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What’s on the board: Four new chess‑inspired puzzlesToday’s article presents a quartet of quirky chess problems curated by We Solve Problems. The puzzles are designed for maths‑loving readers and aim to spark discussion rather than reveal solutions.Odd‑game parity proof – why the count of odd‑play participants is evenIn a tournament where not every player faces every other, some competitors play an odd number of games. The task is to prove that the number of such players must be even, a classic graph‑theoretic result.Knight’s tour from bottom‑right to top‑left – is a full tour possible?Starting from the bottom‑right corner of an 8×8 board, can a knight visit every square exactly once and finish in the top‑left corner? This asks for a Hamiltonian path in the knight‑move graph.Pawn promotion round‑trip – minimal moves to return homeWith both sides cooperating, determine the fewest moves for a pawn to leave its starting square, promote to a queen, and then travel back to its original position.Swapping knights on an irregular grid – the abstract solutionA strangely‑shaped grid holds two pairs of knights. The challenge is to exchange the positions of the black knights with the white knights using only legal knight moves.Why these puzzles matter – boosting maths circles and chess cultureThey promote logical thinking and problem‑solving among secondary‑school students.The article references recent documentaries on Judit Polgár and Hans Niemann, linking popular chess media to educational outreach.We Solve Problems runs free maths circles across the UK, using puzzles like these to engage young learners.Looking ahead – more weekly challenges from the charityThe author will return at 5 pm UK time with solutions and promises future puzzles every alternate Monday. Readers are encouraged to discuss the problems but avoid spoilers.

The Return of Joy in Women's FootballIn an era where football has become increasingly serious and analytical, the World Sevens Football tournament offered a refreshing reminder of the sport's primary purpose: fun. For three sun-kissed afternoons by the Thames, this end-of-season seven-a-side women's football tournament provided the perfect antidote to the depressingly serious elements of the modern game, focusing on goals, laughter, and entertainment rather than VAR controversies or financial fair play regulations.The World Sevens Tournament: A Celebration of FootballThe third edition of World Sevens Football, featuring eight English teams, culminated in an 11-goal thriller final won by Chelsea. The tournament format allowed flair players such as Manchester United's Melvine Malard and Jess Park to thrive, showcasing their stepovers and lethal finishing. Chelsea's Aggie-Beever-Jones emerged as the top scorer with eight goals, demonstrating her clinical finishing ability against defenders. Teams also embraced the entertainment aspect with meticulously prepared walk-on routines, ranging from the hilarious to the bizarre. Everton's players pretended to give birth, Chelsea manager Sonia Bompastor was carried onto the pitch by her players, and Manchester United's head coach Marc Skinner made an entrance in a dressing gown. Even the referees participated in the festivities with their own walkout routine.Financial Impact and Fan EngagementThe tournament delivered significant financial benefits to women's football, with $500,000 (£372,000) awarded to the winners. This injection of capital is particularly valuable for a sport that often struggles for funding. Fan engagement was equally impressive, with a sold-out crowd of 3,000 attending Saturday's final. One clip on Dazn's Instagram featuring Beever-Jones's amusing walk-on garnered over 11 million views, demonstrating the tournament's ability to expand the reach of women's football beyond traditional audiences. Jennifer Mackesy, a co-founder of World Sevens Football, expressed her delight with the fan response: "The response from fans in London has blown us away."Challenging Conventions in Women's FootballThe tournament's entertainment-focused approach sparked debate within the women's football community. Some observers, perhaps mistaking the event for the World Cup final, expressed annoyance on social media about players not acting in a "serious" manner. One X user wrote: "Organisers of this have genuinely set them back years." Another commented: "Unfortunately, women's football won't ever be taken seriously." Even respected figures in the women's game were critical, with former Everton women's manager Andy Spence responding to a viral video of Everton's players acting out a mock funeral by writing: "What is going on? The Women's game has made such progress but these 'tournaments' and the narrative around them are threatening the integrity & hard work people have put in to make the Women's game be accepted by a wider audience. Please stop. Embarrassing."The Future of Entertainment in Women's FootballDespite the criticisms, the World Sevens Football tournament demonstrated several crucial benefits for women's football. It broadened teams' reach through social media exposure, brought much-needed investment into the sport, and attracted new fans. The tournament also provided light relief to players after a gruelling season, allowing them to express themselves in football's "purest form." While the event is not without its challenges—inequitable facilities for different teams and injuries to players like West Ham's Tuva Hansen and Manchester United's Phallon Tullis-Joyce—the engagement from broadcast partners Sky Sports and Dazn suggests this format has a bright future. As women's football continues to grow, events like the World Sevens may play an important role in diversifying the sport's appeal and finding new ways to engage audiences while maintaining the competitive spirit that makes football compelling.

