Category: Skills and Learning

Sudoku, the classic number puzzle, has captured the minds of puzzle enthusiasts around the globe. Originating from Japan, the name Sudoku translates to “single number,” reflecting the puzzle’s core principle: each number should appear only once in each row, column, and grid. This article will introduce you to the basics of Sudoku, walk you through some examples, and offer tips to enhance your solving skills.

What is Sudoku?

Sudoku is a logic-based puzzle game typically played on a 9×9 grid divided into nine 3×3 subgrids. The objective is to fill the grid with numbers from 1 to 9, ensuring that each row, each column, and each 3×3 subgrid contains all the numbers from 1 to 9 without repetition.

Basic Rules of Sudoku:

1. Each row must contain the numbers 1 to 9, without repetition.
2. Each column must contain the numbers 1 to 9, without repetition.
3. Each 3×3 subgrid must contain the numbers 1 to 9, without repetition.

How to Solve a Sudoku Puzzle

Step-by-Step Example

Let’s walk through a simple Sudoku puzzle to understand the solving process.

Initial Puzzle:

5 3 _ | _ 7 _ | _ _ _

6 _ _ | 1 9 5 | _ _ _

_ 9 8 | _ _ _ | _ 6 _

——+——-+——

8 _ _ | _ 6 _ | _ _ 3

4 _ _ | 8 _ 3 | _ _ 1

7 _ _ | _ 2 _ | _ _ 6

——+——-+——

_ 6 _ | _ _ _ | 2 8 _

_ _ _ | 4 1 9 | _ _ 5

_ _ _ | _ 8 _ | _ 7 9

Step 1: Start with the Easy Ones

Begin by looking for rows, columns, or subgrids where only one number is missing. For instance, in the first row, the missing numbers are 1, 2, 4, 6, 8, and 9. However, given the other numbers in the row and subgrid, you can often narrow down the possibilities.

Step 2: Use the Process of Elimination

In the first 3×3 subgrid (top left), we are missing the numbers 1, 2, 4, 6, 7. By checking the rows and columns intersecting the empty cells, we can often deduce which numbers go where.

Example Fill-In:

• For the empty cell in the first row, third column, the possible numbers are 1, 2, 4, 6, 8, 9. However, since 6 and 9 are in the same column and 8 is in the same row, the number for this cell must be 2.

Step 3: Repeat the Process

Continue using the process of elimination and logical deduction for the remaining cells. Let’s fill in a few more:

• For the second row, second column, the missing numbers are 2, 3, 4, 7, 8. Since 3 and 8 are already in the same subgrid, we need to see which other numbers fit based on the column and row.

Intermediate Puzzle:

5 3 2 | _ 7 _ | _ _ _

6 _ _ | 1 9 5 | _ _ _

_ 9 8 | _ _ _ | _ 6 _

——+——-+——

8 _ _ | _ 6 _ | _ _ 3

4 _ _ | 8 _ 3 | _ _ 1

7 _ _ | _ 2 _ | _ _ 6

——+——-+——

_ 6 _ | _ _ _ | 2 8 _

_ _ _ | 4 1 9 | _ _ 5

_ _ _ | _ 8 _ | _ 7 9

Step 4: Solve the Puzzle

By continuing to apply these methods, you will gradually fill in the entire grid. Here’s the completed puzzle for reference:

5 3 4 | 6 7 8 | 9 1 2

6 7 2 | 1 9 5 | 3 4 8

1 9 8 | 3 4 2 | 5 6 7

——+——-+——

8 5 9 | 7 6 1 | 4 2 3

4 2 6 | 8 5 3 | 7 9 1

7 1 3 | 9 2 4 | 8 5 6

——+——-+——

9 6 1 | 5 3 7 | 2 8 4

2 8 7 | 4 1 9 | 6 3 5

3 4 5 | 2 8 6 | 1 7 9

Tips for Solving Sudoku

1. Start with the obvious: Fill in the easy cells first to gain momentum.
2. Use pencil marks: Write possible numbers in cells to keep track of your thoughts.
3. Look for patterns: Familiarize yourself with common patterns and techniques, such as naked pairs, hidden pairs, and X-Wing.
4. Stay organized: Work methodically through rows, columns, and sub grids.
5. Practice regularly: The more you practice, the better you’ll get at spotting solutions quickly.

Conclusion

Sudoku is a fantastic way to challenge your brain, improve your logical thinking, and enjoy a bit of quiet time. Whether you’re a beginner or an experienced solver, the satisfaction of completing a Sudoku puzzle is a reward in itself. So, grab a puzzle, follow these steps, and immerse yourself in the fascinating world of Sudoku!

