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Tanya Khovanova has long been a leading voice in the world of recreational mathematics, authoring a popular blog and running the Number Gossip site where enthusiasts can inquire about any number.She now debuts her first book, Mathematical Puzzles and Curiosities, co‑written with fellow puzzle‑enthusiasts Ivo David and Yogev Shpilman. Published by World Scientific, the volume offers fresh challenges and inventive twists on classic problems.To give readers a taste, Khovanova selected three puzzles for the Guardian column:1. Battleships – As a naval admiral you must choose between sending a single vessel with a success probability of P % or two vessels each with a P/2 % chance, where the mission succeeds if at least one ship does. Which strategy maximises the odds?2. The Two Oracles – You face two respondents, Randie, who answers randomly, and Rando, who randomly decides to tell the truth or lie before each answer. Is there a method to distinguish them?3. Bad Maths – A student computes 5548 − 5489 as 59, claiming the “548” cancels out. He then experiments with a subtraction of the form XXYZ − XYZW (distinct digits) and observes the result is XW. How many digits in the new expression match those in the original calculation?Solutions to the trio were later posted online (update), but readers were urged to avoid spoilers until the scheduled reveal.Khovanova also reminds fans that she has been curating puzzles for the Guardian every other Monday since 2015 and welcomes new submissions via email.

Conversations about mundane topics are often dreaded, but research suggests that people enjoy them more than they anticipate. A study involving 1,800 volunteers found that participants enjoyed chatting about dull subjects, such as the stock market or vegan diets, more than they expected. The findings indicate that by avoiding potentially dull exchanges, individuals may miss out on the mood boost and health benefits that come with connecting with others. Elizabeth Trinh, a PhD candidate in management and organisations at the University of Michigan, noted that people tend to underestimate how interesting and enjoyable conversations about boring topics can be. Trinh and her colleagues conducted a series of experiments to gauge how people responded to conversations on dull topics. Participants were asked to predict how much they would enjoy discussing topics they considered boring, and then engage in brief conversations about those topics. The results, published in the Journal of Personality and Social Psychology, showed that people consistently found the conversations more enjoyable than they expected. The surprise enjoyment arises because individuals base their expectations on static components, such as the topic and the other person, rather than dynamic components, like the conversation itself. Trinh suggests that people may benefit from lowering the bar for what makes a conversation worth having and reframing conversational goals. Instead of focusing on enjoyment, individuals could think about what they might learn from the conversation. Nicholas Epley, a professor of behavioural science at the University of Chicago, added that the fear of a boring conversation shouldn’t prevent people from starting one. “If a conversation is boring once you’re in it, you also have a surprising amount of power to make it better!”

Meta, the company behind Facebook and Instagram, is reportedly working on an AI version of its CEO, Mark Zuckerberg. This AI clone is being trained on Zuckerberg's mannerisms, tone, and public statements to allow employees to interact with a digital version of their boss.The rationale behind this project is to make Meta's 79,000 employees feel more connected to one of the most influential figures in Silicon Valley. The AI character will be developed using images and the voice of Zuckerberg, with the CEO reportedly taking part in the training process.This move is part of Meta's broader effort to integrate AI into its business operations. The company aims to use AI to lower costs and accelerate work pace. Zuckerberg has emphasized the importance of efficiency, stating that the goal is to 'get more done' by elevating individual contributors and flattening teams.The development of this AI character follows Meta's previous experiments with digital avatars. In 2022, Zuckerberg shared his own avatar in the metaverse, which received public criticism for its graphic quality. The company has since scaled back its metaverse vision, focusing on AI-generated 3D characters for everyday conversations.Meta's investment in AI is part of a larger strategy to remain competitive with tech rivals. The company is pouring billions of dollars into AI research to create 'superintelligence,' a system capable of performing any cognitive task far better than a human.