The Lead The next chapter of Marcus Rashford's dysfunctional relationship with Manchester United may involve a long summer waiting to discover where he plays next season. A state of limbo for a forward expected to start England's World Cup opener against Croatia on 17 June in Dallas is an unusual predicament. Barcelona's Title Clinching Performance Yet this is the latest juncture in a period of career uncertainty that began when the former head coach Ruben Amorim excluded Rashford from his first-team plans. That was in December 2024, loans at Aston Villa and Barcelona followed, and Rashford is still looking to put down roots, perhaps in Catalonia, something he may well have expected to transpire after scoring a free-kick against Real Madrid that proved pivotal in Barcelona's La Liga-clinching victory earlier this month. Financial Complications in Potential Transfer Having enjoyed a generally successful spell under Hansi Flick last season, Rashford's stated preference would be to sign permanently for Barcelona. "I am not a magician but if I was, I would stay," he said after scoring against Real on 10 May. "We will see." The problem is Barça's interest in the 28-year-old is opaque. Anthony Gordon's £69m arrival from Newcastle last week confuses the picture further given he, too, is a left-sided attacker. And if Barcelona want Rashford at all it seems it would again only be on a temporary basis. United, meanwhile, would insist on a £26m permanent fee as they attempt to make money on a player reared in their academy before his contract expires in May 2028. Behind the Transfer Saga The answer to why the price is low for a footballer in his peak years offers a clue to the whole saga: behind the sum is Rashford's £17.5m a-year salary, or the total £35m left to pay on his current terms. United want to offload the cost of the high wage. If Rashford is loaned again, the recruiting club will have to cover all or most of the cost. A permanent transfer will, too, surely feature a raise. As things stand, Barcelona do not appear minded to make any move for Rashford permanent. Potential Destinations Beyond Barcelona What are Rashford's other options? With the caveat of never saying never, there seems no way back for him at United, despite Amorim's departure and the appointment of Michael Carrick as his permanent successor. The lad from Wythenshawe remains firmly persona non grata for Sir Jim Ratcliffe, United's minority owner and controller of football policy, as well as for his senior management team: Jason Wilcox, the director of football, and Omar Berrada, the chief executive. When Rashford's loan move to Villa ended last summer, his aim was to join a Champions League-qualified club but not one in London. If this position has changed, Arsenal may be a potential destination. Mikel Arteta would surely categorise Rashford as an upgrade on Leandro Trossard and Gabriel Martinelli as a left-sided attacking option for the Premier League champions. Rashford's ability to operate at No 9 would also offer a further permutation there, alongside Kai Havertz and Viktor Gyökeres. The same holds at Liverpool, where Cody Gakpo is Liverpool's only senior left-sided option and whose output last season was, at best, middling. If they came calling, would Rashford's disaffection with United prove searing enough for him to ignore tribal loyalties and move to Anfield? Villa, too, may be a desirable destination – Rashford lit up Unai Emery's side when there, especially in the Champions League – while another move abroad also remains a possibility. Paris Saint-Germain have been admirers, albeit it feels unlikely the two-time Champions League winners would move for Rashford given they have the world-class Khvicha Kvaratskhelia operating on the left-hand side of their attack. At Bayern Munich, meanwhile, Luis Díaz is established in the position and at Real Madrid there is Vinícius Júnior. World Cup as Career Turning Point Rashford's next destination is likely to become clearer when the transfer window opens on 15 June but maybe only slowly due to the complexities of his situation, the different agendas of different parties and the World Cup, which should be Rashford's prime focus. United could stymie any deal not deemed desirable to them. But Rashford could also refuse any move he does not want. Assessing this fraught dynamic is a cast of admirers who may well want to add a player who helped Barcelona retain the La Liga title but wonder if they can actually afford him. Rashford remains an enigma. A return of eight goals and nine assists in La Liga last season was a relatively modest return and may explain Barcelona's caution regarding a permanent deal for him. This may change. Imagine, for instance, an England World Cup campaign lit up by Rashford. In this scenario, a £26m fee plus a high-end salary may seem cut-price.

