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Category: Mathematics

  • OpenAI and Google Outperform Top Math Students But Not one Another

    OpenAI and Google Outperform Top Math Students But Not one Another

    Image Credit: Pixabay

    AI systems from OpenAI and Google DeepMind have earned gold-medal scores in the 2025 International Math Olympiad (IMO), one of the most prestigious and difficult math competitions for high school students, the companies announced separately in recent days.

    The achievement highlights the rapid progress of AI and how closely matched OpenAI and Google remain in the competition for AI dominance. In a race where perception matters as much as performance, these milestones can influence who attracts the best AI talent, particularly since many top researchers come from competitive math backgrounds, making IMO results especially meaningful.

    Last year, Google’s entry earned a silver medal using a “formal” approach, which required human input to translate problems into a format the AI could understand. This year, both companies used “informal” systems that processed natural language directly, solving and explaining five out of six problems, outperforming most human competitors and Google’s previous system, with no human translation needed.

    IMO Success Highlights Breakthroughs in AI Reasoning Beyond Clear-Cut Tasks

    In interviews with TechCrunch, researchers from OpenAI and Google explained that their AI models’ gold-medal performances in the IMO mark significant progress in AI reasoning within areas where solutions can’t be easily verified. While AI typically performs well on tasks with clear-cut answers—like basic math or coding—it’s much more challenged by open-ended problems, such as offering furniture recommendations or assisting in complex research.

    However, tensions are rising over how OpenAI handled the announcement of its IMO success. In a move reminiscent of high school rivalries, Google is now questioning the timing and validation of OpenAI’s claims.

    Soon after OpenAI shared its results on Saturday morning, just hours after the IMO revealed its top student winners on Friday night, Google DeepMind’s CEO and researchers criticized the announcement. They argued that OpenAI jumped the gun by declaring a gold medal before having its model’s results officially reviewed by the IMO.

    Thang Luong, senior researcher at Google DeepMind and lead on the IMO project, told TechCrunch that the company chose to delay its announcement out of respect for the students competing in the event.

    Luong explained that Google had been collaborating with IMO organizers since last year to prepare for the competition and chose to wait for the IMO president’s approval and official grading before making its announcement, which came on Monday morning.

    “The IMO organizers have specific grading guidelines,” Luong said. “So any evaluation not aligned with those standards can’t credibly claim a gold-medal performance.”

    OpenAI Focused on Language Models, Unaware of IMO’s Informal Test with Google

    Meanwhile, Noam Brown, a senior researcher at OpenAI who worked on its IMO model, told TechCrunch that IMO had contacted OpenAI months ago about joining a formal math competition, but the company declined, focusing instead on developing natural language-based systems. According to Brown, OpenAI was unaware that IMO was conducting an informal evaluation with Google.

    To assess its own model, OpenAI hired three former IMO medalists familiar with the grading criteria to serve as independent evaluators. After determining the model had achieved a gold-medal–level score, OpenAI contacted IMO, which advised the company to hold off on announcing results until after the official student awards ceremony on Friday night.

    IMO did not respond to TechCrunch’s request for comment.

    Although Google followed a more formal and vetted process, the broader takeaway may be more important: leading AI labs are making rapid progress. At this year’s IMO, only a small fraction of the world’s brightest students matched the scores achieved by the AI models from OpenAI and Google.

    While OpenAI once held a clear edge in the field, the competition appears tighter than ever—though few in the industry may want to admit it. With GPT-5 expected soon, OpenAI is undoubtedly aiming to reinforce its position at the forefront of the AI race.

    Read the original article on: TechCrunch

    Read more: Need to Tackle a Complex Problem? Applied Mathematics Can Provide the Solution

  • It’s Pi Day! Here’s Why This Unique Number Deserves Worldwide Recognition.

    It’s Pi Day! Here’s Why This Unique Number Deserves Worldwide Recognition.

    (a4ndreas/iStock/Getty Images Plus)

    Each year on March 14, math enthusiasts celebrate Pi Day, honoring one of the most famous irrational numbers. Represented by the Greek letter π and approximated as 3.14, pi highlights both the practicality and beauty of mathematics.

    Pi defines the ratio of a circle’s circumference to its diameter, an unending, non-repeating decimal. Unlike other constants like e or the golden ratio, pi has gained widespread recognition, though some argue tau deserves more attention.

