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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 Lead: Introduction to Chess PuzzlesEarlier today I set these four chess puzzles. Here they are again with solutions.The Event Details: Analysis of Puzzle Solutions1. OdditiesA chess tournament is taking place with several participants. Not every player played against every other player, and some players may have played many more games than others.Some of the players played an odd number of games. Prove that the number of such players must be even.Solution:The total number of games played by everyone must be even, since every game has two players. When you add up odd and even numbers to make an even number, there must be an even number of odd ones, because if you have an odd number of odd numbers the total will be odd.2. L of a tripA knight in chess moves in an "L" pattern - two squares in one direction and one square in a perpendicular direction. Starting in the bottom right corner of a regular 8×8 chessboard, is it possible for a knight to visit every square on the chessboard exactly once and end up in the top left corner?Solution: No.A knight move goes from a white to a black square, or vice versa. To visit every square on the board exactly once requires 63 moves. If you start on white, you will end on black, or vice versa. You cannot start on one corner and end on the opposite corner, since opposite corners of a chess board are the same colour.3. Pawn returnTake a chessboard with the standard initial setup of pieces. What's the fewest number of moves needed for a pawn to leave its initial place, get promoted/queened, and then return to its original position?(Assuming the two players are collaborating to achieve this, not that the one is scuppering the other).Solution: 6Here's one way. The pawn begins on B2. (second column, second row.)White: B2-4. Pawn moves two in knight column.Black: A7-5. Pawn moves two in adjacent rook column.White: B4-A5. Pawn takes pawn.Black: B7-6. Pawn moves one in knight columnWhite: A5-B6. Pawn takes pawnBlack: B8 – A6. Knight moves out of way.For the next three moves, white's pawn advances one by one in the B column, queens and then returns to B2 in the sixth move.4. Four knightsShow how to swap the two pairs of knights on the following strangely-shaped grid.The knights make one move at a time. You're trying to get the black nights to where the white knights are, and the white knights to where the black knights are.If you try to solve this problem using knights on a physical grid, you will get very confused. Try to think abstractly. With one simple(ish) insight, the problem is quickly solvable.Solution:The positions that the knights can move to are very constrained. Here are all possible moves and positions;This looks like a mess! However, if we untangle it, we can see the pattern. If we number boxes from the top row, and from left to right, so the white knights are on positions 1 and 5, and the black knights on 7 and 9, the board now looks like this:To exchange the positions of the knights is now a train shunting problem.Move the black knights to 8 and 6Move the white knight at 5 into the '"side track"' at 9Move the black knights back to 5 and 7.Move the white knight at 9 to 3Move the black knights back to 6 and 8Tuck away the white knight at 1 to square 9Move the black knights to 1 and 5, which is where we want them.Finally, move the white knight at 3 to 7, and we're done.The Mathematical Principles: Logic and Problem SolvingThese chess puzzles demonstrate fundamental mathematical principles including parity (odd and even numbers), graph theory (knight's tours), and optimization (minimum moves). The solutions require abstract thinking and pattern recognition, skills that are essential in both mathematics and chess strategy.The Impact on Problem-Solving: Developing Critical ThinkingChess puzzles like these help develop critical thinking skills that extend beyond the chessboard. They teach players to think several moves ahead, recognize patterns, and approach problems from multiple angles. These cognitive skills are valuable in academic pursuits, professional challenges, and everyday decision-making.Future of Chess Puzzles: Digital and Educational ApplicationsAs technology advances, chess puzzles continue to evolve with digital platforms offering interactive experiences and adaptive difficulty levels. Educational institutions increasingly recognize the value of chess in developing mathematical and logical reasoning skills. Organizations like We Solve Problems are expanding their reach, offering free math circles and chess programs to students across multiple cities, fostering the next generation of problem solvers.