The Lead: A Dangerous New Threshold in Aerial CombatThe conflict between Russia and Ukraine has entered a highly volatile phase characterized by massive, reciprocal drone bombardments and deep-strike capabilities. A recent Ukrainian drone attack struck a passenger bus in the Russian-controlled Donetsk region, resulting in at least seven fatalities. This event punctuates a 48-hour period of unprecedented aerial warfare, signaling a shift from frontline trench warfare to strategic, long-range infrastructure and psychological targeting.Deadly Strike on Civilian Transport in DonetskThe focal point of the latest escalation occurred in the contested Donetsk region. According to Russian-installed regional head Denis Pushilin, a drone targeted a bus traveling between Moscow and Simferopol in Crimea. The human cost was severe:Casualties: At least seven people were killed in the strike.Injuries: An additional 11 individuals sustained varying injuries and are receiving medical care.Legal Action: Russia’s Investigative Committee has launched a criminal investigation, classifying the incident as a “terrorist attack.”The Unprecedented Scale of Aerial BarragesThe Donetsk tragedy is part of a much larger tit-for-tat escalation that is pushing both nations' air defense systems to their limits. The sheer volume of unmanned aerial vehicles (UAVs) deployed over the last 48 hours represents a new scale of warfare:Russian Interceptions: The Russian Ministry of Defence reported intercepting and destroying 354 Ukrainian drones overnight across multiple regions.Leningrad Region: Governor Alexander Drozdenko confirmed that 50 drones were shot down over the Leningrad region alone.Previous Ukrainian Toll: This follows a massive Russian barrage the day prior, during which Moscow launched 656 drones and missiles, resulting in 23 deaths in Ukraine.Strategic Targeting of the Russian Economic ForumBeyond the immediate border regions, Ukraine has demonstrated an extended operational reach, striking deep into the Russian heartland. Ukrainian drones successfully hit infrastructure in several districts of St. Petersburg, wounding several people. This is highly symbolic and strategically timed, occurring while Russian President Vladimir Putin hosts the “Russian Davos” economic forum in the city. Furthermore, strikes reached the central Russian city of Michurinsk in the Tambov region, damaging an industrial facility, an apartment building, and a library.The Trajectory of Deep-Strike AttritionThe immediate future of this conflict points toward a war of attrition fought primarily in the skies. Both sides are heavily invested in saturating enemy air defenses. Russia claims its recent 656-drone barrage was retaliation for a dormitory attack in Luhansk, while Ukraine's deep strikes into St. Petersburg indicate a strategy of bringing the war directly to the Russian public and economic centers. As both nations mass-produce and deploy long-range UAVs, civilian infrastructure and transport networks will remain highly vulnerable, making a near-term de-escalation increasingly unlikely.

