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Lead: A Fresh Set of Chess‑Inspired Brain TeasersToday the Guardian presents four quirky chess puzzles, curated by We Solve Problems, a UK charity that runs free maths circles for secondary‑school students. The puzzles range from graph‑theoretic parity proofs to knight’s tours, inviting readers to engage with mathematics through the lens of chess.Odd‑Numbered Games: Proving an Even Count of PlayersThe first puzzle asks participants to demonstrate that in any tournament where some players have played an odd number of games, the number of such players must be even. This classic result stems from the handshaking lemma in graph theory, where each game contributes two to the total degree sum.Knight’s Tour Challenge: From Bottom‑Right to Top‑LeftThe second puzzle explores whether a knight can start on the bottom‑right corner of an 8×8 board, visit every square exactly once, and finish on the top‑left corner. While a closed knight’s tour exists, the specific start‑end constraint makes the problem a subtle variation that tests spatial reasoning.Pawn Promotion Loop: Minimal Moves to Return HomeThe third puzzle asks for the fewest moves required for a pawn to leave its starting square, promote to a queen, and then travel back to its original position, assuming both players cooperate. Solving it involves coordinating pawn advancement, promotion, and a reverse queen’s path.Four‑Knight Swap on an Irregular GridThe final puzzle presents a strangely‑shaped grid where two pairs of knights must exchange places. A single insightful observation about symmetry unlocks a solution, illustrating how abstract thinking can simplify seemingly complex board problems.Scale of the Maths‑Circle InitiativeMore than a dozen cities across the UK host weekly maths circles.Each circle runs from September to May, targeting pupils aged 7‑11.Volunteer mentors are typically post‑graduates or PhD students.Why Chess‑Based Puzzles Matter for Youth EducationIntegrating chess puzzles into community programmes leverages the game’s universal appeal to foster logical reasoning, combinatorial thinking, and collaborative problem‑solving. By linking puzzles to popular documentaries about Judit Polgár and Hans Niemann, the charity taps into current cultural interest, boosting participation.Looking Ahead: Expanding Collaborative Math OutreachGiven the positive response, We Solve Problems plans to broaden its reach, potentially adding new puzzle formats and digital platforms. Continued media coverage could attract more volunteers and funding, ensuring that quirky challenges like these remain a staple of UK maths education.

The Bridge Between Art and ScienceWhy do we find things beautiful? More precisely, why do some paintings of coloured dots in rippling patterns inspire in me something like revelation? The idea that beauty is the feeling you get when encountering truth is unfashionable in the arts, but lingers in the sciences. The physicist Paul Dirac once proposed that it is more important that a formula is beautiful than that it can be proven: when a perfectly beautiful theory produces results that cannot be real, he argued, then we should not discard the theory but reconsider what is real.Since the 1970s, Terry Winters has been rebuilding that bridge between art and science. Taking inspiration from disciplines including botany – his early paintings, particularly, evoke sprouting pods and tangled roots – engineering, computer modelling and cybernetics, his paintings might be understood as diagrammatic approximations of the patterns that govern everything from the division of cells to the constellation of stars. If every era has to renew its standards of beauty to reflect new understandings of how the world is constructed, then Winters comes as close to providing that model as any living painter.Mathematical Patterns in Visual ArtThese eight new works take their titles from the language of geometry and mathematics: Area, Array, Field, Locus, Point, Scope, Sequence and Set. Each is composed of overlapping patterns that pull each other out of shape according to invisible laws of attraction and repulsion. Field is indicative: on a sooty pink ground, a dense grid of dusty blue cells bends inwards like a trampoline beneath a bowling ball, while an intersecting arrangement of larger circles swells outwards. The disorienting effect of this push-pull is exaggerated by an optical illusion, created by ragged phosphorescent orange haloes around the blue circles, that makes them appear to be craters sunk into the crust of paint. You have to walk up to the surface to be reassured that it is flat.Point shows a teeming landscape of cells