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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 LeadThe Samsung Galaxy S26 is a compact flagship Android phone that hasn't changed much in a year, but it's still one of the best smaller handsets available. The Event DetailsThe S26 has a 6.3in FHD+ Dynamic Amoled 2X 120Hz screen, Samsung Exynos 2600 chip, 12GB RAM, and 256 or 512GB storage. It runs on One UI 8.5 (Android 16) and features a 50MP + 12MP 0.6x + 10MP 3x camera, 12MP front-facing camera, and a 4000mAh battery. The Data AnalysisThe S26's price has increased by £80 to £879 (€949/$899/A$1,349), but it offers double the starting storage. The phone's performance is snappy, and the battery life is decent, lasting about 40 hours between charges with average use. The Impact AnalysisThe S26's compact design and improved specs make it a great option for those looking for a smaller flagship phone. The phone's camera app is also feature-rich, with tools like text, transcription, and image editing utilities. The PredictionThe S26 is expected to be a popular choice for those who want a compact flagship phone with top-notch specs. With software updates until 28 February 2033, the S26 is a safe bet for long-term use.

The Lead: Derby's Decline and Ambitious RevivalEpsom's prestigious Derby faces a critical moment as organizers implement a five-year plan to reverse declining attendance. With just 22,312 spectators at last year's race—considered a rock-bottom moment—track officials have introduced sweeping changes including a £2m prize fund, free entry for under-18s, and eliminated parking fees. The 247th running of this historic race serves as the first test of these ambitious measures to restore the Derby's grandeur and popularity.The Revival Strategy: Key Changes to the ClassicThe plan, spearheaded by Epsom's general manager Jim Allen, focuses on multiple fronts to rejuvenate the Derby experience. The most significant change is the substantial boost in prize money to £2m, with £1m allocated to the winner. Accessibility improvements include free admission for under-18s to the main enclosure and the elimination of the £30 car parking charge in the Hill enclosure. Additionally, temporary 'bleacher' seats along the inside rail will provide racegoers with a premium 'bird's eye' view of the crucial closing stages of the race.The Attendance Challenge: Numbers and ExpectationsLast year's attendance of 22,312 paying spectators represented a concerning low for the prestigious event. The current five-year plan aims to more than double the aggregate attendance to over 100,000 across the two-day Classic meeting, up from 37,500 in 2025. While weather conditions affected last year's turnout with a yellow weather warning reducing 'walk-up' attendance, the organizers recognize that immediate improvement is necessary to prevent further erosion of the Derby's status as a premier sporting occasion.The Royal Factor and Industry ResponseA significant boost for this year's Derby comes from the announcement that the King and Queen will attend, recalling the event's historic connection to royalty. The royal couple's decision to leave a family wedding 90 miles away to attend demonstrates the Derby's continued importance. However, the absence of Aidan O'Brien's Constitution River, Europe's top three-year-old colt, from the Derby—instead competing in and winning the French Derby—presents a challenge. Despite this, O'Brien, who holds the record with 11 Derby victories, still has seven possible runners in contention, including the 7-4 favorite Benvenuto Cellini.The Future Outlook: Balancing Tradition and InnovationThe Derby's revival strategy represents a delicate balance between preserving its historic appeal and adapting to modern expectations. By maintaining free access to common land while enhancing the main enclosure experience, organizers hope to recreate the vibrant atmosphere that characterized the Derby in its heyday. The success of this approach may determine whether the Classic can recapture its place as a must-attend sporting event, drawing not just dedicated racing fans but also those seeking a grand day out. As the 247th running approaches, all eyes will be on whether these changes can reverse the Derby's declining fortunes and restore its status as the highlight of the British racing calendar.

The Leadership Challenge England's Test cricket team is at a crossroads. After a disappointing Ashes series, coach Brendon McCullum and managing director Rob Key have been tasked with rebuilding the team. A quote from James Kerr's book Legacy resonates: 'Our values decide our character, our character decides our value.' Culture Reset The team's culture was found wanting during their winter tour of Australia. However, this setback presents an opportunity for a reset. McCullum, appointed in 2022, and Key must now lead the rebuild. They are 'extremely lucky' to have kept their roles and must capitalize on this chance. The Coaching Conundrum McCullum's coaching style, while effective in some areas, has been questioned. He is a people person with great empathy for players but may lack technical expertise. The appointment of new coaches raises questions about whether the right people have been brought in to support the players. The Road Ahead McCullum's messaging must change, shifting from 'running towards danger' to 'winning key moments and playing smart cricket.' The upcoming series against New Zealand and Pakistan will be crucial tests. With an experienced team, McCullum must bring them along as he implements changes to restore positivity and success. The Verdict If McCullum and Key can successfully rebuild the team, England could be in a great place to win upcoming series and face bigger challenges with confidence. The pressure is on, but with intelligent players and a renewed approach, they may just turn the corner.

