a “safe space” in which students can ask homework questions without judgment. The tool is open to students worldwide to help them build their math skills in a way that works best for them. Students can sign up for a free trial to get started. Subscription plans start at just $3 per user, per month.

● Offers a familiar chatbot with a conversational text style.

● Uses the latest in AI technology to help students learn.

● Can be more affordable than hiring a math tutor.

● Accuracy may be reduced for complex math problems.

SnapXam doesn't just offer a one-size-fits-all math tool. Instead, it has more than 100 online math calculators that perform a wide range of mathematical functions, ranging from a condensing logarithms calculator to a polynomial factorization calculator.

The powerful AI math solver also offers step-by-step examples that walk students through different types of math problems. Students can also help SnapXam to explore different ways to solve the problem.

● Provides access to YouTube math videos and tutorials.

● Students who get different answers than the one shown can verify their answers.

● Allows students to share solutions with their friends.

● Better suited for older math students.

● Additional features are only available with a Premium account.

The ability to solve complex problems anywhere is appealing. Air Math is unique in that it offers an Air Math Homework Helper Chrome Extension that students can install, snap, and crop a math problem from the web, and instantly generate a solution.

Air Math can solve many types of problems, including challenging word problems. If the math AI solver cannot provide an accurate answer to your math question, it will recommend connecting with a real tutor who will walk you through the solution. All live tutors at Air Math are tested, certified, and consistently monitored.

● Students can type in math questions or upload photos of questions.

● The tool is available for the web and iOS and Android devices.

● A “Bookmarks” feature enables students to bookmark any solution.

● Some features are only available on the mobile version.

AI math problem solver tools are transforming the way that students approach homework assignments and prepare for exams. Leading AI math solvers like AI Math help create an engaging and intuitive learning experience that makes math a more approachable subject. Try AI Math for yourself and see how it can help you build your math skills.

Disclaimer: The above sponsored content is non-editorial and has been sourced from a third party. NDTV does not guarantee, vouch for or necessarily endorse any of the above content, nor is responsible for it in any manner whatsoever.

March 12, 2024

The Simplest Math Problem Could Be Unsolvable

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

By Manon Bischoff

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

James Brey/Getty Images

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

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

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

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

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

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

Credit: Keenan Pepper/Public domain via Wikimedia Commons

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

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

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

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

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

Empirical Evidence for the Collatz Conjecture

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

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

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

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

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

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

Credit: Cirne/Public domain via Wikimedia Commons

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

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

A Proof for Almost All Numbers

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

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

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

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

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

Can AI Solve the Math Mysteries Stumping the Field’s Brightest Minds?

Artificial Intelligence already won a gold ribbon in the most elite high school math competition. It could help humans conduct boundary-pushing math research next.

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Each year since 1959, high school math students from more than 100 countries have competed to solve a wide variety of math problems involving algebra , geometry , and number theory quickly and elegantly. Many IMO winners have secured prestigious math awards as adults, including the coveted Fields Medal .

In essence, IMO is a benchmark for students to see if they have what it takes to succeed in the field of mathematics. Now, artificial intelligence has aced the test—well, the geometry part at least.

In a paper published this January in Nature , a team of scientists from Google’s DeepMind have introduced a new AI called AlphaGeometry that’s capable of passing the geometry section of the International Math Olympiad without relying on human examples.

“We’ve made a lot of progress with models like ChatGPT … but when it comes to mathematical problems, these [large language models] essentially score zero,” Thang Luong , Ph.D., a senior staff research scientist at Google DeepMind and a senior author of the AlphaGeometry paper, tells Popular Mechanics . “When you ask [math] questions, the model will give you what looks like an answer, but [it actually] doesn’t make sense.”

For example, things get messy when AI tries to solve an algebraic word problem or a combinatorics problem that asks it to find the number of permutations (or versions) of a number sequence.

To answer math questions of this caliber , AlphaGeometry relies on a combination of symbolic AI—which Luong describes as being precise but slow—and a neural network more similar to large language models (LLMs) that is responsible for the quick, creative side of problem-solving.

Yet, math experts aren’t convinced that an AI made to solve high school-level math problems is ready to take off the training wheels and tackle more difficult subjects, e.g. advanced number theory or combinatorics, let alone boundary-pushing math research.