Three factors—the needs of the subject, the child, and the society—have influenced what mathematics is to be taught in schools. Many people think that “math is math” and never changes. In this three-part series, we briefly discuss these three factors and paint a different picture: mathematics is a subject that is ever-changing.

In this third and final part, we discuss the-

Needs of the Society

The usefulness of mathematics in everyday life and in many vocations has also affected what is taught and when it is taught. In early America, mathematics was considered necessary primarily for clerks and bookkeepers. The curriculum was limited to counting, the simpler procedures for addition, subtraction, and multiplication, and some facts about measures and fractions. By the late nineteenth century, business and commerce had advanced to the point. that mathematics was considered important for everyone. The arithmetic curriculum expanded to include such topics as percentages, ratios and proportions, powers, roots, and series.

This emphasis on social utility, on teaching what was needed for use in occupations, continued into the twentieth century. One of the most vocal advocates of social utility was Guy Wilson. He and his students conducted numerous surveys to determine what arithmetic was actually used by carpenters, shopkeepers, and other workers. He believed that the dominating aim of the school mathematics program should be to teach those skills and only those skills.

In the 1950s, the outburst of public concern over the “space race” resulted in a wave of research and development in mathematics curricula. Much of this effort was focused on teaching the mathematically talented student. By the mid-1960s, however, concern was also being expressed for the disadvantaged student as U.S. society renewed its commitment to equality of opportunity. With each of these changes, more and better mathematical achievement was promised.

In the 1970s, when it became apparent that the promise of greater achievement had not fully materialised, another swing in curriculum development occurred. Emphasis was again placed on the skills needed for success in the real world. The minimal competency movement stressed the basics. As embodied in sets of objectives and in tests, the basics were considered to be primarily addition, subtraction, multiplication, and division with whole numbers and fractions. Thus, the skills needed in colonial times were again being considered by many to be the sole necessities, even though children were now living in a world with calculators, computers, and other features of a much more technological society.

By the 1980s, it was acknowledged that no one knew exactly what skills were needed for the future but that everyone needed to be able to solve problems. The emphasis on problem-solving matured through the last 20 years of the century to the point where problem-solving was not seen as a separate topic but as a way to learn and use mathematics.

Today, one need of our society is for a workforce that is competitive in the world. There is a call for school mathematics to ensure that students are ready for workforce training programs or college.

Conclusion

International Mathematics Olympiad play a vital role in shaping the intellectual and analytical landscape of society. They not only foster critical thinking, problem-solving skills, and creativity among students but also prepare them to tackle complex real-world issues. By encouraging young minds to engage with challenging mathematical concepts, Olympiads help cultivate a future generation of scientists, engineers, economists, and leaders who can drive innovation and progress. Moreover, the collaborative and competitive nature of these competitions promotes a culture of academic excellence and perseverance.

As we face increasingly complex global challenges, the importance of nurturing a strong foundation in mathematics through Olympiads cannot be overstated. They are not just competitions; they are essential platforms for equipping society with the tools and mindset needed to build a better, more informed, and innovative world.

 

 

Three factors—the needs of the subject, the child, and the society—have influenced what mathematics is to be taught in schools. Many people think that “math is math” and never changes. This three-part series briefly discusses these three factors and paints a different picture: mathematics is an ever-changing subject.

We have already discussed the Needs of the Subject in the previous blog. In this second part, we discuss-

Needs of the Child

The mathematics curriculum has been influenced by beliefs and knowledge about how children learn and, ultimately, about how they should be taught. Before the early years of the twentieth century, mathematics was taught to train “mental faculties” or provide “mental discipline.” Struggling with mathematical procedures was thought to exercise the mind (like muscles are exercised), helping children’s brains work more effectively. Around the turn of the twentieth century, “mental discipline” was replaced by connectionism, the belief that learning established bonds, or connections, between a stimulus and responses. This led teachers to the endless use of drills aimed at establishing important mathematical connections.

In the 1920s, the Progressive movement advocated incidental learning, reflecting the belief that children would learn as much arithmetic as they needed and would learn it better if it was not systematically taught. The teacher’s role was to take advantage of situations when they occurred naturally as well as to create situations in which arithmetic would arise.

During the late 1920s, the Committee of Seven, a committee of school superintendents and principals from midwestern cities, surveyed pupils to find out when they mastered various topics. Based on that survey, the committee recommended teaching mathematics topics according to students’ mental age. For example, subtraction facts under 10 were to be taught to children with a mental age of 6 years 7 months and facts over 10 at 7 years 8 months; subtraction with borrowing or carrying was to be taught at 8 years 9 months. The recommendations of the Committee of Seven had a strong impact on the sequencing of the curriculum for years afterward.