The Lead One Nation leader Pauline Hanson has unveiled a gas policy inspired by Norway's successful model of resource management, proposing government equity stakes in oil and gas production and a sovereign wealth fund. However, experts point out that while One Nation has adopted some elements of Norway's approach, it has notably excluded the high taxation on profits that is central to Norway's success. The Norwegian Model Explained Norway's approach to managing its oil and gas resources has been globally recognized as "the gold standard." The Norwegian government holds ownership interests in approximately 30% of the nation's oil and gas reserves, with direct equity stakes in 187 production licenses, 48 producing fields, and 16 joint ventures. Crucially, the government also owns two-thirds of Equinor, Norway's largest oil and gas firm. What makes the Norwegian model unique is its combination of extensive public ownership with a 78% marginal tax rate on oil and gas company profits (resulting from a 71.8% "special" tax plus the standard 22% company tax). This approach generates approximately $100 billion annually for the Norwegian government, which is transferred to the Government Pension Fund Global, now worth $2.9 trillion—equivalent to about $500,000 per Norwegian citizen. One Nation's Policy: Selective Adoption One Nation's proposal includes two key elements from the Norwegian model: offering a 30% rebate on oil and gas exploration in Commonwealth waters in exchange for up to 30% equity in production licenses, and creating a sovereign wealth fund to reinvest profits. However, the party has notably excluded Norway's high taxation approach, instead proposing a simple 10% royalty on production to replace Australia's petroleum resource rent tax (PRRT). Pauline Hanson has criticized opponents for suggesting a 25% gas export levy, claiming it would be "industry-destroying." She argues that the Norway model has succeeded because "government and industry partner together supported by generous tax incentives," rather than through high taxation. Financial Impact Analysis Experts have raised concerns that One Nation's proposed 10% royalty may actually deliver less revenue than the current PRRT. Additionally, the opt-in approach to government partnership means only companies that choose to participate would be subject to the equity arrangement, potentially limiting the breadth of public ownership. Josh Runciman, lead gas analyst at the Institute for Energy Economics and Financial Analysis, questions whether it's ideal for taxpayers to be exposed to exploration and appraisal risk when the government lacks expertise in this area. The policy also includes a provision for the government to direct its share of oil and gas production to "Australia's greatest benefit," which could include selling to domestic industries or exporting to pay down debt. Industry and Regional Impact One Nation's policy comes amid growing public unrest over successive governments' failure to secure a "fair share" of Australia's natural resource wealth. The party positions its approach as addressing this concern by ensuring that profits from Australia's resources benefit the nation through both direct ownership and a sovereign wealth fund. The policy has sparked debate within Australia's energy sector, with some experts questioning whether the selective adoption of Norway's model without the high taxation component will actually deliver the benefits claimed. The approach could potentially lead to increased government involvement in the energy sector while maintaining relatively low tax rates on industry profits. Long-Term Outlook and Predictions According to analysts, it would likely take a decade or more before early-stage gas projects under One Nation's policy would begin generating additional revenue for Australians. If implemented after the next election, Australians would not start receiving any extra tax windfall until the late 2030s at the earliest. The timeline for the proposed sovereign wealth fund to accumulate meaningful resources could be even longer, potentially delaying any significant impact on Australia's finances. This extended timeframe raises questions about whether the policy will deliver on its promise of securing a "fair share" for Australians within a reasonable period, especially as global energy markets continue to evolve.

The Tragic Accident At least 18 people, including women and children, were killed when a cargo truck carrying recently returned Afghan refugees from Pakistan overturned on a major highway in eastern Afghanistan, authorities said. Taliban government spokesperson Zabihullah Mujahid said the crash took place in the Qarghayi district of Laghman province on Saturday. Circumstances of the Crash The vehicle, heavily loaded with displaced families and their household belongings, veered off the road at approximately 5:30am local time (01:00 GMT) near the Surkhakan intersection in Qarghayi district. The provincial Director of Public Health Aminullah Sharif said the accident occurred when the truck fell into a ditch after the driver fell asleep. The Victims and Injuries Authorities said at least 10 children were among the dead. Abdul Malik Niazay, a spokesperson for the Laghman provincial governor, said more than 30 other passengers were injured, some critically. The families had been temporarily staying in eastern Kunar province and were en route to the capital, Kabul. Aftermath and Response Emergency services quickly transferred the wounded to medical facilities in neighbouring Nangarhar province, where several remain in intensive care. The central government expressed formal condolences to the families of the victims. The Taliban’s Mujahid said in a post on X, “we pray for the speedy recovery of the injured”, adding that he was “deeply saddened” by the tragedy which took place at the end of the Muslim holiday of Eid al-Adha. Meanwhile, the National Disaster Management Authority announced 730,000 afghanis ($10,000) in emergency financial assistance for the affected families. A Growing Concern Deadly traffic accidents are common in Afghanistan, where highways are severely degraded after decades of conflict, vehicles are poorly maintained, and traffic regulations are seldom enforced. At least 20 people have been killed in three separate traffic accidents reported across Afghanistan over the past 10 days, according to local Taliban authorities and media reports. The tragedy highlights the growing strain on transportation infrastructure as hundreds of thousands of Afghans return from neighbouring countries. According to United Nations figures, more than 447,000 Afghans have crossed back from Pakistan this year alone following a sustained crackdown on undocumented migrants by Islamabad. International aid groups warn that forced expulsions are compelling families to travel in hazardous, packed commercial cargo trucks.

