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The Puzzle Today's offering is for fans of the number 4. It's a cute puzzle that offers up its solution in an elegant way. The Problem Statement Nose to tail There is a number N beginning with 4 such that moving the 4 to the end of it creates a new number that is a quarter of N. In other words N is of the form 4[…], where […] is a sequence of digits, and N ÷ 4 = […]4 What is the lowest possible value of N? The Hint HINT. Suppose that N has two digits. If you can't find a solution, suppose that N has three digits. Repeat until done. The Update I'll be back at 5pm UK with the solution. PLEASE NO SPOILERS. Instead discuss snouts. UPDATE: Solution is here. Source: Moscow Mathematical Olympiad 1983, via @mathematicsproblems and Kevin Gately I've been setting a puzzle here on alternate Mondays since 2015. I'm always on the look-out for great puzzles. If you would like to suggest one, email me.

The Plight of Young Jobseekers in the UK About 1 million 16- to 24-year-olds in the UK are not in employment, education, or training, and a recent report warned that this number could rise to 1.25 million by the early 2030s without urgent government action. Readers' Tips for Young Jobseekers We asked readers for their advice on finding employment. Here are some of their responses: 'Don’t Be Afraid of Being Seen as Cringey' My advice would be to make your own luck. Put yourself out there, meet as many different people as you can, post content online, don’t be afraid of being seen as cringey. I messaged so many people at the start of my career for coffees, for help with interviews. Many ignored me. But some helped me along the way. – Christopher, 28, London 'Find Things That Make You Feel Good About Yourself' I was unemployed for a couple of years, with no savings as a safety net, and the constant stream of rejection on top of the worry about my finances was incredibly hard to cope with. My advice is to find cheap hobbies and things that make you feel joyful, connected to nature, and/or creative, to keep yourself sane. – Kay, 35, mathematics tutor, Staffordshire 'Visibility Is Everything' My advice for young people is to talk to as many people as they can, from different backgrounds. Having a LinkedIn profile and being active is essential. Tell people that you’re at the beginning of your career journey. Own it! – Nikola Doanova, 31, Brighton 'Cast Your Net Wide' If you’re struggling to find work, focus on securing a job for now, not for life. Core professional skills, such as communication and teamwork, are highly transferable, and so taking any opportunity to develop those is key. – Ryan McKiernan, 31, managing director of a charity, London 'You Absolutely Have Value to Offer' You can create your own job opportunity; you just need to get a foot in the door. This could be adding value to a business by offering to work on a commission-only basis for a few months. – Tim Castle, author, Singapore 'Take Up the Job Offer' Not having any particular skills or specialties, I came to realise that there would only ever be one job offered to me during a search and my advice to anyone is to accept that one. – Christine, 75, Kent 'Don’t Be Embarrassed to Network' Network. Network. Network. Use every available network you and the people around you have. Follow up! If someone has said they will have a word/email, thank them and don’t end the conversation until you have pinned down a date and time to reconnect about it. – Kuldip, semi-retired teacher, Sheffield

Opening the Escher Metaverse at Somerset HouseThe new MC Escher exhibition, running from 5 June‑6 September 2026, transforms the historic London venue into a mind‑bending journey through the artist’s “metaverse” of impossible architecture and mathematical wonder.Immersive Installations Reveal the Mathematical Roots of Escher’s WorkVisitors encounter large‑scale video projections, giant metal spheres, chessboard floors and interactive sculptures that let them step inside iconic prints such as Belvedere (1958) and Waterfall (1961). The show also highlights Escher’s wartime diploma design of 1945 and his early fascination with tessellation after seeing the Alhambra.Video walls that animate the shifting staircases of RelativityMetal spheres echoing the convex‑mirror motifChessboard floor that reacts to foot trafficWhy the Exhibition Redefines the Intersection of Art and ScienceBy linking Escher’s visual paradoxes to the insights of physicist Roger Penrose and the broader pop‑culture legacy (e.g., Pink Floyd’s Ummagumma cover), the show demonstrates how mathematical concepts can inspire both fine art and popular media.What the Future Holds for Escher’s Legacy in Digital SpacesCurators suggest that the immersive format could evolve into virtual‑reality experiences, allowing global audiences to explore Escher’s impossible worlds online, keeping his “language of mathematics” alive for new generations.

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.

The Lead: Title Race Upended by Controversial CallA controversial penalty awarded to Celtic against Motherwell following a VAR review has dramatically altered the Scottish Premiership title race, setting up a potentially decisive showdown between league leaders Hearts and Celtic at Celtic Park. The decision, which has sparked widespread debate and criticism, has left Hearts frustrated as they now face the prospect of losing their title lead in what could be the biggest Scottish game in a generation.The Event Details: A Night of VAR ControversyThe incident that led to Celtic being awarded a spot-kick against Motherwell has been intensely scrutinized, with referee John Beaton initially declining to award a penalty before being sent to his screen by officials at McStockley Park. The decision has been labeled in some quarters as the most controversial in Scottish football history, with even English pundits like Gary Lineker and Jeff Stelling weighing in on the debate. Football Daily's analysis suggests that while the ball may have hit Motherwell defender Sam Nicholson's arm, the initial decision not to award a penalty was not a clear-and-obvious error, making the subsequent VAR intervention particularly controversial.The Data Analysis: Title Race MathematicsWith the controversial penalty, Celtic secured a crucial victory that kept them in the title hunt, creating a scenario where Hearts now need to avoid defeat at Celtic Park to secure the championship. The mathematical implications are stark: Hearts, having seen off Falkirk at Tynecastle's final match of the season, were left seething after watching the Fir Park incident unfold. The result has set up a winner-takes-all showdown at Celtic Park, with Hearts knowing that even a draw might be insufficient depending on other results in the league.The Impact Analysis: Scottish Football in the SpotlightThe controversy surrounding this decision has once again brought Scottish football under scrutiny, with critics questioning the consistency and application of VAR in the league. Former Rangers player Ally McCoist described it as "a horrendous decision that has once again embarrassed Scottish football," while Hearts head coach Derek McInnes expressed frustration at what he perceived as bias against his team. The incident has reignited debates about the role of technology in football and whether it is enhancing or detracting from the game, particularly in a league where resources for implementing VAR may not match those in top European competitions.The Prediction: Paradise Awaits for CelticAs the title race reaches its climax, all eyes turn to the upcoming match at Celtic Park, where Hearts will need to overcome both their opponents and the weight of history. Recent encounters between these two sides suggest that the main post-match talking points may extend beyond the football itself, with potential disputes and controversies likely to overshadow the action on the pitch. For Celtic, the opportunity to reclaim the title at home presents both a chance for redemption and a platform to silence their critics, while Hearts will be aiming to prove they can handle the pressure and secure their first championship in decades.