Another change in thinking occurred in the mid-1930s, under the influence of field theory, or Gestalt theory. A 1954 article by William A. Brownell (2006), a prominent mathematics education researcher, showed the benefits of encouraging insight and the understanding of relationships, structures, patterns, interpretations, and principles. His research contributed to an increased focus on learning as a process that led to meaning and understanding. The value of drill was acknowledged, but it was given less importance than understanding; drill was no longer the major means of providing instruction.

The relative importance of drill and understanding is still debated today. In this debate, people often treat understanding and learning skills as if they are opposites, but this is not the case. The drill is necessary to build speed and accuracy and to make skills automatic. But equally clearly, you need to know why as well as how. Both skills and understanding must be developed, and they can be developed together with the help of International Maths Challenge sample questions.

Changes in the field of psychology have continued to affect education. During the second half of the twentieth century, educators came to understand that the developmental level of the child is a major factor in determining the sequence of the curriculum. Topics cannot be taught until children are developmentally ready to learn them. Or, from another point of view, topics must be taught in such a way that children at a given developmental level are ready to learn them.

Research has provided increasing evidence that children construct their own knowledge. In so doing, they make sense of the mathematics and feel that they can tackle new problems. Thus, helping children learn mathematics means being aware of how children have constructed mathematics from their experiences both in and out of school.

End Note

As we have explored, a child’s journey through mathematics is deeply intertwined with their cognitive development, critical thinking skills, and overall academic success. By addressing their individual needs, providing appropriate support, and fostering a positive learning environment, we lay the foundation for a lifelong appreciation and understanding of mathematics. But what about the broader context? How does mathematics serve society at large, and what influences has it made in history? In our next blog, we will delve into these questions, examining the societal needs in mathematics and its profound impact on the course of human history.

Three factors—the needs of the subject, the child, and the society—have influenced what mathematics is to be taught in schools. Many people think that “math is math” and never changes. This three-part series briefly discusses these three factors and paints a different picture: mathematics is an ever-changing subject.

In the first part, we discuss-

Needs of the Subject

The nature of mathematics helps determine what is taught and when it is taught in elementary grades. For example, number work begins with whole numbers, then fractions and decimals. Length is studied before area. Such seemingly natural sequences are the result of long years of curricular evolution. This process has involved much analysis of what constitutes a progression from easy to difficult, based in part on what is deemed necessary at one level to develop ideas at later levels. Once a curriculum is in place for a long time, however, people tend to consider it the only proper sequence. Thus, omitting a topic or changing the sequence of issues often involves a struggle for acceptance. However, research shows that all students do not always learn in the sequence that has been ingrained in our curriculum.

Sometimes, the process of change is the result of an event, such as when the Soviet Union sent the first Sputnik into orbit. The shock of this evidence of another country’s technological superiority sped curriculum change in the United States. The “new math” of the 1950s and 1960s was the result, and millions of dollars were channeled into mathematics and science education to strengthen school programs. Mathematicians became integrally involved. Because of their interests and the perceived weaknesses of previous curricula, they developed curricula based on the needs of the subject. The emphasis shifted from social usefulness to such unifying themes as the structure of mathematics, operations and their inverses, systems of notation, properties of numbers, and set language. New content was added at the elementary school level, and other topics were introduced at earlier grade levels.

Mathematics continues to change; new mathematics is created, and new uses of mathematics are discovered. As part of this change, technology has made some mathematics obsolete and has opened the door for other mathematics to be accessible to students. Think about all the mathematics you learned in elementary school. How much of this can be done on a simple calculator? What mathematics is now important because of the technology available today?

As mathematical research progresses and new theories and applications emerge, the curriculum must adapt to incorporate these advancements. For example, the development of computer science has introduced concepts such as algorithms and computational thinking into mathematics education. These topics were not traditionally part of the elementary curriculum but have become essential due to their relevance in today’s technology-driven world. Additionally, as interdisciplinary fields like data science and quantitative biology grow, there is a pressing need to equip students with skills in statistics, probability, and data analysis from an early age, and here, the International Maths Challenge is playing a crucial role. This integration ensures that students are prepared for future academic and career opportunities that increasingly rely on mathematical literacy. Furthermore, globalization and the interconnected nature of modern societies require students to understand complex systems and patterns, necessitating the introduction of topics such as systems theory and network analysis. Consequently, the curriculum evolves not only to preserve the integrity and progression of mathematical concepts but also to reflect the dynamic and ever-expanding landscape of mathematical applications in the real world.