    The 18th-century mathematician William Jones introduced the symbol π, likely short for “periphery.” Before that, fractions and descriptions attempted to capture its infinite nature. While Jones suspected pi’s exact value could never be fully expressed, Johann Lambert finally proved its irrationality in the 1760s.

    For thousands of years, civilizations estimated pi’s value. Around 4,000 years ago, the Babylonians used 3.125, while an Egyptian text from 1650 BCE suggested 3.16049. Archimedes refined the calculation, estimating pi between 3 1/7 and 3 10/71 using polygons.

    Why Pi Matters

    (British Museum Department of Ancient Egypt and Sudan/PD)

    As a fundamental mathematical constant, pi appears in countless equations, from geometry to physics. It describes natural phenomena like river meandering and atomic structures. Even in unrelated areas, pi surfaces in surprising ways—for example, the probability that two random numbers are relatively prime involves 6/π².

    Beyond math, pi captivates people with its poetic nature. In 1988, physicist Larry Shaw noticed that March 14 (3/14) mirrored pi’s first digits, inspiring the first Pi Day celebration. Now, people worldwide mark the occasion by sharing math trivia, solving problems, and, of course, baking circular pies.

    (ScienceAlert)

    Read Original Article: Science Alert

    Read More: Mathematician Solves Long-Standing Moving Sofa Puzzle

  • Amateur Discovers The Largest Known Prime Number, And it’s Gigantic

    Amateur Discovers The Largest Known Prime Number, And it’s Gigantic

    Credit: Pixabay

    Breaking a Six-Year Drought in Prime Discoveries

    A recent participant in the Great Internet Mersenne Prime Search (GIMPS) has broken a six-year stretch without new discoveries in the quest for large prime numbers, among the usual composites.

    This new prime number has an impressive 41,024,320 decimal digits—it would take months to write out in full. To simplify, it’s 1 less than 2 raised to the power of 136,279,841. Officially named M136279841, it’s the latest milestone in mathematics.

    Luke Durant, a former NVIDIA employee, only began contributing to the search last October, but had more than beginner’s luck on his side. He used thousands of graphics processing unit servers spread across 24 datacenter regions in 17 countries to run the software that helped in the discovery.

    Confirmed as a Prime Giant

    On October 11 of this year, a server in Dublin identified M136279841 as a possible prime. The next day, another server in Texas confirmed its authenticity, sealing its legendary status.

    Prime numbers are integers greater than 1 that are not products of two smaller numbers. Although they seem simple at first, like 2, 3, and 5, indivisible numbers become rarer as we count higher, raising the question of whether they eventually run out. To the relief of mathematicians, the answer is no: primes are infinite, though hard to find.

    Although technology has evolved, the search for massive prime numbers hasn’t changed much since the 17th-century French friar Marin Mersenne popularized a method for finding primes of a specific type that bears his name.

    What Are Mersenne Primes?

    “Mersenne primes” follow the formula 2ⁿ – 1. However, not all numbers in this form are prime—an example is 2 raised to the 4th power minus 1, which results in 15, a composite number (divisible by 3 and 5). Likewise, not all primes are of Mersenne type.

    But this approach is particularly effective, and GIMPS, founded in 1996, adopted it to identify 18 of these giants so far, bringing the known total to 52.

    The previous record, discovered in 2018 by Patrick Laroche from Ocala, Florida, had almost 25 million digits. He used his own equipment for the search, while Durant relied on a GPU network, marking a new era in the hunt for Mersenne primes.

    The Appeal of Massive Primes

    Why search for such large numbers? Beyond prestige, cash prizes, and the admiration of fellow enthusiasts, there aren’t many practical reasons. George Woltman, co-founder of GIMPS, explained that it’s essentially entertainment for math fans.

    Prime numbers are useful for certain types of encryption, though quantum computing may soon limit this application. Even so, primes are the “atoms” of all positive integers and possess a unique beauty.

    Soon, a new Mersenne prime may appear within the expanding global network of technology. And it will be number 53 on the list. A prime number.