bulging outwards at the centre, as if a scum of frogspawn had been skimmed off a pond and placed under a paperweight microscope. In Sequence, a storm of pink interference skims through a yellow circle split like a brain into hemispheres, while a nebulous weather system sweeps in from the right. In Scope, vaguely symbolic arrangements of freshwater blue and desert orange nodules move across the familiar circle-within-a-square geometry of Leonardo's Vitruvian Man. Where that famous demonstration of sacred geometries exudes stillness and calm, the systems that run through Winters' worlds are wild and strange. Here, as much importance is given to the subjective factors of perception and consciousness as the objective principles of logic and proportion.Optical Illusions and Sensory ExperienceAll of this should not distract from the pure sensory pleasure afforded by these pictures. In Locus, another optical illusion seems to lift the red edge of the painting off the canvas like a clumsy wooden frame, squeezing the pockmarked sphere at its centre so that it balloons outwards, threatening to burst. The same cadmium red, so sandy that the pigment seems barely to have been suspended in oil, turns carmine pockets into rock formations that climb off the surface of Set. These sleights of hand evoke the more secular movement of op art, in which patterns are manipulated to exploit the idiosyncrasies of human perception rather than to uncover deeper truths. Which begs the question: is this all just a conjuring trick?Rediscovering the Renaissance ApproachThere is something magical about these works. Even though Winters is generally credited with having extended the lineage of modernist American painting into the present, his practice is in this sense pre-modern. In its rejection of the idea that art should be separated from science, it resembles the Renaissance attitude according to which painting is no less a tool for understanding the world than mathematics, and magic is just the name for things we don't yet understand. His commitment to synthesising diverse spheres of knowledge, his alchemist's commitment to materials, and his sheer technical ability certainly mark him out in a scene recently overwhelmed by chancers and charlatans, bandwagoners and snake-oil salesmen. That his work is enjoying a revival might be taken as an encouraging sign.Beauty as a Path to TruthDirac, incidentally, was proved right. He formulated a theory that was so beautiful it couldn't possibly be wrong, even though it implied the existence of phenomena that everyone agreed must be impossible. He stood by it, and a few years later, someone else discovered anti-matter. Winters' paintings likewise offer a flash of those secret patterns that underpin the physical world, and which science has yet to illuminate. Which is to say, they're beautiful.Exhibition DetailsTerry Winters: Along the River is at Modern Art, London, until 11 July

The Master of Mind-Bending Staircases Maurits Cornelis Escher, a master of the impossible, has been fascinating audiences for decades with his mind-bending staircases and buildings that twist and turn in impossible ways. A new exhibition at Somerset House in London takes visitors on a journey through Escher's metaverse, showcasing his most iconic works and exploring the mathematical concepts that underpin his art. The Art of Mathematics Escher's work is often described as surreal, but he was not a surrealist. Instead, he was a traditional artist with a deep understanding of mathematics and geometry. His use of tessellations, symmetry, and perspective created a unique visual language that continues to inspire artists, scientists, and mathematicians today. A Journey Through Escher's Career The exhibition at Somerset House features a wide range of Escher's works, from his early designs to his most famous prints. Visitors can explore his use of different techniques, such as lithography and woodcuts, and see how his style evolved over time. The exhibition also includes interactive installations and videos that bring Escher's art to life. The Influence of Escher's Art Escher's art has had a profound influence on contemporary culture, inspiring musicians, architects, and artists. His work has been referenced in music, film, and literature, and continues to be a source of inspiration for creatives around the world. The Exhibition Experience The MC Escher exhibition at Somerset House is a must-see for anyone interested in art, mathematics, or simply the impossible. With its interactive installations, giant metal spheres, and chessboard floors, the exhibition is a fun and engaging experience that will leave visitors of all ages inspired and amazed.