Why AI Struggles With Math

While LLM-powered AI tools have exploded in the past two years, these models have routinely struggled to handle math problems. This is part of what makes AlphaGeometry stand out from the crowd. But even so, that doesn’t necessarily mean it’s ready to tackle higher-level math yet.

.css-2l0eat{font-family:UnitedSans,UnitedSans-roboto,UnitedSans-local,Helvetica,Arial,Sans-serif;font-size:1.625rem;line-height:1.2;margin:0rem;padding:0.9rem 1rem 1rem;}@media(max-width: 48rem){.css-2l0eat{font-size:1.75rem;line-height:1;}}@media(min-width: 48rem){.css-2l0eat{font-size:1.875rem;line-height:1;}}@media(min-width: 64rem){.css-2l0eat{font-size:2.25rem;line-height:1;}}.css-2l0eat b,.css-2l0eat strong{font-family:inherit;font-weight:bold;}.css-2l0eat em,.css-2l0eat i{font-style:italic;font-family:inherit;} “The challenge of AI is that [it] cannot come up with new concepts.”

Marijin Heule , Ph.D., is an associate professor of computer science at Carnegie Mellon University whose work focuses on another kind of automated theorem prover called SAT solvers. In this case, “SAT” refers to a measure of validity called “satisfiability” and not the math section of the high school SAT.

“When it comes down to solving math problems or problems in general, the challenge of AI is that [it] cannot come up with new concepts,” Heule tells Popular Mechanics .

This limitation impacts symbolic AI and neural networks in different ways, Heule explains, but both stem from the issue that these AI rely on an existing bank of human knowledge. However, this isn’t exactly true for AlphaGeometry because it relies on synthetic data , which isn’t based on human examples but is made to mimic them.

While AIs might not be effective mathematicians on their own, that doesn’t necessarily mean they can’t be great apprentices to human mathematicians.

“At least for the foreseeable future, [AI will] be mostly assisting,” Heule says. “One of the other things that these machines can do really well is they can tell you if there is an incorrect argument and [offer] a counterexample.”

These AI-powered nudges can help researchers distinguish research dead-ends from promising paths.

Why Geometry?

Of all the math fields the AlphaGeometry team could have tackled, Luong says there were a few factors that helped them zero in on geometry .

“I think geometry is visually appealing [and] we do geometry as kids,” he says. “And geometry is everywhere in design and architecture, so it’s very important.”

Geometry also offered a unique challenge as being one of the International Math Olympiad fields with the fewest number of proof examples written in a computer-friendly format (e.g. without pictures).

While Heule agrees that AlphaGeometry is “really cool work,” he admits that designing a geometry solver is one of the easier tasks for a math AI.

While human computer scientists did work behind the scenes to formalize geometry problems in a way that computers can reason about, Heule says the reasoning is pretty straightforward once that preparation work is complete.

In part, this is because the considerations of geometry problems (e.g. the relationship between angles, points, and lines) are fairly contained compared to more complex areas, he says.

Take for example Fermat’s Last Theorem . This number theory problem took over three centuries to solve, and Heule says it would be extremely difficult to explain its solution to AI, let alone ask AI to solve it.

“Large-scale fields of modern mathematics … are so big that any one of them contains multitudes,” says Heather Macbeth , Ph.D., an assistant professor of mathematics at Fordham University with a focus on geometry. “I think, maybe a more precise question would be to talk about the styles of problems, which might occur within any mathematical field that some of these AI systems are useful for,” she tells Popular Mechanics.

For example, AI could be useful for pattern recognition or so-called needle-in-a-haystack problems where mathematicians are looking for something with a very particular property, Macbeth says.

Toward General AI

While AI likely won’t be solving centuries-old math problems in the near future, Luong is confident there are still existing advancements on the horizon for AlphaGeometry and its ilk. Perhaps these models could even graduate high school and take on the Putnam Mathematical Competition for undergraduate students.

But beyond math tests themselves, Luong is hopeful about what models like AlphaGeometry could mean for the field of AI at large—in particular, researchers’ goals of designing a generalized AI.

“If we want to talk about building an artificial general intelligence, where we want the AI to be as smart as a human, I think the AI needs to be able to perform deep reasoning,” Luong says. “This means that the AI needs to be able to plan ahead for many, many steps [and] see the big picture of how things connect together … the IMO is the perfect test for that.”

Sarah is a science and technology journalist based in Boston interested in how innovation and research intersect with our daily lives. She has written for a number of national publications and covers innovation news at Inverse .