The Reopening of Asylum Cases Athens, Greece – Bashir, a Syrian Muslim who has lived in Greece since 2014, had his asylum case reopened in February. He, along with 1,200 other Syrians, received a notice to restate his reasons for coming to Greece and why he should not return to Syria. Bashir's lawyer, Angeliki Theodoropoulou, said that only men are currently receiving such notices, and not just from Syria but also from Afghanistan, another country whose civil war is deemed to have ended. The Concerns Over Safety However, neither Syria nor Afghanistan is considered safe to return to. Theodoropoulou argued that the entire regime of international protection is being tightened for these two nationalities, with few asylum cases being granted and many rejections. Bashir expressed his concerns, saying, “I don’t understand how this can happen. If they decide I should leave the country, should my family stay here?” Greece's Shift in Migration Policy Greek Migration Minister Thanos Plevris announced in February that he had ordered a reopening of any asylum cases that could be revoked. The move is part of a broader effort to tighten migration policy in Greece. Last year, Greece revoked the asylum of almost 200 people, compared with 400 in the previous decade. Dozens more cases are under review this year. The Broader Context Europe is undergoing a transition as it prepares to put into force an Asylum and Migration Pact next month. The pact demands a hard-border policy and a returns policy for rejected asylum seekers, both of which each member state must manage itself. Kristin Fabbe, chair in Business and Comparative Politics at the European University Institute, noted that Europe has not yet figured out how to do returns at scale, which is a major bottleneck in reforming asylum and migration policies.

Australian Cycling History Made at Giro d'ItaliaJai Hindley has roared again in the "pinnacle of cycling", making history as just the second Australian to enjoy three overall podium finishes in Grand Tours as he rode home for a valiant third place in the Giro d'Italia. As new champion Jonas Vingegaard confirmed his place among the sport's all-time elite in Rome by completing his set of the three Grand Tour triumphs and dominating the 109th Giro with five magnificent stage wins, Hindley underlined why he is one of Australia's greatest.Hindley's Remarkable Comeback JourneyThe 30-year-old had been ill during the race, needing to take antibiotics in the second of the three weeks, yet recovered spectacularly to complete his own set of Giro podium finishes, having won in 2022, been runner-up in 2021 and now third. Only the great former Tour de France winner Cadel Evans has had more Grand Tour podium finishes among Australians, with five.The often unsung Perth rider Hindley said on the eve of the race from Bulgaria that he and his Red Bull BORA-hansgrohe team "aren't here to lick stamps". That was his 2026 version of his famous but bewildering affirmation "we're not here to put socks on centipedes" in the year he won. The translation was that he and his team weren't once again there to mess around, and after he proved stronger over the race than his talented young co-team leader Giulio Pellizzari to assume the role of main man, Hindley lived up to his promise."The second week was very hard for me, I was really quite sick. I even had to take antibiotics, but I seem to be back in time," Hindley had smiled after a remarkable third-week resurgence saw him move into third place overall on the 19th of the 21 stages.Australian Cyclists Dominate Final StandingsOn Sunday, the largely processional ride into the capital ended with Italian Jonathan Milan winning the final-stage sprint and Vingegaard clinching his fourth Grand Tour, having twice won the Tour de France and last year's Vuelta, where Hindley had signalled he was recovering his best form with a fourth-place finish. Hindley did suffer a mechanical mid-race on Sunday but swiftly got back to the peloton. He ended 6 minutes 25 seconds behind Vingegaard, and 1:03 behind runner-up, Austrian Felix Gall."I haven't had a good result in a Grand Tour for a long time, and to be fighting for the podium again was really nice," said Hindley. "For me, the Grand Tours are the pinnacle of cycling, and being competitive in them is what I train for."It was only the second Giro in a decade in which there wasn't a stage won by an Australian, following the early abandonments of the luckless Kaden Groves and Jay Vine with crashes, but, impressively, there were four Australian finishers in the top 17. Michael Storer (Tudor Pro) finished a career-best seventh at 10:13 down, Ben O'Connor (Jayco AlUla) was 16th at 24:12 and Chris Harper (Pinarello Q36.5 Pro) 17th at 30:43.Elevating Australian Cycling on World StageHindley's achievement places him among the elite of Australian cycling history, with only Cadel Evans boasting more Grand Tour podium finishes. His consistent performances across multiple Grand Tours demonstrate the growing strength of Australian cycling on the international stage. The presence of four Australians in the top 17 of the Giro d'Italia further highlights the depth of talent coming from Australia.Future Prospects for Australian CyclingWith Hindley establishing himself as a consistent contender in Grand Tours and younger Australian riders like Pellizzari showing promise, the future looks bright for Australian cycling. The country's ability to produce multiple top-tier cyclists suggests that Hindley's historic achievement may be just the beginning of a new era for Australian cycling excellence on the world stage.