End Note

In the next blog, we will move towards the second factor: understanding the need for mathematics in a child’s education can set a foundation for problem-solving, logical thinking, and even everyday decision-making. In the following blog, we will delve into why mathematics is not just a subject but a vital tool for a child’s overall development and future success. Stay tuned for further updates!

By rapidly analysing large amounts of data and making accurate predictions, artificial intelligence (AI) tools could help to answer many long-standing research questions. For instance, they could help to identify new materials to fabricate electronics or the patterns in brain activity associated with specific human behaviours.

One area in which AI has so far been rarely applied is number theory, a branch of mathematics focusing on the study of integers and arithmetic functions. Most research questions in this field are solved by human mathematicians, often years or decades after their initial introduction.

Researchers at the Israel Institute of Technology (Technion) recently set out to explore the possibility of tackling long-standing problems in number theory using state-of-the-art computational models.

In a recent paper, published in the Proceedings of the National Academy of Sciences, they demonstrated that such a computational approach can support the work of mathematicians, helping them to make new exciting discoveries.

“Computer algorithms are increasingly dominant in scientific research, a practice now broadly called ‘AI for Science,'” Rotem Elimelech and Ido Kaminer, authors of the paper, told Phys.org.

“However, in fields like number theory, advances are often attributed to creativity or human intuition. In these fields, questions can remain unresolved for hundreds of years, and while finding an answer can be as simple as discovering the correct formula, there is no clear path for doing so.”

Elimelech, Kaminer and their colleagues have been exploring the possibility that computer algorithms could automate or augment mathematical intuition. This inspired them to establish the Ramanujan Machine research group, a new collaborative effort aimed at developing algorithms to accelerate mathematical research.

Their research group for this study also included Ofir David, Carlos de la Cruz Mengual, Rotem Kalisch, Wolfram Berndt, Michael Shalyt, Mark Silberstein, and Yaron Hadad.

“On a philosophical level, our work explores the interplay between algorithms and mathematicians,” Elimelech and Kaminer explained. “Our new paper indeed shows that algorithms can provide the necessary data to inspire creative insights, leading to discoveries of new formulas and new connections between mathematical constants.”

The first objective of the recent study by Elimelech, Kaminer and their colleagues was to make new discoveries about mathematical constants. While working toward this goal, they also set out to test and promote alternative approaches for conducting research in pure mathematics.

“The ‘conservative matrix field’ is a structure analogous to the conservative vector field that every math or physics student learns about in first year of undergrad,” Elimelech and Kaminer explained. “In a conservative vector field, such as the electric field created by a charged particle, we can calculate the change in potential using line integrals.

“Similarly, in conservative matrix fields, we define a potential over a discrete space and calculate it through matrix multiplications rather than using line integrals. Traveling between two points is equivalent to calculating the change in the potential and it involves a series of matrix multiplications.”

In contrast with the conservative vector field, the so-called conservative matrix field is a new discovery. An important advantage of this structure is that it can generalize the formulas of each mathematical constant, generating infinitely many new formulas of the same kind.

“The way by which the conservative matrix field creates a formula is by traveling between two points (or actually, traveling from one point all the way to infinity inside its discrete space),” Elimelech and Kaminer said. “Finding non-trivial matrix fields that are also conservative is challenging.”

As part of their study, Elimelech, Kaminer and their colleagues used large-scale distributed computing, which entails the use of multiple interconnected nodes working together to solve complex problems. This approach allowed them to discover new rational sequences that converge to fundamental constants (i.e., formulas for these constants).

“Each sequence represents a path hidden in the conservative matrix field,” Elimelech and Kaminer explained. “From the variety of such paths, we reverse-engineered the conservative matrix field. Our algorithms were distributed using BOINC, an infrastructure for volunteer computing. We are grateful to the contribution by hundreds of users worldwide who donated computation time over the past two and a half years, making this discovery possible.”

The recent work by the research team at the Technion demonstrates that mathematicians can benefit more broadly from the use of computational tools and algorithms to provide them with a “virtual lab.” Such labs provide an opportunity to try ideas experimentally in a computer, resembling the real experiments available in physics and in other fields of science. Specifically, algorithms can carry out mathematical experiments providing formulas that can be used to formulate new mathematical hypotheses.

“Such hypotheses, or conjectures, are what drives mathematical research forward,” Elimelech and Kaminer said. “The more examples supporting a hypothesis, the stronger it becomes, increasing the likelihood to be correct. Algorithms can also discover anomalies, pointing to phenomena that are the building-blocks for new hypotheses. Such discoveries would not be possible without large-scale mathematical experiments that use distributed computing.”

Another interesting aspect of this recent study is that it demonstrates the advantages of building communities to tackle problems. In fact, the researchers published their code online from their project’s early days and relied on contributions by a large network of volunteers.