Keiko Fujimori has moved from the shadow of her father’s legacy to become the focal point of Peru’s 2026 presidential race, commanding significant public attention and party resources. Keiko Fujimori’s Political Trajectory from First Lady to Party Leader 1990s: Served as first lady during Alberto Fujimori’s presidency. 2009: Elected president of the Popular Force party. 2011, 2016, 2021: Ran for president, finishing second in each election. 2024‑2025: Oversaw a resurgence of Popular Force in congressional elections, securing 28 seats. Polling Data Shows Continued Voter Support National Ipsos poll (May 2026): 31% intention to vote for Fujimori, ahead of the nearest rival at 24%. Urban vs. rural split: 38% support in Lima, 24% in Andean highlands. Demographic trends: Strong backing among voters aged 35‑55 who cite economic stability. Implications for Peru’s Democratic Stability Polarization: Fujimori’s candidacy deepens the divide between Fujimorista supporters and anti‑Fujimori movements. Judicial scrutiny: Ongoing investigations into alleged campaign‑finance irregularities could affect public perception. International outlook: The United States and European partners monitor the election for signs of democratic backsliding. Scenarios for the 2026 Presidential Race First‑round victory: If poll momentum holds, Fujimori could secure the presidency outright, reshaping policy on mining, security, and foreign investment. Run‑off dynamics: A second‑round contest may force coalition‑building with centrist parties, potentially moderating her platform. Electoral setbacks: Legal challenges or a surge in opposition turnout could keep Fujimori out of the final ballot, reinforcing a fragmented Congress.

The Context of a Warm-Up MatchLet's be honest with ourselves and each other: this really doesn't matter that much. Not only is it a World Cup warm-up game, contested by two teams trying out a few things and making sure nobody pulls up lame, it's also a World Cup warm-up game between England and New Zealand … and history suggests those two nations aren't much of a match.Current Form and Tournament ExpectationsBut first up, consider the state of play as is. England will go into the tournament as third-favourites to win, behind just Spain and France and ahead of five-time winners Brazil and reigning champions Argentina. New Zealand meanwhile made it to the finals having beaten Tahiti, Vanuatu, Samoa, Fiji and New Caledonia in qualifying, to the cumulative score of 29-1, and have subsequently lost eight of their last ten matches, the latest an embarrassing 4-0 defeat by Haiti. Their only win during that sequence was an admittedly good-looking 4-1 victory over Chile … who went down to ten men after 27 minutes.Historical DominanceAnd then there's our good old friend The Past. England and New Zealand have officially met twice previously, both matches held within the space of five days in the summer of 1991. Graham Taylor's team won both games, 1-0 and 2-0, Gary Lineker, Stuart Pearce and David Hirst doing the damage. There were another six unofficial games in the sixties: between June 1961 and June 1969, England won every one of them, running up an aggregate score of 35-2.Match Preview and ExpectationsSo, y'know. But then England's last two outings weren't that impressive – a 1-1 draw with Uruguay and a 1-0 defeat to Japan, both at Wembley – so it's probably best for everyone not to get too far ahead of themselves. Yet despite the generally tepid nature of pre-tournament warm-ups, and with all their concomitant line-up experiments and substitutions, this game tonight should be a shoo-in for Thomas Tuchel's team. Hey, if they don't win, the internet won't explode … but it may start gently rocking and convulsing with mirth. Not that anyone should be too euphoric/embarrassed. Because let's be honest with ourselves and each other: this really doesn't matter that much. Kick-off is at 9pm BST. It's on!