The Lead The Edinburgh Festival 2026 promises to be an exciting event with a wide range of theatre productions. From feminist explorations to epic dramas, there's something for everyone. Roleplay and Mayflies Producer Francesca Moody brings 'Roleplay', a new offering by Australia's Hannah Reilly, about a feminist podcaster who becomes an online 'slutfluencer'. Meanwhile, Andrew O'Hagan's coming-of-age novel 'Mayflies' is adapted by Ben Harrison and directed by Grid Iron. 'Roleplay' - Summerhall, 6-31 August 'Mayflies' - Brown's of Leith, 7-30 August Epic Dramas Paines Plough's Katie Posner directs 'After Party', an epic drama by Morna Young, which explores family secrets on the night of an actor's 80th birthday celebrations. Additionally, Internationaal Theater Amsterdam revives Tony Kushner's 'Angels in America', a masterpiece about AIDS in the 1980s, directed by Ivo van Hove. 'After Party' - Traverse, 6-30 August 'Angels in America' - King's theatre, 15-20 August Unique Performances Other notable productions include '116 Grams: A Play to Lose Weight' by Letícia Rodrigues, 'Cathy' starring Elaine C Smith, and 'The Jolly Fisherman' by John Dinneen. '116 Grams: A Play to Lose Weight' - Zoo Southside, 7-30 August 'Cathy' - Traverse, 31 July-30 August 'The Jolly Fisherman' - Underbelly, 5-31 August Experimental and Innovative The festival also features experimental and innovative productions like 'Bigfoot Ripped My Dog in Half I Saw It' by Xhloe and Natasha, 'The Singer' with music and lyrics by KT Tunstall, and 'Concerts of the Future' by Ciaran Frame. 'Bigfoot Ripped My Dog in Half I Saw It' - Summerhall, 6-30 August 'The Singer' - Traverse, 4-30 August 'Concerts of the Future' - Summerhall, 6-31 August Powerful Messages Lastly, productions like 'Hang Time' by Zora Howard and 'A Trial – After An Enemy of the People' tackle serious issues like racialised violence and social justice. 'Hang Time' - Royal Lyceum, 20-23 August

The LeadJamaica has emerged from a rare islandwide power outage that struck the Caribbean Island overnight, with Energy Minister Daryl Vaz announcing that all affected customers have had their power restored. Both Vaz and Prime Minister Andrew Holness described the situation as 'unacceptable,' highlighting concerns about the nation's electrical grid vulnerability as the Atlantic hurricane season begins.The Islandwide Power CrisisThe blackout began at approximately 9pm local time on Friday (2:00 GMT Saturday), affecting all of Jamaica's 2.8 million residents. The Jamaica Public Service Company (JPS), the sole electricity distributor on the island, reported the outage and began restoration efforts. By 2am Saturday (7:00 GMT), the company had restored power to 20 percent of customers, or approximately 140,000 people in areas including Kingston, St Andrew, and Clarendon. Three hours later, Minister Vaz announced that electricity had been returned to 500,000 of JPS's 700,000 customers overnight, with the remainder to be restored in the following hours.The Government ResponseMinister Vaz took immediate action, calling an emergency meeting with government and JPS officials to discuss the blackout. He committed to keeping the nation informed throughout the restoration process, stating on social media: 'I have been closely monitoring the situation all night and will continue to do so until full restoration is completed.' Vaz has formally demanded a full report from JPS within 24 hours, including a detailed explanation of what caused the power outage. The minister also encouraged residents experiencing any 'isolated issues' related to the blackout to contact him directly.The Grid Vulnerability ConcernsThe timing of the blackout has raised particular alarm, occurring at the start of the Atlantic hurricane season. This has intensified concerns about how Jamaica's electrical grid might withstand future weather events. The island's reliance on a single electricity provider, JPS—which was briefly nationalized before returning to private hands—has drawn scrutiny following this widespread failure. This incident is unusual for Jamaica, which typically only experiences islandwide outages during weather emergencies, such as last year's Hurricane Melissa, which caused billions of dollars in damage and dozens of deaths.The Aftermath and InvestigationAs of Saturday morning, JPS stated it was 'investigating the cause of the cause of this incident,' with no official explanation provided yet. The company had assured customers that its teams would be working throughout the night to restore power 'as safely and quickly as possible.' The government's strong reaction suggests potential regulatory consequences for JPS, as the nation seeks to prevent similar disruptions in the future. With hurricane season now underway, the resilience of Jamaica's energy infrastructure will likely face increased scrutiny and testing in the coming months.