The Historic VictoryIn a stunning upset at the Norway Chess tournament in Oslo, French grandmaster Alireza Firouzja defeated world No. 1 Magnus Carlsen in the opening round. The 22-year-old secured his first victory against Carlsen in classical chess despite playing with a sprained ankle sustained from falling off a stage at the previous week's Bucharest Grand Chess Tour event.This victory marks one of the most high-profile successes by a physically injured grandmaster since Tilburg 1985, when England's Tony Miles shared first prize while playing prone from a massage table due to a back injury. Carlsen, who has won the Norway Chess tournament six of the past seven years, was visibly unsettled by Firouzja's resilient performance.Tournament DynamicsNorway Chess employs a unique scoring system that adds strategic complexity to the competition. A win in classical chess counts for three points, while a loss yields zero. Draws are immediately replayed as Armageddon games, with White receiving 10 minutes on the clock and Black seven minutes. A draw on the board counts as a win for Black, with the Armageddon winner scoring 1.5 points and the loser 1.This system creates immediate pressure, as Carlsen's classical loss in round one put him at a disadvantage despite nine rounds remaining. Firouzja continued his strong form by winning his classical game in round two against India's Praggnanandhaa Rameshbabu, reclaiming his place in the world top 10 in live ratings.Carlsen's StrugglesCarlsen faced further challenges in the tournament, suffering losses in both round two and three. His defeat by Praggnanandhaa in round three was particularly chaotic, with the Indian grandmaster describing the result as "like tossing a coin." Carlsen spent excessive time reacting to White's surprise novelty 6 h4!? in the Najdorf Sicilian, responding weakly after 28 minutes with 6...h6?! and remaining under pressure for much of the game.The tournament's time control proved particularly brutal, with only a 10-second increment at move 41, where players typically have an additional half hour. Carlsen eventually showed his championship form in round four, defeating world champion Gukesh Dommaraju in a repeat of their matchup from last year's infamous table punch incident.Women's Chess in the SpotlightThe Norway Chess tournament also aims to elevate the profile of women's chess, with the women's event offering equal prize money—a significant break from longstanding tradition. Women's world champion Ju Wenjun leads the women's section, though early results show Bibisara Assaubayeva of Kazakhstan in the lead with 7 points after four rounds.This commitment to gender equality in chess prizes reflects broader efforts to increase visibility and support for women in the sport, addressing the historical disparity in recognition and opportunities between male and female players.International MatchupA significant development in the chess world is the upcoming eight-game classical match between American grandmaster Hans Niemann and Russian Ian Nepomniachtchi in Belgrade. This rare encounter between top players from the US and Russia is arguably the most important East-West match since the Boris Spassky vs. Bobby Fischer confrontation in 1972.Both players are ambitious to reach the world's top 10—Niemann for the first time and Nepomniachtchi after recently dropping below it. Their match comes amid heightened geopolitical tensions, adding another layer of significance to this competition between chess representatives from opposing nations.