“Our study shows that scientific research can be conducted without exclusive access to supercomputers, taking a substantial step toward the democratization of scientific research,” Elimelech and Kaminer said. “We regularly post unproven hypotheses generated by our algorithms, challenging other math enthusiasts to try proving these hypotheses, which when validated are posted on our project website. This happened on several occasions so far. One of the community contributors, Wolfgang Berndt, got so involved that he is now part of our core team and a co-author on the paper.”

The collaborative and open nature of this study allowed Elimelech, Kaminer and the rest of the team to establish new collaborations with other mathematicians worldwide. In addition, their work attracted the interest of some children and young people, showing them how algorithms and mathematics can be combined in fascinating ways.

In their next studies, the researchers plan to further develop the theory of conservative matrix fields. These matrix fields are a highly powerful tool for generating irrationality proofs for fundamental constants, which Elimelech, Kaminer and the team plan to continue experimenting with.

“Our current aim is to address questions regarding the irrationality of famous constants whose irrationality is unknown, sometimes remaining an open question for over a hundred years, like in the case of the Catalan constant,” Elimelech and Kaminer said.

“Another example is the Riemann zeta function, central in number theory, with its zeros at the heart of the Riemann hypothesis, which is perhaps the most important unsolved problem in pure mathematics. There are many open questions about the values of this function, including the irrationality of its values. Specifically, whether ζ(5) is irrational is an open question that attracts the efforts of great mathematicians.”

The ultimate goal of this team of researchers is to successfully use their experimental mathematics approach to prove the irrationality of one of these constants. In the future, they also hope to systematically apply their approach to a broader range of problems in mathematics and physics. Their physics-inspired hands-on research style arises from the interdisciplinary nature of the team, which combines people specialized in CS, EE, math, and physics.

“Our Ramanujan Machine group can help other researchers create search algorithms for their important problems and then use distributed computing to search over large spaces that cannot be attempted otherwise,” Elimelech and Kaminer added. “Each such algorithm, if successful, will help point to new phenomena and eventually new hypotheses in mathematics, helping to choose promising research directions. We are now considering pushing forward this strategy by setting up a virtual user facility for experimental mathematics,” inspired by the long history and impact of user facilities for experimental physics.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Ingrid Fadelli , Phys.org

What does “equals” mean? For mathematicians, this simple question has more than one answer, which is causing issues when it comes to using computers to check proofs. The solution might be to tear up the foundations of maths.

When you see “2 + 2 = 4”, what does “=” mean? It turns out that’s a complicated question, because mathematicians can’t agree on the definition of what makes two things equal.

While this argument has been quietly simmering for decades, a recent push to make mathematical proofs checkable by computer programs, called formalisation, has given the argument new significance.

“Mathematicians use equality to mean two different things, and I was fine with that,” says Kevin Buzzard at Imperial College London. “Then I started doing maths on a computer.” Working with computer proof assistants made him realise that mathematicians must now confront what was, until recently, a useful ambiguity, he says – and it could force them to completely redefine the foundations of their subject.

The first definition of equality will be a familiar one. Most mathematicians take it to mean that each side of an equation represents the same mathematical object, which can be proven through a series of logical transformations from one side to the other. While “=”, the equals sign, only emerged in the 16th century, this concept of equality dates back to antiquity.

It was the late 19th century when things began to change, with the development of set theory, which provides the logical foundations for most modern mathematics. Set theory deals with collections, or sets, of mathematical objects, and introduced another definition of equality: if two sets contain the same elements, then they are equal, similar to the original mathematical definition. For example, the sets {1, 2, 3} and {3, 2, 1} are equal, because the order of the elements in a set doesn’t matter.

But as set theory developed, mathematicians started saying that two sets were equal if there was an obvious way to map between them, even if they didn’t contain exactly the same elements, says Buzzard.

To understand why, take the sets {1, 2, 3} and {a, b, c}. Clearly, the elements of each set are different, so the sets aren’t equal. But there are also ways of mapping between the two sets, by identifying each letter with a number. Mathematicians call this an isomorphism. In this case, there are multiple isomorphisms because you have a choice of which number to assign to each letter, but in many cases, there is only one clear choice, called the canonical isomorphism.

Because a canonical isomorphism of two sets is the only possible way to link them, many mathematicians now take this to mean they are equal, even though it isn’t technically the same concept of equality that most of us are used to. “These sets match up with each other in a completely natural way and mathematicians realised it would be really convenient if we just call those equal as well,” says Buzzard.