The LeadThe Guardian's recent publication of four chess puzzles highlights the fascinating intersection of chess and mathematics. These puzzles, while appearing simple on the surface, reveal deep mathematical principles that govern both chess gameplay and broader problem-solving strategies.Tournament Mathematics: The Oddities PuzzleThe first puzzle, 'Oddities,' demonstrates a fundamental principle of tournament mathematics: in any chess tournament, the number of players who have played an odd number of games must be even. This isn't just a chess rule but a mathematical certainty that applies to any pairing system where each game involves two participants.Knight's Movement and Chessboard ColoringThe second puzzle, 'L of a trip,' explores the knight's unique movement pattern and its relationship to the chessboard's coloring. The solution reveals how the knight's L-shaped movement alternates between white and black squares, making certain journeys impossible—particularly traveling from one corner to its opposite corner.Mathematical Optimization in Chess ProblemsThe third puzzle, 'Pawn return,' showcases how mathematical optimization can solve seemingly complex problems. By finding the minimal sequence of moves (just six), we see how chess problems can be approached systematically rather than through trial and error.Abstraction and Spatial ReasoningThe fourth puzzle, 'Four knights,' demonstrates the power of abstraction in problem-solving. By converting the spatial problem into a numbered sequence, the solution becomes manageable—illustrating how mathematical representation can simplify complex spatial challenges.Educational Value of Chess PuzzlesThese puzzles, provided by We Solve Problems, highlight chess's value as an educational tool. They develop not just chess skills but mathematical thinking, pattern recognition, and logical reasoning—skills transferable to numerous academic and professional fields.The puzzles demonstrate how chess serves as a gateway to mathematical thinking, making abstract concepts tangible through gameplay. This approach aligns with modern educational trends emphasizing experiential learning and interdisciplinary connections.The Future of Chess and Mathematical EducationAs educational methods evolve, chess puzzles are likely to play an increasingly important role in STEM education. Digital platforms can provide interactive versions of these puzzles, while educational initiatives can use them to demonstrate mathematical principles in engaging ways.We may also see the development of new chess variants specifically designed to teach particular mathematical concepts, further cementing chess's role as both a game and a learning tool. The tradition of challenging chess puzzles will continue, adapted to new technologies and educational needs while maintaining their core appeal of testing and developing human reasoning abilities.

Sunak’s Financial Literacy Initiative Stirs ControversyPrime Minister Rishi Sunak has called for a nationwide push to teach children how to handle money, insisting that the UK lags behind countries such as Germany. His broader vision ties financial literacy to an ambitious plan to keep maths in the classroom until the age of 18, sparking a heated debate among educators, former ministers and commentators.Proposed Extension of Maths to Age 18 and Its RationaleSunak’s proposal frames mathematics as the gateway to sound financial decisions. He argues that without a solid grounding in arithmetic, percentages and interest rates, young people cannot navigate inflation, assess risk or detect scams. The plan would make advanced maths a compulsory subject through the end of secondary education, effectively reshaping the national curriculum.Youth Unemployment and Education Gaps: The Numbers Behind the DebateApproximately 1 million 16‑24‑year‑olds are currently not in education, training or employment – roughly one in seven of them hold university degrees.This inactivity rate is double that of Ireland and three times higher than the Netherlands.Recent government measures aim to create 200,000 new apprenticeships, yet the overall transition support for school leavers remains weak.Why the Curriculum Push Could Reshape UK Education and EconomyThe emphasis on compulsory financial numeracy challenges the long‑standing “academic‑first” model of British schooling, which prioritises examinations over practical life skills. Critics warn that making advanced maths mandatory may marginalise students who would benefit more from broader competencies such as health literacy, civic engagement and basic budgeting. If adopted, the policy could influence employer expectations, apprenticeship uptake and long‑term economic productivity.What the Next Five Years May Hold for Financial Literacy in SchoolsShould the government follow through, we can expect a phased rollout of new curricula, teacher training programmes and assessment frameworks centred on real‑world financial scenarios. However, resistance from teachers’ unions and concerns over curriculum overload could delay implementation. In the medium term, successful integration may lower youth financial insecurity and improve labour‑market readiness, while failure could reinforce the gap between academic qualifications and employability.