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Netflix's hit sci-fi series '3 Body Problem' is based on a real math problem that is so complex it's impossible to solve

• The three-body problem is a centuries-old physics question that puzzled Isaac Newton .
• It describes the orbits of three bodies, like planets or stars, trapped in each other's gravity.
• The problem is unsolvable and led to the development of chaos theory.

While Netflix's "3 Body Problem" is a science-fiction show, its name comes from a real math problem that's puzzled scientists since the late 1600s.

In physics, the three-body problem refers to the motion of three bodies trapped in each other's gravitational grip — like a three-star system.

It might sound simple enough, but once you dig into the mathematics, the orbital paths of each object get complicated very quickly.

Two-body vs. three- and multi-body systems

A simpler version is a two-body system like binary stars. Two-body systems have periodic orbits, meaning they are mathematically predictable because they follow the same trajectory over and over. So, if you have the stars' initial positions and velocities, you can calculate where they've been or will be in space far into the past and future.

However, "throwing in a third body that's close enough to interact leads to chaos," Shane Ross, an aerospace and ocean engineering professor at Virginia Tech, told Business Insider. In fact, it's nearly impossible to precisely predict the orbital paths of any system with three bodies or more.

While two orbiting planets might look like a ven diagram with ovular paths overlapping, the paths of three bodies interacting often resemble tangled spaghetti. Their trajectories usually aren't as stable as systems with only two bodies.

All that uncertainty makes what's known as the three-body problem largely unsolvable, Ross said. But there are certain exceptions.

The three-body problem is over 300 years old

The three-body problem dates back to Isaac Newton , who published his "Principia" in 1687.

In the book, the mathematician noted that the planets move in elliptical orbits around the sun. Yet the gravitational pull from Jupiter seemed to affect Saturn's orbital path.

The three-body problem didn't just affect distant planets. Trying to understand the variations in the moon's movements caused Newton literal headaches, he complained.

But Newton never fully figured out the three-body problem. And it remained a mathematical mystery for nearly 200 years.

In 1889, a Swedish journal awarded mathematician Henri Poincaré a gold medal and 2,500 Swedish crowns, roughly half a year's salary for a professor at the time, for his essay about the three-body problem that outlined the basis for an entirely new mathematical theory called chaos theory .

According to chaos theory, when there is uncertainty about a system's initial conditions, like an object's mass or velocity, that uncertainty ripples out, making the future more and more unpredictable.

Think of it like taking a wrong turn on a trip. If you make a left instead of a right at the end of your journey, you're probably closer to your destination than if you made the mistake at the very beginning.

Can you solve the three-body problem?

Cracking the three-body problem would help scientists chart the movements of meteors and planets, including Earth, into the extremely far future. Even comparatively small movements of our planet could have large impacts on our climate, Ross said.

Though the three-body problem is considered mathematically unsolvable, there are solutions to specific scenarios. In fact, there are a few that mathematicians have found.

For example, three bodies could stably orbit in a figure eight or equally spaced around a ring. Both are possible depending on the initial positions and velocities of the bodies.

One way researchers look for solutions is with " restricted " three-body problems, where two main bodies (like the sun and Earth) interact and a third object with much smaller mass (like the moon) offers less gravitational interference. In this case, the three-body problem looks a lot like a two-body problem since the sun and Earth comprise the majority of mass in the system.

However, if you're looking at a three-star system, like the one in Netflix's show "3 Body Problem," that's a lot more complicated.

Computers can also run simulations far more efficiently than humans, though due to the inherent uncertainties, the results are typically approximate orbits instead of exact.

Finding solutions to three-body problems is also essential to space travel, Ross said. For his work, he inputs data about the Earth, moon, and spacecraft into a computer. "We can build up a whole library of possible trajectories," he said, "and that gives us an idea of the types of motion that are possible."

If you enjoyed this story, be sure to follow Business Insider on Microsoft Start.

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Computer Science > Computation and Language

Title: large language models are unconscious of unreasonability in math problems.