Having two definitions for equality is of no real concern to mathematicians when they write papers or give lectures, as the meaning is always clear from the context, but they present problems for computer programs that need strict, precise instructions, says Chris Birkbeck at the University of East Anglia, UK. “We’re finding that we were a little bit sloppy all along, and that maybe we should fix a few things.”

To address this, Buzzard has been investigating the way some mathematicians widely use canonical isomorphism as equality, and the problems this can cause with formal computer proof systems.

In particular, the work of Alexander Grothendieck, one of the leading mathematicians of the 20th century, is currently extremely difficult to formalise. “None of the systems that exist so far capture the way that mathematicians such as Grothendieck use the equal symbol,” says Buzzard.

The problem has its roots in the way mathematicians put together proofs. To begin proving anything, you must first make assumptions called axioms that are taken to be true without proof, providing a logical framework to build upon. Since the early 20^th century, mathematicians have settled on a collection of axioms within set theory that provide a firm foundation. This means they don’t generally have to use axioms directly in their day-to-day business, because common tools can be assumed to work correctly – in the same way you probably don’t worry about the inner workings of your kitchen before cooking a recipe.

“As a mathematician, you somehow know well enough what you’re doing that you don’t worry too much about it,” says Birkbeck. That falls down, however, when computers get involved, carrying out maths in a way that is similar to building a kitchen from scratch for every meal. “Once you have a computer checking everything you say, you can’t really be vague at all, you really have to be very precise,” says Birkbeck.

To solve the problem, some mathematicians argue we should just redefine the foundations of mathematics to make canonical isomorphisms and equality one and the same. Then, we can make computer programs work around that. “Isomorphism is equality,” says Thorsten Altenkirch at the University of Nottingham, UK. “I mean, what else? If you cannot distinguish two isomorphic objects, what else would it be? What else would you call this relationship?”

Efforts are already under way to do this in a mathematical field called homotopy type theory, in which traditional equality and canonical isomorphism are defined identically. Rather than trying to contort existing proof assistants to fit canonical isomorphism, says Altenkirch, mathematicians should adopt type theory and use alternative proof assistants that work with it directly.

Buzzard isn’t a fan of this suggestion, having already spent considerable effort using current tools to formalise mathematical proofs that are needed to check more advanced work, such as a proof of Fermat’s last theorem. The axioms of mathematics should be left as they are, rather than adopting type theory, and existing systems should be tweaked instead, he says. “Probably the way to fix it is just to leave mathematicians as they are,” says Buzzard. “It’s very difficult to change mathematicians. You have to make the computer systems better.”

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Alex Wilkins*

Mathematicians are celebrating a 1000-page proof of the geometric Langlands conjecture, a problem so complicated that even other mathematicians struggle to understand it. Despite that, it is hoped the proof can provide key insights across maths and physics.

The Langlands programme aims to link different areas of mathematics

Mathematicians have proved a key building block of the Langlands programme, sometimes referred to as a “grand unified theory” of maths due to the deep links it proposes between seemingly distant disciplines within the field.

While the proof is the culmination of decades of work by dozens of mathematicians and is being hailed as a dazzling achievement, it is also so obscure and complex that it is “impossible to explain the significance of the result to non-mathematicians”, says Vladimir Drinfeld at the University of Chicago. “To tell the truth, explaining this to mathematicians is also very hard, almost impossible.”

The programme has its origins in a 1967 letter from Robert Langlands to fellow mathematician Andre Weil that proposed the radical idea that two apparently distinct areas of mathematics, number theory and harmonic analysis, were in fact deeply linked. But Langlands couldn’t actually prove this, and was unsure whether he was right. “If you are willing to read it as pure speculation I would appreciate that,” wrote Langlands. “If not — I am sure you have a waste basket handy.”

This mysterious link promised answers to problems that mathematicians were struggling with, says Edward Frenkel at the University of California, Berkeley. “Langlands had an insight that difficult questions in number theory could be formulated as more tractable questions in harmonic analysis,” he says.

In other words, translating a problem from one area of maths to another, via Langlands’s proposed connections, could provide real breakthroughs. Such translation has a long history in maths – for example, Pythagoras’s theorem relating the three sides of a triangle can be proved using geometry, by looking at shapes, or with algebra, by manipulating equations.

As such, proving Langlands’s proposed connections has become the goal for multiple generations of researchers and led to countless discoveries, including the mathematical toolkit used by Andrew Wiles to prove the infamous Fermat’s last theorem. It has also inspired mathematicians to look elsewhere for analogous links that might help. “A lot of people would love to understand the original formulation of the Langlands programme, but it’s hard and we still don’t know how to do it,” says Frenkel.

One analogy that has yielded progress is reformulating Langlands’s idea into one written in the mathematics of geometry, called the geometric Langlands conjecture. However, even this reformulation has baffled mathematicians for decades and was itself considered fiendishly difficult to prove.