The Evolutionary Perspective on Gender DifferencesAccording to evolutionary psychologist Steve Stewart-Williams, almost everyone gets sex wrong. Traditionalists tend to exaggerate natural differences between men and women, while progressives tend to minimize them and assume that nurture and socialization play a decisive role. In his book "A Billion Years of Sex Differences," Stewart-Williams promotes a more nuanced, scientifically rigorous public conversation about why and how men and women differ to guide better policymaking.Understanding the Spectrum of Sex DifferencesStewart-Williams identifies varying degrees of sex differences. Some are relatively pronounced, such as attraction patterns, upper body strength, height, likelihood of committing violence, and occupational interests. Others, like ability in mathematics or conscientiousness, are much more modest. These differences are best visualized as two overlapping bell curves. For example, while the average man is taller than the average woman, there is considerable common ground—knowing someone is 5ft 8in doesn't enable confident gender identification.Research Evidence and Scientific BiasesAs a professor of psychology at the University of Nottingham Malaysia, Stewart-Williams examines various physical, psychological, and cognitive sex differences through an evolutionary lens. He identifies common biases in his field, including "gamma bias" (minimizing differences that paint men in a better light while highlighting those that do the opposite) and "delta bias" (an aversion to traditional sex differences and a preference for the reverse).His research found that when presented with fictitious studies showing men drawing better, lying less, or being more intelligent, people rated them as lower in quality, more harmful, and more worthy of censorship than studies showing women possessed these positive traits.Implications for Gender Equality and PolicyStewart-Williams argues that innate differences don't imply one sex is better than the other, nor do they suggest a moral imperative to enforce sex differences. However, he believes that when given freedom of choice, men and women tend to gravitate toward different directions—men toward working with things and status, women toward working with people and relationships.Interestingly, various studies find that many sex differences—from occupational preference to personality traits—are more pronounced in more gender-equal countries, suggesting that when social constraints are removed, natural tendencies become more evident.The Future of Gender ResearchWhile Stewart-Williams presents compelling evolutionary explanations for sex differences, the reviewer notes that evolutionary psychology isn't always the most useful lens. For example, understanding domestic violence as primarily a manifestation of male aggression rather than patriarchy may overlook how socialization and culture shape behavior in different societies.The book contributes valuable insights to the nature versus nurture debate, but ultimately represents only one part of a complex story about human development and gender relations. Future research will likely need to integrate biological, psychological, and sociological perspectives to fully understand the intricate interplay of factors that shape who we are.

The High-Stakes Showdown at Tottenham Hotspur Stadium Oh Tottenham. How has it come to this? Champions League finalists in 2019, and Europa League winners just 12 months ago, one of the grandest clubs in England are now possibly 90 minutes plus fingernail-and-artery-bothering add-ons away from relegation to the second tier for the first time in nearly half a century. The Mathematics of Survival Ah but let's leave those questions for later. And they'll be a lot less pointed should Spurs escape the executioner this afternoon. The maths are simple: beat Everton and they stay up. A draw will almost certainly be enough, unless West Ham beat Leeds 12-0, which, given that scoreline would tie a top-flight record set 134 years ago* and matched only once since, 117 years ago†, is not going to happen. But a defeat … hoo boy. Spurs' Path to Safety The good news for Spurs: under Roberto de Zerbi, they've lost just one of their last five fixtures. And even if the worst was to happen, West Ham would still need to win to leapfrog them to safety. However, Spurs have only won two matches at home all season, against Burnley and Brentford, while Everton have the sixth-best away record in the division, winning seven of 18. Also in David Moyes, Everton have a manager who is on record saying he'd "love to keep West Ham in the league if I can", having previously managed them to European glory. The Tension Builds at N17 So imagine the tension in N17 should Everton score first this afternoon. But Spurs beat the Toffees 3-0 on their own patch last October, so glee is as likely an outcome for the hosts as misery. At least that's what they've got to tell themselves as they go into their most important game for a generation … and the bean counters will tell you that yes, that does include last year's Europa League final. Kick-off is at 4pm BST. It's on! *: West Bromwich Albion 12-0 Darwen (April 1892) †: Nottingham Forest 12–0 Leicester Fosse (April 1909)