    Read the original article on: Science Alert

    Read more: Number Theory’s Genetic Insight

  • Need to Tackle a Complex Problem? Applied Mathematics Can Provide the Solution

    Need to Tackle a Complex Problem? Applied Mathematics Can Provide the Solution

    You’ve likely used math to solve common tasks, like figuring out a restaurant tip or calculating a room's square footage. But how does math contribute to tackling more complex challenges, such as finding a cure for a disease?
    Credit: Pixabay

    You’ve likely used math to solve common tasks, like figuring out a restaurant tip or calculating a room’s square footage. But how does math contribute to tackling more complex challenges, such as finding a cure for a disease?

    As an applied mathematician, I use mathematical tools to analyze and solve complex biological problems. My work has involved studying gene and neural networks, such as cell interactions and decision-making processes. To tackle these challenges, I translate real-world scenarios into mathematical terms—a process known as modeling.

    Mathematical Modeling in the Real World

    If you’ve ever solved an arithmetic problem about train speeds or grocery costs, you’ve engaged in mathematical modeling. However, with more complex issues, simply framing the real-world scenario as a math problem can be challenging. This process demands creativity and a deep understanding of the problem, often involving collaboration between applied mathematicians and scientists from various fields.

    For instance, a game of Sudoku can be represented as a mathematical model. In Sudoku, the player fills empty boxes with numbers from 1 to 9, following rules such as no repeated numbers in any row or column.

    Once a mathematical problem is created to represent a real-world situation, the next step in modeling is solving that problem.

    Diverse Mathematical Tools for Complex Problems

    For Sudoku, this involves solving a system of equations with 81 variables. In the aspirin example, it requires solving an equation that describes how concentrations change over time. This is where various areas of math, such as algebra, analysis, combinatorics, and others, come into play. These fields, often used in combination, help tackle the complex math problems that arise from real-world applications.

    The third step of modeling is translating the mathematical solution back into the context of the original problem. For Sudoku, the solution provides the correct number for each box in the puzzle. In the aspirin example, the solution would be a set of curves showing the concentration of aspirin in the digestive system and bloodstream over time. This is the essence of applied mathematics.

    The Challenge of Unsolvable Problems in Applied Math

    Or is it? While this three-step process represents the ideal workflow in applied math, reality is often more complex. When I reach the second step and need a solution for the math problem, it’s common—if not typical—to discover that no one knows how to solve it. In some cases, the necessary mathematical tools to analyze the problem don’t even exist yet.

    For instance, analyzing cancer models is challenging because the interactions between genes, proteins, and chemicals are far more complex than the relationships between boxes in Sudoku. The main challenge is that these interactions are “nonlinear,” meaning the combined effect of two inputs isn’t just the sum of their individual effects. To tackle this, I’ve been developing new methods to study nonlinear systems, such as using Boolean network theory and polynomial algebra. With these and more traditional approaches, my colleagues and I have explored areas like decision-making, gene networks, cellular differentiation, and limb regeneration.

    When dealing with unsolved problems in applied math, the line between applied and pure mathematics often blurs. Concepts once seen as highly abstract have proven essential for addressing modern challenges. This underscores the value of math for everyone—today’s pure mathematics could become tomorrow’s applied math, offering the tools needed to solve complex, real-world problems.

    Read the original article on: Phys Org

    Read more: Math Behind Meeting Scheduling Headaches

  • Math Behind Meeting Scheduling Headaches

    Math Behind Meeting Scheduling Headaches

    "What's the date three years from July 28?"
    “What’s the date three years from July 28?”
    Image credit: Andrey_Popov/Shutterstock.com

    A new study by physicists explores a common frustration: scheduling meetings. The central question is how challenging it is to find a time when everyone is available. As expected, the results weren’t encouraging.

    “We wanted to understand the odds,” said Harsh Mathur, a physics professor at Case Western Reserve University. “Probability theory, which started with gambling, is just as relevant to scheduling.” The team used mathematical modeling to determine how the likelihood of scheduling a meeting decreases as the number of participants increases.

    Their findings revealed that the number of potential meeting times to consider grows exponentially with more participants. “Initially, the project seemed like a joke, but this exponential growth highlighted how complex planning is, akin to major computer science problems,” Mathur noted.

    The study also uncovered a tipping point where scheduling becomes nearly impossible with just four or five participants, depending on available timeslots. This sudden shift is similar to physical phase transitions, like ice melting into water. “It’s remarkable how scheduling complexity parallels phase transitions,” Mathur remarked.