Now, Sam Raskin at Yale University and his colleagues claim to have proved the conjecture in a series of five papers that total more than 1000 pages. “It’s really a tremendous amount of work,” says Frenkel.

The conjecture concerns objects that are similar to those in one half of the original Langlands programme, harmonic analysis, which describes how complex structures can be mathematically broken down into their component parts, like picking individual instruments out of an orchestra. But instead of looking at these with harmonic analysis, it uses other mathematical ideas, such as sheaves and moduli stacks, that describe concepts relating to shapes like spheres and doughnuts.

While it wasn’t in the setting that Langlands originally envisioned, it is a sign that his original hunch was correct, says Raskin. “Something I find exciting about the work is it’s a kind of validation of the Langlands programme more broadly.”

“It’s the first time we have a really complete understanding of one corner of the Langlands programme, and that’s inspiring,” says David Ben-Zvi at the University of Texas, who wasn’t involved in the work. “That kind of gives you confidence that we understand what its main issues are. There are a lot of subtleties and bells and whistles and complications that appear, and this is the first place where they’ve all been kind of systematically resolved.”

Proving this conjecture will give confidence to other mathematicians hoping to make inroads on the original Langlands programme, says Ben-Zvi, but it might also attract the attention of theoretical physicists, he says. This is because in 2007, physicists Edward Witten and Anton Kapustin found that the geometric Langlands conjecture appeared to describe an apparent symmetry between certain physical forces or theories, called S-duality.

The most basic example of this in the real world is in electricity and magnetism, which are mirror images of one another and interchangeable in many scenarios, but S-duality was also used by Witten to famously unite five competing string theory models into a single theory called M-theory.

But before anything like that, there is much more work to be done, including helping other mathematicians to actually understand the proof. “Currently, there’s a very small group of people who can really understand all the details here. But that changes the game, that changes the whole expectation and changes what you think is possible,” says Ben-Zvi.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Alex Wilkins*

An analysis of a mathematical economic game suggests that even learning from past mistakes will almost never help us optimise our decision-making – with implications for our ability to make the biggest financial gains.

When people trade stocks, they don’t always learn from experience

Even when we learn from past mistakes, we may never become optimal decision-makers. The finding comes from an analysis of a mathematical game that simulates a large economy, and suggests we may need to rethink some of the common assumptions built into existing economic theories.

In such theories, people are typically represented as rational agents who learn from past experiences to optimise their performance, eventually reaching a stable state in which they know how to maximise their earnings. This assumption surprised Jérôme Garnier-Brun at École Polytechnique in France because, as a physicist, he knew that interactions in nature – such as those between atoms – often result in chaos rather than stability. He and his colleagues mathematically tested whether economists are correct to assume that learning from the past can help people avoid chaos.

They devised a mathematical model for a game featuring hundreds of players. Each of these theoretical players can choose between two actions, like buying or selling a stock. They also interact with each other, and each player’s decision-making is influenced by what they have done before – meaning each player can learn from experience. The researchers could adjust the precise extent to which a player’s past experiences influenced their subsequent decision-making. They could also control the interactions between the players to make them either cooperate or compete with each other more.

With all these control knobs available to them, Garnier-Brun and his colleagues used methods from statistical physics to simulate different game scenarios on a computer. The researchers expected that in some scenarios the game would always result in chaos, with players unable to learn how to optimise their performance. Economic theory would also suggest that, given the right set of parameters, the virtual players would settle into a stable state where they have mastered the game – but the researchers found that this wasn’t really the case. The most likely outcome was a state that never settled.

Jean-Philippe Bouchaud at École Polytechnique, who worked on the project, says that in the absence of one centralised, omniscient, god-like player that could coordinate everyone, regular players could only learn how to reach “satisficing” states. That is, they could reach a level that satisfied minimum expectations, but not much more. Players gained more than they would have done by playing at random, so learning was not useless, but they still gained less than they would have if past experience had allowed them to truly optimise their performance.

“This work is such a powerful new way of looking at the problem of learning complex games and these questions are fundamental to the construction of models of economic decision-making,” says Tobias Galla at the Institute for Cross-Disciplinary Physics and Complex Systems in Spain. He says the finding that learning typically does not lead to outcomes better than satisficing could also be important for processes like foraging decisions by animals or for some machine learning applications.

Bouchaud says his team’s game model is too simple to be immediately adopted for making predictions about the real world, but he sees the study as a challenge to economists to drop many assumptions that currently go into theorising processes like merchants choosing suppliers or banks setting interest rates.