OpenAI has reported a major advance in AI reasoning after its model successfully challenged an 80‑year‑old conjecture in discrete geometry, the planar unit distance problem first posed by Paul Erdős in 1946.OpenAI’s Model Cracks the 80‑Year‑Old Planar Unit Distance ConjectureThe conjecture suggested that the number of equal‑distance dot pairs on a plane grows only slightly faster than the number of dots.OpenAI's reasoning system generated a family of point arrangements that exceed Erdős’s proposed limit.The result was announced on X and confirmed in a companion paper co‑authored by mathematician Thomas Bloom.Quantifying the Breakthrough: No Monetary Figures, but Scientific SignificanceWhile the article provides no financial data, the achievement is described as a “milestone in AI mathematics” by Tim Gowers.The validation by experts underscores the credibility of AI‑generated proofs, contrasting with a prior, unverified claim from last year.Implications for AI‑Driven Mathematical ResearchThe model’s ability to explore unconventional solution paths highlights AI’s potential to augment human intuition.Researchers, including Andrew Rogoyski, note that AI is becoming a fundamental tool for future scientific inquiry.The breakthrough may accelerate AI involvement in other open problems across mathematics.What the Next Steps Could Mean for AI and MathematicsFurther collaboration between AI systems and mathematicians is expected to refine the new constructions and explore their consequences.OpenAI’s upcoming IPO could bring additional resources to expand its reasoning capabilities.The community anticipates more AI‑driven insights that could eventually resolve the broader Erdős problems.

The Lead: Guardiola's Immediate FocusDespite securing Manchester City's second trophy of the season with a 1-0 FA Cup final victory over Chelsea, manager Pep Guardiola has made it clear there will be no immediate celebrations. The Spanish tactician emphasized that his team's attention remains firmly on the Premier League title race, where City currently trails leaders Arsenal by two points with just two matches remaining.The Event Details: Semenyo's Moment of MagicAntoine Semenyo, who joined City from Bournemouth in January, emerged as the unlikely hero with a stunning 72nd-minute backheel goal that settled the final at Wembley. The strike came after City had limited preparation time compared to Chelsea, with Guardiola complaining about the challenges of travel logistics in England.This victory marks City's second consecutive cup triumph following their Carabao Cup win over Arsenal in April, and the 20th trophy of Guardiola's decade-long tenure at the club, including three Community Shields.The Data Analysis: Title Race MathematicsThe Premier League standings present a compelling narrative as the season approaches its conclusion:Arsenal leads with 79 points after 36 matchesCity sits on 77 points after 36 matchesCity faces Bournemouth (away) and Aston Villa (home) in their final two matchesArsenal hosts Burnley and travels to EvertonAny slip from Arsenal could allow City to claim their seventh title under GuardiolaThe mathematical reality is that City must win both matches and hope for an Arsenal stumble to retain their crown.The Impact Analysis: Managerial Uncertainty and Club CultureGuardiola's refusal to celebrate extends beyond mere tactical focus, potentially hinting at his future plans with the club. When pressed about his future, the manager offered an opaque response, stating: "The moment the fire decreases, then we will go home. I am a funny guy." This cryptic comment has fueled speculation that he may depart this summer after a decade at the helm.The manager also emphasized the club's commitment to celebrating with both the men's and women's teams, suggesting a broader cultural approach to success that transcends individual trophies.The Prediction: Title Race Decider and Potential Guardiola Era ConclusionWith the title race entering its final week, City's fate rests not just on their own results but on Arsenal's performance. The psychological advantage may lie with City, who have experience in title deciders and have overcome similar deficits in previous seasons.Meanwhile, Guardiola's future remains the subject of intense speculation. If City fails to retain the Premier League title, it could potentially signal the end of his transformative era at the club, having established them as a dominant force in English and European football.For Semenyo, the young forward who has quickly adapted to City's demands, the focus remains clear: "Hopefully, we can finish the job off [in the league]." The player's journey from Bournemouth to potential title winner encapsulates the club's ability to identify and develop talent, even as they pursue the highest honors.