    Beyond scheduling, the study’s models have broader implications. “Our sophisticated models could have wider applications,” Mathur said. Katherine Brown, co-author and Associate Professor at Hamilton College, suggests that these models could benefit any consensus-driven problem, such as international climate agreements. Ultimately, the study confirms what many already know: scheduling meetings can be a significant challenge. “Consensus-building is hard,” Mathur concluded.

    Read the Original Article: IFL SCIENCE

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  • Mathematician Explains Equals Has Multiple Meanings

    Mathematician Explains Equals Has Multiple Meanings

    Mathematics has many abstract concepts that are often difficult to grasp, but we assumed the meaning of 'equals' was well understood. However, it appears that mathematicians disagree on the exact definition of equality, which could pose challenges for computer programs increasingly used to verify mathematical proofs.
    Credit: Depositphotos

    Mathematics has many abstract concepts that are often difficult to grasp, but we assumed the meaning of ‘equals’ was well understood. However, it appears that mathematicians disagree on the exact definition of equality, which could pose challenges for computer programs increasingly used to verify mathematical proofs.

    This academic debate has simmered for decades but has become critical because computer programs used to formalize or verify proofs require precise and unambiguous definitions, not ones open to interpretation or context that computers lack.

    British mathematician Kevin Buzzard of Imperial College London encountered this issue while collaborating with programmers, leading him to reevaluate the definition of ‘equals‘ to challenge common assumptions about equality.

    Buzzard’s Revelation on Mathematical Equality

    Buzzard reflects in his preprint on the arXiv server that six years ago he believed he understood mathematical equality as a well-defined concept. However, working with computer theorem provers at the master’s level revealed that equality is a more complex and thorny issue than he had realized.

    The equals sign (=), created by Welsh mathematician Robert Recorde in 1557 to symbolize equality between objects, was initially slow to gain acceptance. It eventually replaced the Latin term ‘aequalis’ and set the stage for computer science, making its debut in the programming language FORTRAN I exactly 400 years later, in 1957.

    The concept of equality dates back to ancient Greece, but modern mathematicians use it somewhat loosely, according to Buzzard.

    Traditionally, mathematicians use the equals sign in equations to demonstrate that different mathematical objects share the same value or meaning, a relationship verified through transformations. For instance, the integer 2 can represent a pair of objects, just like 1 + 1 does.

    Set Theory’s Influence on Equality

    Since the late 19th century, another definition of equality has been in use, originating with set theory. As set theory developed, the notion of equality expanded. For example, mathematicians can consider the set {1, 2, 3} equal to the set {a, b, c} because of canonical isomorphism, which evaluates the structural similarities between groups.

    Buzzard explains that mathematicians found it practical to call such sets equal because they align naturally, as he noted to Alex Wilkins of New Scientist.

    However, this approach to equality, known as canonical isomorphism, is now causing issues for mathematicians trying to formalize proofs with computers, even affecting foundational concepts established long ago.

    “None of the existing computer systems capture how mathematicians like Grothendieck use the equals symbol,” Buzzard told Wilkins, referring to Alexander Grothendieck, a key 20th-century mathematician who used set theory to describe equality.

    Some mathematicians propose redefining concepts to formally equate canonical isomorphism with equality.

    Buzzard disagrees, urging mathematicians to reevaluate foundational concepts like equality to bridge the gap between their understanding and what computers can process.

    “When forced to clearly define what one means without relying on vague terms,” Buzzard writes, “one sometimes has to do additional work or rethink how certain ideas should be presented.”

    Read the original article on: Science Alert

    Read more: How to Make Math Lectures More Fun

  • How to Make Math Lectures More Fun

    How to Make Math Lectures More Fun

    Credit: Canvas

    In high school math classes, play often disappears from the learning process. However, Kathy Sun is finding new ways to make math playful in high school.

    According to Sun, students usually focus on mastering procedures in math, rather than understanding math’s broader contributions and community aspects. Sun, who is a researcher and professor at Santa Clara University, believes that making math playful is not just a fun teaching method but also helps students deeply understand math and make connections.

    Dan Finkel, a math curriculum writer and former teacher, sees play in math as rigorous, even at advanced levels. He says that the best mathematicians are those who can play with complex ideas.

