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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.

Liverpool FC is gearing up for a crucial Champions League showdown against Paris Saint-Germain, with manager Arne Slot expressing confidence in his team's ability to turn the tide in their favor. The Reds are facing a daunting task, having lost 2-0 in the first leg at Parc des Princes.Slot emphasized that his team must deliver a great performance to overcome the European champions, stating that while the task is complicated, it is not impossible. He drew on his team's past successes, highlighting that in 36 of their 50 home games under his management, Liverpool has scored two or more goals.The Liverpool head coach is instilling belief in his players, citing examples of their ability to perform at a high level, even against strong opponents. He noted that his team has shown resilience in the face of adversity, such as coming back from 4-0 down against Manchester City in the FA Cup.Dominik Szoboszlai, Liverpool's midfielder, echoed Slot's sentiments, urging his teammates to go 'all in' from the first minute to the final whistle to knock out PSG. Szoboszlai expressed his desire to secure a win, stating that anything can happen at Anfield in a one-off game.Meanwhile, PSG's manager, Luis Enrique, warned his players not to underestimate Liverpool, cautioning that there could be pitfalls and that the game can change quickly. He emphasized the importance of taking care of details and not allowing Liverpool to benefit from their home crowd and noise.

Rory McIlroy's sixth major victory at the Masters Tournament has cemented his status as a golfing great, drawing comparisons to Seve Ballesteros and other legends of the sport.McIlroy's achievement is not just about the numbers; it's about the style and excitement he brings to the game. His thrilling performances have captivated audiences, making him a must-watch for fans and a key figure in golf's popularity.The Northern Irish golfer's journey to success has been inspiring, from his humble roots in Belfast to his rise as a global superstar. His relatable personality and dedication to hard work have endeared him to fans worldwide.McIlroy's comeback at the Masters was nothing short of remarkable. Despite struggling with consistency, he managed to secure a dramatic win, showcasing his resilience and creativity on the course.His influence on golf extends beyond his own performances. McIlroy is a key figure in shaping the sport's future, and his involvement in high-profile events like the Ryder Cup has made him a leader in the golfing world.

The 2018 World Cup in Russia served as a high‑profile platform for Vladimir Putin, showcasing his nation and bolstering his personal legitimacy. The tournament was effectively a diplomatic bow to the Kremlin’s ambitions.Fast‑forward to the summer of 2026, and the buildup to the event has taken on a distinctly American flavour, with the competition becoming a backdrop for Donald Trump’s political narrative.The next edition, slated for 2034 in Saudi Arabia, presents a fresh set of challenges. Despite the kingdom’s controversial human‑rights record, the event offers Crown Prince Mohammed bin Salman an opportunity to polish his and the nation’s image. FIFA’s current reluctance to enforce independent oversight of migrant‑worker conditions raises fears that construction could be as deadly as the 2022 Qatar experience.These developments underscore a pressing need to insulate the World Cup from the whims of powerful leaders. One proposed solution is to fragment the tournament—treating it like a monopoly that has become too dominant.Evidence that this approach is feasible already exists: the 2026 World Cup will be co‑hosted by three nations, and the 2030 edition is set to span six countries across three continents (Spain, Portugal, Morocco, Argentina, Paraguay, Uruguay).Building on that, the author suggests a radical redesign: allocate each group stage to a different global city—Paris, Rio de Janeiro, Tokyo, Sydney, Johannesburg, London, the Basque Country, and so on. Knock‑out rounds could be broken into three‑match clusters and scattered worldwide, with the semi‑finals, final, and third‑place match awarded to the highest‑bidding venue.Carbon‑footprint concerns are addressed by noting that teams already travel long distances to a single host nation; distributing groups based on the median distance to participating teams would not significantly increase emissions.Financially, the cost of staging a traditional, single‑host World Cup has ballooned, limiting the pool of viable bidders to those seeking political or economic leverage. A decentralized format would dilute any single leader’s ability—whether Trump, Putin, or the Saudi crown prince—to manipulate the event for personal gain.Decentralisation would still align with FIFA’s stated objectives: expanding the sport’s reach, creating a truly global spectacle, and bringing football closer to fans worldwide.While FIFA claims a fiduciary duty to maximise revenue for its 211 member associations—justifying steep ticket prices and controversial sponsorships—the proposed model could actually enhance revenue by turning each small cluster of matches into premium, high‑value events.Precedent exists in the form of Euro 2020, which, despite being postponed by the pandemic, successfully unfolded across 11 European cities, delivering record‑breaking goal tallies and strong attendance figures.In sum, the most effective way to protect the World Cup’s cultural significance and prevent its exploitation by authoritarian figures may be to deconstruct and disperse it globally, turning a single‑host behemoth into a series of interconnected, locally hosted celebrations of the sport.

Rory McIlroy has revealed that weeks of preparation at 'home course' Augusta National played a crucial role in his successful Masters defence. The Northern Irish golfer spent considerable time at Augusta in the lead-up to the Masters, rather than participating in PGA Tour events, despite struggling with a back injury.McIlroy's preparation involved simulating a tournament by playing with a single ball, a strategy inspired by advice from golf legend Jack Nicklaus. This approach allowed him to feel comfortable with the course and make informed decisions about his shots.After securing his sixth major win, McIlroy expressed his satisfaction with his preparation, stating, “I joked last week that this place feels like my home course”. He emphasized the importance of getting to know the course and being prepared for various situations.McIlroy's victory joins an exclusive group of golfers who have retained the Green Jacket, including Jack Nicklaus, Nick Faldo, and Tiger Woods. With this win, McIlroy believes he can focus on playing his game and increasing his chances of winning more majors.The article also mentions Scottie Scheffler's comments on the course conditions, particularly on Friday afternoon when McIlroy birdied six out of Augusta's closing seven holes. Scheffler suggested that the firmness of the course was not equal on Thursday and Friday, but acknowledged that the weather and wind conditions also played a role.