“The idea that people are always making complicated economic computations and learn how to become the most rational agents, our paper invites everyone to move on [from that],” he says.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Karmela Padavic-Callaghan*

 

For all the students wondering why they would ever need to use the Pythagorean theorem, Katie Steckles is delighted to report on a real-world encounter.

Recently, I was building a flat-pack wardrobe when I noticed something odd in the instructions. Before you assembled the wardrobe, they said, you needed to measure the height of the ceiling in the room you were going to put it in. If it was less than 244 centimetres high, there was a different set of directions to follow.

These separate instructions asked you to build the wardrobe in a vertical orientation, holding the side panels upright while you attached them to the base. The first set of directions gave you a much easier job, building the wardrobe flat on the floor before lifting it up into place. I was intrigued by the value of 244 cm: this wasn’t the same as the height of the wardrobe, or any other dimension on the package, and I briefly wondered where that number had come from. Then I realised: Pythagoras.

The wardrobe was 236 cm high and 60 cm deep. Looking at it side-on, the length of the diagonal line from corner to corner can be calculated using Pythagoras’s theorem. The vertical and horizontal sides meet at a right angle, meaning if we square the length of each then add them together, we get the well-known “square of the hypotenuse”. Taking the square root of this number gives the length of the diagonal.

In this case, we get a diagonal length a shade under 244 cm. If you wanted to build the wardrobe flat and then stand it up, you would need that full diagonal length to fit between the floor and the ceiling to make sure it wouldn’t crash into the ceiling as it swung past – so 244 cm is the safe ceiling height. It is a victory for maths in the real world, and vindication for maths teachers everywhere being asked, “When am I going to use this?”

This isn’t the only way we can connect Pythagoras to daily tasks. If you have ever needed to construct something that is a right angle – like a corner in joinery, or when laying out cones to delineate the boundaries of a sports pitch – you can use the Pythagorean theorem in reverse. This takes advantage of the fact that a right-angled triangle with sides of length 3 and 4 has a hypotenuse of 5 – a so-called 3-4-5 triangle.

If you measure 3 units along one side from the corner, and 4 along the other, and join them with a diagonal, the diagonal’s length will be precisely 5 units, if the corner is an exact right angle. Ancient cultures used loops of string with knots spaced 3, 4 and 5 units apart – when held out in a triangle shape, with a knot at each vertex, they would have a right angle at one corner. This technique is still used as a spot check by builders today.

Engineers, artists and scientists might use geometrical thinking all the time, but my satisfaction in building a wardrobe, and finding the maths checked out perfectly, is hard to beat.

Katie Steckles is a mathematician, lecturer, YouTuber and author based in Manchester, UK. She is also puzzle adviser for New Scientist’s puzzle column, BrainTwister. Follow her @stecks

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Peter Rowlett*

A mathematical model of a particle that remembers its past so that it never travels the same path twice produces stunningly complex patterns.

A beautiful and surprisingly complex pattern produced by ‘mathematical billiards’

Albers et al. PRL 2024

In a mathematical version of billiards, particles that avoid retracing their paths get trapped in intricate and hard-to-predict patterns – which might eventually help us understand the complex movement patterns of living organisms.

When searching for food, animals including ants and slime moulds leave chemical trails in their environment, which helps them avoid accidentally retracing their steps. This behaviour is not uncommon in biology, but when Maziyar Jalaal at the University of Amsterdam in the Netherlands and his colleagues modelled it as a simple mathematical problem, they uncovered an unexpected amount of complexity and chaos.

They used the framework of mathematical billiards, where an infinitely small particle bounces between the edges of a polygonal “table” without friction. Additionally, they gave the particle “spatial memory” – if it reached a point where it had already been before, it would reflect off it as if there was a wall there.

The researchers derived equations describing the motion of the particle and then used them to simulate this motion on a computer. They ran over 200 million simulations to see the path the particle would take inside different polygons – like a triangle and a hexagon – over time. Jalaal says that though the model was simple, idealised and deterministic, what they found was extremely intricate.

Within each polygon, the team identified regions where the particle was likely to become trapped after bouncing around for a long time due to its “remembering” its past trajectories, but zooming in on those regions revealed yet more patterns of motion.

“So, the patterns that you see if you keep zooming in, there is no end to them. And they don’t repeat, they’re not like fractals,” says Jalaal.

Katherine Newhall at the University of North Carolina at Chapel Hill says the study is an “interesting mental exercise” but would have to include more detail to accurately represent organisms and objects that have spatial memory in the real world. For instance, she says that a realistic particle would eventually travel in an imperfectly straight line or experience friction, which could radically change or even eradicate the patterns that the researchers found.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Karmela Padavic-Callaghan*