    Math Concepts

    When high school students play with math concepts, they move beyond memorization. Sun highlights three elements of play in math classes: exploration, creativity, and collaboration. When math activities involve these elements, even quiet students tend to participate more, feeling validated and belonging, which improves their persistence in learning math.

    Exploration in math means teachers help students take ownership of the material. Sun suggests increasing cognitive demand by moving away from memorization and procedural tasks. Instead, students can make creative connections and understand why things work, leading to deeper understanding over time.

    Creativity in Math

    Creativity in math allows students to discover rules in unconventional ways. Sun gives an example of a teacher who asked students to find various percentage points of a number without using a formula, leading them to discover mathematical principles through their creativity.

    Collaboration is key in math learning. Sun recommends activities like card sorting and group problem-solving to deepen students’ understanding and encourage creative engagement with math concepts.

    To incorporate play in math classes, Sun advises starting with small bursts of play and gradually increasing them. Online resources like Math Equals Love and YouCubed offer creative activity ideas for math teachers looking to make their classes more playful.

    Read the Original Article KQED

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  • Study Suggests Math Homework May Harm More Than Help

    Study Suggests Math Homework May Harm More Than Help

    A recent study indicates that assigning math homework to students can have adverse effects, especially when the tasks are overly complex, making it difficult for children to complete them even with parental assistance.
    Credit: Pixabay

    A recent study indicates that assigning math homework to students can have adverse effects, especially when the tasks are overly complex, making it difficult for children to complete them even with parental assistance.

    Researchers from the University of South Australia and St. Francis Xavier University in Canada conducted interviews with eight Canadian families. These interviews focused on the families’ experiences with mathematics homework and its impact on their dynamics.

    All the families included in the study had a child in grade 3, typically aged 8 or 9, coinciding with the introduction of standardized math tests in the region.

    Generally, participants perceived mathematics as a disliked subject that demanded excessive additional effort.

    Reevaluating the Role of Homework in Academic Achievement

    Lisa O’Keeffe, a senior lecturer in mathematics education at the University of South Australia, states, Traditionally, we have regarded homework as a method to reinforce children’s learning and enhance academic achievement. “However, when assignments prove excessively challenging for students, even with parental support, it prompts questions about their appropriateness.”

    The study identified various issues, including homework being overly difficult, encroaching on bedtime and family time, and inducing feelings of inadequacy and frustration.

    As with many subjects, teaching methods in mathematics can evolve over time, causing frustration for parents accustomed to different approaches than their children.

    “Mathematics teaching methods have evolved,” explains O’Keeffe. “But parents may struggle to adapt to new methods, adding additional pressure.”

    This dynamic can perpetuate negativity across generations, particularly as mothers are often primarily responsible for assisting with homework. When they too find assignments challenging, it can reinforce stereotypes about mathematics being a subject where girls may not naturally excel, impacting their academic performance and career aspirations.

    Advocating for Appropriate Math Homework Practices

    While this study’s sample size is limited, its findings align with common narratives in education.

    The researchers advocate for setting math homework in an appropriate manner to prevent early disengagement with the subject, particularly considering the recent pandemic-induced surge in at-home learning and parental involvement in assignments.

    Study author Sarah McDonald, an education lecturer at the University of South Australia, emphasizes that teachers aim to avoid disadvantaging girls in cultivating strong mathematical identities. Therefore, she stresses the necessity for a better comprehension of homework policies and expectations.

    McDonald further notes that homework is commonly believed to offer non-academic advantages, such as promoting independence and enhancing organizational skills and self-discipline. However, the family experiences documented in their study do not necessarily support this notion.

    Read the original article on: Science Alert

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  • The Layers of Logic: A Journey Through Reasoning

    The Layers of Logic: A Journey Through Reasoning

    Logic, as we delve into its depths, reveals itself as more than just a set of rules; it's a systematic exploration of reasoning governed by strict principles.
    Logic, as we delve into its depths, reveals itself as more than just a set of rules; it’s a systematic exploration of reasoning governed by strict principles.

    Logic, as we delve into its depths, reveals itself as more than just a set of rules; it’s a systematic exploration of reasoning governed by strict principles. In this blog, we embark on a journey through the intricacies of logic, from deductive to inductive reasoning, formal to informal logic, and the evolution of logical thought throughout history.