On a grey, rain‑spattered Saturday at Lord’s, the much‑talked‑about steam‑treated surface behaved like a mystery, offering erratic bounce and variable movement that left batters struggling and spectators disappointed. A Rain‑Soaked Day Turns Lord’s Pitch into a Nightmare London’s turn‑around weather forced a brief window of play around midday, only for clouds to return as soon as the lunch break ended. The limited sunshine was spent largely on a 40‑minute early lunch, after which intermittent showers dictated the rhythm of the game. Steam Treatment Backfires: The Pitch’s Unpredictable Bounce Despite recent outfield relaying, the playing strip felt “like something rolled up by the bins”. The MCC’s attempt to “purify the soil” with steam has produced the worst conditions yet, with variable bounce that saw a ball hit Jacob Bethell on the head at one end and an ankle‑height delivery at the other. Numbers on the Day: 58 Balls, 2 Wickets, 80 Minutes of Play 58 balls bowled over 80 minutes of intermittent cricket 2 wickets fell – Rachin Ravindra clean‑bowled and Daryl Mitchell lbw, both to Ollie Robinson Target of 218 runs for New Zealand remained out of reach What This Means for MCC and International Cricket The pitch has been described as the most unpredictable wicket in England since records began. Its failure undermines the reputation of MCC as the custodian of the game and raises concerns for upcoming international fixtures at the venue. Looking Ahead: Drop‑In Pitches and the 2028 Solution In response, the MCC is developing “drop‑in” pitches to be stored on the Nursery Ground and installed for future Tests. The rollout is targeted for 2028, a timeline the club hopes will restore confidence in Lord’s as the home of cricket.

The Impact of Rain on Day ThreeJust 58 legal deliveries – plus one no ball – were bowled as rain dominated the third day of the first Test match between England and New Zealand. The limited action allowed England to upgrade their chances of victory from probable to overwhelmingly likely.England's Bowling EffortsOllie Robinson struck twice, bowling Rachin Ravindra for eight and trapping Daryl Mitchell lbw. These wickets left New Zealand on 55 for five and still 199 runs from victory.The Challenge of Batting on a Difficult PitchThe New Zealand batters faced significant challenges on a capricious surface, with the ball nipping quicker and more when the cloud was in. Nathan Smith noted that the pitch was easier to bat on when the sun was out, which is why New Zealand seemed to be waiting for better batting conditions on Sunday.The Frustration of a Rain-Halted DayPlay started at 12:59 pm, stopped at 1:07 pm, resumed at 1:21 pm, was curtailed again at 1:37 pm, recommenced at 1:41 pm, and concluded for the last time at 2:09 pm. The day was officially abandoned after the umpires inspected the pitch in thick mizzle at 5:20 pm.The Current State of the MatchNew Zealand's Devon Conway ended on 19 off 55, surviving this abbreviated day along with Tom Blundell and a dwindling amount of hope. England made good use of the limited time they got in the field, but players and crowd alike spent most of the day watching the rain.

The Stunning Upset Russian teenager Mirra Andreeva was already a tennis phenom at age 15. At 19, she's a Grand Slam champion. The eighth-ranked Andreeva ended the run of 114th-ranked Polish qualifier Maja Chwalinska by 6-3, 6-2 in the French Open final on Saturday. Andreeva's Historic Win Andreeva became the youngest player to win the women's singles title since Monica Seles, who was 18 when she landed her third straight French Open in 1992. During the trophy presentation, Andreeva took the unusual step of thanking herself “for believing in myself, always giving my 100 percent, even when it’s tough, trying every day to be better as a person and as a player, believing that I can do this, fighting so many demons inside of me.” The Final Match The final was played under mostly sunny skies, but wind was a factor in the first Grand Slam final for both players. Chwalinska double-faulted on the opening point of the match, but she was the first player to hold serve in the fifth game for a 3-2 lead. But then Andreeva won nine straight games to take control as she found a way to hit through the wind and answer Chwalinska’s array of spins and drop shots. Andreeva's Background Andreeva was born in Siberia, moved to Sochi, and eventually France, to develop her tennis career. She drew a loud applause from the crowd on Court Philippe-Chatrier when she spoke a few words of French during the trophy presentation. “Thanks for your support today and over these past two marvellous weeks here in Paris,” Andreeva said. “It was very important for me.” The Future of Tennis Andreeva has been considered a Grand Slam contender since she burst onto the scene as a 15-year-old at the 2023 Madrid Open, where she became the third youngest player to win a main draw match at a WTA 1000 tournament and made the quarterfinals. Lately, Andreeva has had to contend with playing under neutral status and without her country’s flag due to the war with Ukraine.