Chala Chala Nav Bala (Maze Baal, 1943) marks the debut of Asha Bhosle, who entered the film world at ten years old. Paired with her sister Lata Mangeshkar, her bright falsetto captures the youthful innocence of the Marathi romance, foreshadowing the emotive style that would define her career. Aaiye Meherbaan (Howrah Bridge, 1958) showcases Bhosle’s rise during Hindi cinema’s golden age, thanks to her partnership with composer O.P. Nayyar. The song’s sultry vibrato and lush orchestration set the tone for the film’s noir atmosphere, establishing her as a leading‑lady playback voice. Aao Huzoor Tumko (Kismat, 1968) became a chart‑topping hit, featuring intricate vocal runs over a flamenco‑style guitar. Bhosle’s lower‑register chorus broke the conventional shrillness of female playback, while her nuanced phrasing added depth to the on‑screen heroine’s drunken allure. Dum Maro Dum (Hare Rama, Hare Krishna, 1971) stands out as her most successful crossover, later sampled by Western rappers. The track, produced with R.D. Burman—her future husband—blends psychedelic Beatles‑inspired grooves with Hindi lyrics, demonstrating her ability to bridge Eastern and Western pop sensibilities. Piya Tu Ab To Aaja (Caravan, 1971) pushes the fusion further into jazz‑cabaret territory, with bold horn sections and cinematic guitar reverb. Bhosle’s breathy, suggestive delivery sparked controversy, yet the performance remains a masterclass in balancing sensuality with technical agility. Chura Liya Hai Tumne Jo Dil Ko (Yaadon Ki Baaraat, 1973) epitomises the “masala” film soundtrack, merging drama, romance, and crime. Over a gentle guitar backdrop, Bhosle’s tender humming conveys quiet longing, contrasting with the film’s high‑octane narrative. In Ankhon Ki Masti (Umrao Jaan, 1981) sees Bhoske venture into Urdu ghazals with composer Khayyam. Her lower, huskier timbre—adjusted a half‑step down—highlights her continued artistic experimentation even as she approached fifty. Bow Down Mister (1991) illustrates her early 1990s foray into international collaborations, lending wordless, soaring vocals to Boy George’s post‑Culture Club project. The track transforms into a rave‑infused anthem, underscoring Bhosle’s versatility across genres. Radha Kaise Na Jale (Lagaan, 2001) pairs Bhosle with a young A.R. Rahman, reaffirming her status as an elder stateswoman of Indian music. The duet with Udit Narayan blends tabla and flute with powerful vocal runs, marrying traditional Hindustani scales to contemporary film scoring. The Way You Dream (2002) features an unexpected partnership with REM frontman Michael Stipe on the 1 Giant Leap project. The eight‑minute piece weaves tabla rhythms, subtle guitar, and a dramatic breakbeat, proving that Bhosle’s voice can seamlessly inhabit New Age and electronic soundscapes.

Joz Norris' latest show, 'You Wait. Time Passes', is a thought-provoking exploration of artistic character and the choices that define a creative life. In this maverick confection of a show, Norris takes on the role of an unhinged, self-absorbed guru figure, imparting life lessons with a healthy dose of silliness.The show delves into Norris's career as a comedian, actor, and magician, as well as his bid to become Google's number one Joz. Through snippets of his career and interactions with his erratic AI girlfriend, Norris presents a picture of himself as a complex, multifaceted artist.At its core, the show poses big questions about the futility of the niche creative's life and whether weird artists are truly weird or just trying to express themselves in a different way. While some moments feel a bit strenuous or for-its-own-sake, the show's extreme dottiness is backed up by good jokes, twisty philosophy, and a highly energetic performance.Ultimately, 'You Wait. Time Passes' constitutes a curiously rousing defence of experimental art-making in a conformist, capitalist world. The show is a must-see for fans of comedy and experimental art, and it will be touring after its run at the Lowry in Salford.

Barney Ronay set out to test whether a disciplined betting strategy could turn a modest £10 stake into a sizeable profit. Over five days he managed to grow the amount to £120, a return that sounded impressive but left him emotionally flat. His experiment underscores a broader truth: sport betting is engineered to disturb and addict. The personal journey is set against a looming financial shock for English football. Nine Premier League clubs have warned they cannot replace the cash flow previously supplied by gambling sponsors, which will be barred from shirt‑front advertising next season under a voluntary industry agreement. One club executive summed up the anxiety: “Nearly everyone is losing money.” The shortfall is estimated at around £4 million for the affected clubs. These concerns arrive at a time when the gambling sector itself faces scrutiny. Recent data show that up to 1.4 million UK adults may have a gambling problem, a figure that has risen alongside the proliferation of mobile betting apps. The Guardian previously reported that the world‑champion club could incur losses of £335 million in a single season, illustrating the massive financial stakes involved. Ronay’s betting log reads like a sports‑fan’s diary. He began with a £10 wager on a Florida horse race, which paid out modestly. Subsequent bets on high‑profile matches – Manchester City versus Liverpool, Southampton beating Arsenal in the FA Cup – produced a rapid climb to £120. Yet each win felt hollow, prompting him to chase larger, riskier bets such as a four‑way accumulator on the Champions League semi‑finalists, a gamble that ultimately fell short. Beyond the numbers, the piece highlights how gambling permeates the football experience: logos dominate club kits, betting terminology infiltrates fan conversation, and promotional offers tempt even casual viewers. Ronay argues that this saturation turns a simple pastime into a “highly available, stimulating activity designed to hook” users, exploiting the brain’s natural reward pathways. In concluding, Ronay stresses two take‑aways. First, the industry’s promise of “extra money” for clubs is a façade – the money only comes out of fans’ pockets. Second, the impending £4 million sponsorship gap may actually serve as a catalyst for sensible self‑regulation, forcing clubs to reconsider reliance on gambling revenue.