    At its core, logic is the study of arguments’ structure, aiming to uncover truths and establish reliable principles. It emerged from a desire to ensure the correctness of argumentation and find a solid basis for verifying arguments. Renowned logician Godlove Berger aptly described logic’s function as the discovery of truths, making it a fundamental task across all sciences.

    Distinguishing between deductive and inductive validity is crucial. Deductive validity hinges on the certainty that if all premises are true, the conclusion must also be true. On the other hand, inductive reasoning deals with probabilities, where premises lend varying degrees of support to the conclusion.

    Formal logic

    Formal logic delves into structured language arguments, associated with critical thinking, fallacies, and argumentation theory. Informal logic, encompassing deductive and inductive reasoning, highlights the complexity of human thought processes.

    Aristotle’s contribution to deductive reasoning through the syllogism laid the groundwork for logical thinking for centuries, influencing fields like geometry and philosophy. However, with the rise of modern science, Francis Bacon championed inductive reasoning, emphasizing empirical data collection and analysis to derive general conclusions.

    Evolution of Logic

    The evolution of logic led to symbolic logic, applying mathematical symbols to logical analysis. This shift addressed language ambiguities and expanded logical applications beyond traditional Aristotelian logic.

    Non-classical logics

    Non-classical logics challenged binary true or false conclusions, accommodating scenarios with variable answers based on partial knowledge. This flexibility reflects the nuanced nature of reasoning in real-world situations.

    In conclusion, logic is not a rigid set of rules but a dynamic exploration of reasoning methods. It adapts to diverse contexts, from formal systems to everyday reasoning, shaping our understanding of truth and knowledge acquisition. Join us on this journey as we unravel the layers of logic and delve into the essence of reasoning.

    Read more Dispelling the Mirage of Comprehension: MIT Reveals the Fallacy of AI’s Formal Specifications and NASA

  • Do You Know Who Invented Math?

    Do You Know Who Invented Math?

    Credit:Unsplash / Dan Cristian Pădureț.

    Carl Friedrich Gauss famously declared mathematics the “queen of the sciences.” However, as history’s renowned mathematician, he might have been slightly biased. Ask a physicist, and she might retort with the provocative analogy that “physics is to mathematics what sex is to masturbation.”

    But regardless of whether mathematics wears the crown, it undoubtedly stands as the doyenne of the sciences. Its roots extend far back in history, preceding other forms of rational inquiry by millennia. When Ibn al-Haytham laid science foundations in the 10th century, he relied upon centuries of mathematical knowledge and discovery.

    This raises an intriguing question: who initiated this journey?

    Lead us not into temptation

    The earliest “mathematicians” – the first individuals referred to as such in English – were far more formidable than today’s number-crunching enthusiasts.

    In the second quarter of the 15th century, an anonymous translator of Ranulf Higden’s Polychronicon used the term “mathematician” in English for the first time. The passage recounts the reign of Domitian, Emperor of Rome from 81 to 96 CE, notorious for his authoritarian rule. Domitian’s disdain for philosophers, adulterers, mimes, and mathematicians led to their expulsion from Rome.

    Art in heaven

    Saint Augustine of Hippo cautioned against mathematicians, although he likely referred to astrologers rather than mathematicians. Nonetheless, this translation error hints at an earlier era in mathematical history.

    Different ancient civilizations approached mathematics uniquely. The Greeks embraced geometry and logic for theorem development, while the Babylonians focused on astronomy. The oldest Babylonian mathematical records date back to 1600 BCE, reflecting an ancient tradition spanning over 2000 years.

    The first named mathematician

    Egyptians, too, had a profound mathematical legacy, evident in their base-10 numeral system and practical applications in agriculture, administration, and construction, notably demonstrated in the construction of the Great Pyramid at Giza.

    Ahmes, the author of the Rhind Papyrus, emerges as the earliest named mathematician, though little is known about him.

    In the beginning

    However, to trace mathematics’s origins, we must venture beyond written records. The Ishango bone, dating back 20,000 to 25,000 years, bears notches, suggesting an understanding of numerical patterns and concepts such as duplication and prime numbers.

    This artifact, among others, offers a glimpse into the primal origins of mathematics, with its creator representing humanity’s first known mathematician.

    Read the otiginal article on IFL Science.

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