The border that separates Lebanon and Israel has become a self‑reinforcing war machine, where each exchange fuels the next. Recent incidents in 2025‑2026 have revived old grievances, entrenched militia power, and strained regional diplomacy, making a durable cease‑fire increasingly elusive. Escalating Tensions Along the Blue Line in 2025‑2026 April 2025: Hezbollah fired a salvo of rockets toward the Israeli town of Kiryat Shmona, prompting a retaliatory airstrike on a suspected weapons depot in southern Lebanon. January 2026: Israeli drones intercepted a convoy crossing the Blue Line, alleging the transport of advanced missile components. June 2026 (latest): A cross‑border artillery exchange resulted in civilian casualties on both sides, reigniting UNIFIL calls for restraint. Human and Economic Toll of the Stalemate Since 2025, approximately 1,200 civilians have been killed and over 4,500 injured across the border region. UN estimates that the conflict has displaced 150,000 residents in southern Lebanon and the northern Israeli districts. Combined infrastructure damage exceeds $2 billion, with agricultural losses accounting for 30 % of Lebanon’s southern output. Regional Ripple Effects and Diplomatic Gridlock Iran’s continued support for Hezbollah deepens Tehran’s leverage in the broader Middle‑East power balance. U.S. and EU mediation efforts have stalled, as both sides demand pre‑conditions that the other deems unacceptable. UNIFIL’s mandate faces criticism for limited enforcement capability, eroding confidence in multinational peacekeeping. Scenarios Shaping the Next Decade of the Border Conflict Continued Low‑Intensity Warfare: Persistent skirmishes keep the status quo, draining resources and fostering radicalization. Escalation to Full‑Scale Conflict: A miscalculation or external trigger could spark a broader war, drawing in regional powers. Negotiated Freeze: A mutually‑acceptable cease‑fire, backed by robust UNIFIL rules of engagement, could stabilize the border but would require significant concessions. Until a credible security architecture replaces the cycle of retaliation, the Lebanon‑Israel frontier will remain a perpetual engine of conflict, shaping the political and economic landscape of the entire Eastern Mediterranean.

The Bridge Between Art and ScienceWhy do we find things beautiful? More precisely, why do some paintings of coloured dots in rippling patterns inspire in me something like revelation? The idea that beauty is the feeling you get when encountering truth is unfashionable in the arts, but lingers in the sciences. The physicist Paul Dirac once proposed that it is more important that a formula is beautiful than that it can be proven: when a perfectly beautiful theory produces results that cannot be real, he argued, then we should not discard the theory but reconsider what is real.Since the 1970s, Terry Winters has been rebuilding that bridge between art and science. Taking inspiration from disciplines including botany – his early paintings, particularly, evoke sprouting pods and tangled roots – engineering, computer modelling and cybernetics, his paintings might be understood as diagrammatic approximations of the patterns that govern everything from the division of cells to the constellation of stars. If every era has to renew its standards of beauty to reflect new understandings of how the world is constructed, then Winters comes as close to providing that model as any living painter.Mathematical Patterns in Visual ArtThese eight new works take their titles from the language of geometry and mathematics: Area, Array, Field, Locus, Point, Scope, Sequence and Set. Each is composed of overlapping patterns that pull each other out of shape according to invisible laws of attraction and repulsion. Field is indicative: on a sooty pink ground, a dense grid of dusty blue cells bends inwards like a trampoline beneath a bowling ball, while an intersecting arrangement of larger circles swells outwards. The disorienting effect of this push-pull is exaggerated by an optical illusion, created by ragged phosphorescent orange haloes around the blue circles, that makes them appear to be craters sunk into the crust of paint. You have to walk up to the surface to be reassured that it is flat.Point shows a teeming landscape of cells bulging outwards at the centre, as if a scum of frogspawn had been skimmed off a pond and placed under a paperweight microscope. In Sequence, a storm of pink interference skims through a yellow circle split like a brain into hemispheres, while a nebulous weather system sweeps in from the right. In Scope, vaguely symbolic arrangements of freshwater blue and desert orange nodules move across the familiar circle-within-a-square geometry of Leonardo's Vitruvian Man. Where that famous demonstration of sacred geometries exudes stillness and calm, the systems that run through Winters' worlds are wild and strange. Here, as much importance is given to the subjective factors of perception and consciousness as the objective principles of logic and