The Olivier Awards, a prestigious event in the theatre world, recently took place, and among the many memorable moments were the quotes shared by the winners. Host Nick Mohammed kicked off the ceremony with a lighthearted remark about the opening gondola scene from Phantom of the Opera, joking that it wasn't Cunard's boat, but rather Andrew Lloyd Webber's mode of transportation.Cole Escola, playwright of Oh, Mary!, expressed gratitude to a rather unconventional recipient: "my amazing husband, who doesn’t exist". This humorous comment set the tone for a night filled with both laughter and heartfelt acknowledgments.Elaine Paige, a special award winner, shared the secret to her long-lasting career, attributing her success to her father's motto: "perseverance furthers". This insight into her personal life added a touching element to the evening.Rachel Zegler, winner of the best actress in a musical for her role in Evita, recalled her experience performing in the Palladium balcony, calling it "the honour of a lifetime". Her sentiment encapsulated the pride and gratitude felt by many of the winners.Rosamund Pike, best actress winner, reflected on the challenges of balancing life and art, acknowledging the support of her family and partner. Her words resonated with the audience, highlighting the importance of a strong support system.The ceremony also saw Jack Holden win the surprise best actor award, who, upon hearing his name, jokingly asked, "It was definitely me, was it?". This lighthearted moment brought humor to the event.Throughout the evening, winners shared their appreciation for their loved ones and mentors. James Hameed thanked his dad, who had come to London to chase his dreams, showcasing the impact of family on one's journey to success.

CF Montreal, formerly known as Montreal Impact, is facing a tough time in Major League Soccer (MLS). The team's rebranding in 2021 was met with criticism from fans, who felt it was a step backward. The team's performance on the field has been equally disappointing, with five defeats in their opening six games, including a 5-0 drubbing and a pair of 3-0 humblings. The team's owner, Joey Saputo, has invested heavily in Bologna FC, an Italian Serie A club where he has been majority shareholder and chair since October 2014. Despite the close ties between the two clubs, CF Montreal are not benefiting much from their association. The team's academy is underdeveloped, with just three homegrown products having logged a combined 1,737 MLS minutes. In contrast, Bologna FC had a successful season, playing Champions League football and winning the Coppa Italia. The stark difference in performance between the two clubs has raised concerns about the future of CF Montreal. The team's latest struggle led to the departure of head coach Marco Donadel after a dismal start to the season. His replacement has not been announced yet. The team's poor performance has made them a worryingly neglected holdover in the league.

Rory McIlroy, the world's No 2 golfer, secured his second Masters win at Augusta National, a triumph he largely attributes to the support of his parents, Rosie and Gerry McIlroy. In a heartwarming display of family devotion, McIlroy had to convince his parents to attend the tournament, as they feared their presence might jinx his title defense. The McIlroy family has a remarkable story of sacrifice and dedication. Rory's parents worked multiple jobs, including graveyard shifts, to support their son's golfing ambitions. Gerry, 66, managed a bar and worked as a cleaner, while Rosie, 65, worked late nights at a factory. Their unwavering support and refusal to pressure Rory into playing golf have been instrumental in his success. During his final round, McIlroy admitted to thinking about his parents and fighting back tears. After securing his win, he turned to his family, including his wife Erica and daughter Poppy, and expressed his gratitude to his parents. 'Mum and Dad, I owe everything to you. You’re the most wonderful parents,' he said, visibly emotional. This victory cements McIlroy's stature as one of golfing's greats, with his sixth major win. His parents' presence at the tournament made the moment even more special, with Rosie sporting a handbag adorned with newspaper reports from his 2025 win. As McIlroy looks to the future, he hopes to emulate his parents' example, stating, 'If I can be half the parent to Poppy as you were to me then I’ll know I’ve done a good job.'

Taiwan's 'Sunshine Women's Choir' has taken the local box office by storm, but its success has been met with criticism over its emotionally manipulative tactics and lack of depth. The film, adapted from the 2010 Korean movie 'Harmony,' tells the story of Hui-Zhen, a woman who forms a choir with her fellow inmates to raise her infant daughter behind bars.The movie's director, Gavin Lin, has been accused of resorting to manipulative tactics to elicit emotions from the audience, including excessive crying and contrived plot twists. Despite its feel-good/feel-bad tone, the film's relentlessly peppy approach undermines any potential for foundational grit, critics argue.The story follows Hui-Zhen's journey as she navigates life in prison and forms a choir with her fellow inmates. However, the film's sanitized portrayal of prison life and its lack of meaningful character development have been criticized. The movie's use of showstoppers and flashbacks has also been seen as a way to avoid addressing the characters' deeper issues.Despite its criticisms, 'Sunshine Women's Choir' is set to hit UK cinemas on April 17. The film's success has sparked a debate about the role of emotion in filmmaking and the importance of depth in storytelling.