Abstract: Large language models (LLMs) demonstrate substantial capabilities in solving math problems. However, they tend to produce hallucinations when given questions containing unreasonable errors. In this paper, we study the behavior of LLMs when faced with unreasonable math problems and further explore their potential to address these problems. First, we construct the Unreasonable Math Problem (UMP) benchmark to examine the error detection ability of LLMs. Experiments show that LLMs are able to detect unreasonable errors, but still fail in generating non-hallucinatory content. In order to improve their ability of error detection and correction, we further design a strategic prompt template called Critical Calculation and Conclusion(CCC). With CCC, LLMs can better self-evaluate and detect unreasonable errors in math questions, making them more reliable and safe in practical application scenarios.

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The 3-body problem is real, and it’s really unsolvable

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Everybody seems to be talking about 3 Body Problem , the new Netflix series based on Cixin Liu’s Remembrance of Earth’s Past book trilogy . Fewer people are talking about the two series’ namesake: The unsolvable physics problem of the same name.

This makes sense, because it’s confusing . In physics, the three-body problem attempts to find a way to predict the movements of three objects whose gravity interacts with each of the others — like three stars that are close together in space. Sounds simple enough, right? Yet I myself recently pulled up the Wikipedia article on the three-body problem and closed the tab in the same manner that a person might stagger away from a bright light. Apparently the Earth, sun, and moon are a three-body system? Are you telling me we don’t know how the moon moves ? Scientists have published multiple solutions for the three-body problem? Are you telling me Cixin Liu’s books are out of date?

All I’d wanted to know was why the problem was considered unsolvable, and now memories of my one semester of high school physics were swimming before my eyes like so many glowing doom numbers. However, despite my pains, I have readied several ways that we non-physicists can be confident that the three-body problem is, in fact, unsolvable.

Reason 1: This is a special definition of ‘unsolvable’

The three-body problem is extra confusing, because scientists are seemingly constantly finding new solutions to the three-body problem! They just don’t mean a one-solution-for-all solution. Such a formula does exist for a two-body system, and apparently Isaac Newton figured it out in 1687 . But systems with more than two bodies are, according to physicists, too chaotic (i.e., not in the sense of a child’s messy bedroom, but in the sense of “chaos theory”) to be corralled by a single solution.

When physicists say they have a new solution to the three-body problem, they mean that they’ve found a specific solution for three-body systems that have certain theoretical parameters. Don’t ask me to explain those parameters, because they’re all things like “the three masses are collinear at each instant” or “a zero angular momentum solution with three equal masses moving around a figure-eight shape.” But basically: By narrowing the focus of the problem to certain arrangements of three-body systems, physicists have been able to derive formulas that predict the movements of some of them, like in our solar system. The mass of the Earth and the sun create a “ restricted three-body problem ,” where a less-big body (in this case, the moon) moves under the influence of two massive ones (the Earth and the sun).

What physicists mean when they say the three-body problem has no solution is simply that there isn’t a one-formula-fits-all solution to every way that the gravity of three objects might cause those objects to move — which is exactly what Three-Body Problem bases its whole premise on.

Reason 2: 3 Body Problem picked an unsolved three-body system on purpose

Henri Poincaré’s research into a general solution to the three-body problem formed the basis of what would become known as chaos theory (you might know it from its co-starring role in Jurassic Park ). And 3 Body Problem itself isn’t about any old three-body system. It’s specifically about an extremely chaotic three-body system, the exact kind of arrangement of bodies that Poincaré was focused on when he showed that the problem is “unsolvable.”

[ Ed. note: The rest of this section includes some spoilers for 3 Body Problem .]

In both Liu’s books and Netflix’s 3 Body Problem , humanity faces an invasion by aliens (called Trisolarans in the English translation of the books, and San-Ti in the TV series) whose home solar system features three suns in a chaotic three-body relationship. It is a world where, unlike ours, the heavens are fundamentally unpredictable. Periods of icy cold give way to searing heat that give way to swings in gravity that turn into temporary reprieves that can never be trusted. The unpredictable nature of the San-Ti environment is the source of every detail of their physicality, their philosophy, and their desire to claim Earth for their own.

In other words, 3 Body Problem ’s three-body problem is unsolvable because Liu wanted to write a story with an unsolvable three-body system, so he chose one of the three-body systems for which we have not discovered a solution, and might never.

Reason 3: Scientists are still working on the three-body problem

Perhaps the best reason I can give you to believe that the three-body problem is real, and is really unsolvable, is that some scientists published a whole set of new solutions for specific three-body systems very recently .

If physicists are still working on the three-body problem, we can safely assume that it has not been solved. Scientists, after all, are the real experts. And I am definitely not.

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