London-based musician Alex Peringer, known as Reckonwrong, has released his self-titled debut album 'How Long Has It Been?' after years of near silence. The record marks a departure from his early work, characterized by dizzying time signatures and wry tales, instead embracing warm, tinkering sounds and earnest sentiment.The album's theme of winter is reflected in its introspective stories and soft arrangements, punctuated by subtly wonky elements that prevent it from sounding overly polished. Tracks like 'Before and After' and 'Two Lovers' showcase Peringer's ability to blend atmospheric bedroom-pop with left-field flourishes.Influences from Arthur Russell and Robert Wyatt are evident in the album's use of echo and melancholy. Peringer's delightfully off-kilter vocals add to the charm, as he navigates themes of love, loss, and introspection. The album's eclectic sound is both captivating and emotionally resonant.In addition to Reckonwrong's album, other notable releases include Index for Working Musik's 'Bunker Intimations II' and Inrain's 'Rise', showcasing a range of experimental and shoegaze-inflected sounds.

Oppdal, a mountain village in central Norway, hosted its first interspecies council on a snowy morning at the Bjerkeløkkja conference centre. Thirty‑eight local participants each represented a different species – from a northern bat to a birch tree – and voiced the needs of the natural world in a structured decision‑making process.The council draws on the legacy of the Council of All Beings, a practice created in the 1980s by environmentalists John Seed and Joanna Macy. Phoebe Tickell, a scientist and “moral imagination” activist mentored by Macy, adapted the ritual into a governance methodology that expands representation beyond humans.Facilitators first identified the multispecies stakeholders in the region, then briefed human representatives – chosen either randomly or for expertise – on the perspectives they would embody. The session concluded with a manifesto of principles for human governance and an impact‑evaluation plan to track participants’ connection to nature six months later.Oppdal’s mayor, Elisabeth Hals, noted that the village’s population swells from 5,000 to over 30,000 in winter as tourists flock to private cabins (hytter). The municipality plans to add 1,000 new apartments by 2035 to promote year‑round tourism, a move that has sparked debate over land use, farming, and conservation.During the council, a birch “worried there’s too much of me,” while a rockfoil flower urged humans to “slow down and listen to where nature can tolerate more activity.” The River Driva lamented being treated as a mere resource, and a fox enthusiastically pointed at a wader, highlighting the emotional range the exercise seeks to capture.Proponents argue that such empathy‑building exercises are crucial as wildlife populations have fallen by roughly 70 % over the past 50 years. Tickell contends that “imperfect representation beats exclusion” and envisions interspecies councils becoming as routine as environmental impact assessments.Across the UK, 13 councils have recognised river rights since 2023, and a coalition of artists, ecologists, lawyers and policymakers is exploring similar biodiversity governance around the North Sea. In London, multispecies assemblies have already informed stewardship of the River Roding and land‑use consultations.Oppdal’s experiment aims to create “institutional trace” – genuine decision‑making power backed by robust methodology and longitudinal research – rather than a tokenistic green‑washing exercise. If successful, the model could be replicated internationally, giving non‑human nature a formal seat at the policy table.

Compiling a definitive Top 10 of superhero movies proves as elusive as “herding thunder through a spreadsheet.” Critics must decide whether to prioritize cinematic craftsmanship, cultural influence, or the raw emotional resonance that these films generate.Over the past two decades the genre has diversified dramatically, spanning crime‑driven Batman sagas, family‑friendly adventures like The Incredibles and Guardians of the Galaxy, political allegories such as Captain America: The Winter Soldier and Black Panther, and experimental visual feats exemplified by Spider‑Man: Into the Spider‑Verse. This breadth makes any hierarchical ranking inherently subjective.Influence remains a key metric. Richard Donner’s Superman (1978) demonstrated that audiences would pay to see a man fly, paving the way for half a century of cape‑wearing blockbusters. Similarly, Iron Man reshaped the modern blockbuster formula, launching the Marvel Cinematic Universe and redefining franchise economics.Emotional impact also matters. Logan stands out as a bruised, mournful western that subverts the genre’s usual invincibility myth, presenting a weary Wolverine confronting mortality. Its rawness offers a rare catharsis rarely found in superhero narratives.Event status cannot be ignored. Avengers: Endgame turned its opening weekend into a cultural ritual, with multiplexes resembling revival tents where audiences collectively cheered, gasped, and celebrated. Even a “colder” analysis must acknowledge that the film transformed anticipation into a generational sacrament.Visual innovation is another hallmark. The Oscar‑winning animation Spider‑Man: Into the Spider‑Verse made comic‑book ink feel alive, delivering a sensory overload that redefined what animated superhero storytelling could achieve.Cultural relevance caps the discussion. Black Panther emerged as a modern myth that forced critics to confront race, identity, and heritage within blockbuster fantasy, expanding the perceived centre of the superhero universe beyond the traditional Batman, Superman and Spider‑Man triad.While outliers like Alex Garland’s Dredd and Guillermo del Toro’s Hellboy II: The Golden Army deserve mention, the critic’s final Top 10 list reflects a balance of influence, artistry, and cultural weight:1. The Dark Knight2. Spider‑Man: Into the Spider‑Verse3. Superman (1978)4. The Incredibles5. Spider‑Man 26. Dredd7. Spider‑Man: No Way Home8. Avengers: Endgame9. Spider‑Man10. Logan