proportion.Optical Illusions and Sensory ExperienceAll of this should not distract from the pure sensory pleasure afforded by these pictures. In Locus, another optical illusion seems to lift the red edge of the painting off the canvas like a clumsy wooden frame, squeezing the pockmarked sphere at its centre so that it balloons outwards, threatening to burst. The same cadmium red, so sandy that the pigment seems barely to have been suspended in oil, turns carmine pockets into rock formations that climb off the surface of Set. These sleights of hand evoke the more secular movement of op art, in which patterns are manipulated to exploit the idiosyncrasies of human perception rather than to uncover deeper truths. Which begs the question: is this all just a conjuring trick?Rediscovering the Renaissance ApproachThere is something magical about these works. Even though Winters is generally credited with having extended the lineage of modernist American painting into the present, his practice is in this sense pre-modern. In its rejection of the idea that art should be separated from science, it resembles the Renaissance attitude according to which painting is no less a tool for understanding the world than mathematics, and magic is just the name for things we don't yet understand. His commitment to synthesising diverse spheres of knowledge, his alchemist's commitment to materials, and his sheer technical ability certainly mark him out in a scene recently overwhelmed by chancers and charlatans, bandwagoners and snake-oil salesmen. That his work is enjoying a revival might be taken as an encouraging sign.Beauty as a Path to TruthDirac, incidentally, was proved right. He formulated a theory that was so beautiful it couldn't possibly be wrong, even though it implied the existence of phenomena that everyone agreed must be impossible. He stood by it, and a few years later, someone else discovered anti-matter. Winters' paintings likewise offer a flash of those secret patterns that underpin the physical world, and which science has yet to illuminate. Which is to say, they're beautiful.Exhibition DetailsTerry Winters: Along the River is at Modern Art, London, until 11 July

The Energy Advantage In the race against China for AI supremacy, the United States dominates when it comes to access to the most cutting-edge semiconductors. But when it comes to powering the huge data centres that run on AI chips, China holds the clear advantage. That's because data centres, the sprawling computing facilities needed to train and run AI models, require vast amounts of energy. A typical data centre can consume as much electricity as 100,000 households, while next-generation “hyperscale” facilities can gobble up as much power as two million homes, according to the International Energy Agency (IEA). China's Renewable Energy Boom China already generates more than twice as much electricity as the US, a lead that is expected to widen amid an aggressive state-led investment in the country’s energy grid. BloombergNEF, a research provider, estimates that China will add more than six times as much electricity generation capacity as the US over the next five years. Much of that extra capacity will be in the form of renewables such as solar and wind. In 2025 alone, China increased its wind and solar power capacity by more than 430 gigawatts, accounting for more than half of the additional capacity in the renewables added globally that year. The Impact on Data Centres A key element of China’s AI strategy involves integrating its data centres into its rapidly expanding renewables sector. Under the “East Data, West Computing” initiative, China’s government is concentrating the construction of new data centres in the country’s sparsely populated interior, where land and renewable energy sources are abundant compared with the heavily built-up eastern seaboard. Earlier this month, Beijing announced the start of operations at the country’s first “large-scale” renewable energy project to be linked directly to a data centre. Narrowing the Gap For now, the US still has the largest data centre footprint by a wide margin. According to Stanford University’s AI Index, the US had an estimated 5,427 data centres in 2025, compared with 449 in China. But as China constructs data centres at a blistering pace – its number of data centre racks grew 30 percent annually from 2016 to 2023, according to the China Academy of Information and Communications Technology – the gap between the superpowers is rapidly narrowing. The Future Outlook “In the long run, the country that can provide cheap, stable, low-carbon electricity will have a major advantage in AI infrastructure,” Qiyang Xiong, a PhD candidate at Renmin University of China who specialises in AI and energy policy, told Al Jazeera. “China is a global leader in solar, wind and ultra-high-voltage transmission,” Xiong said. “This gives it an advantage in supplying western data centre clusters with large volumes of relatively cheap, clean electricity.”

Escalating El Niño Probability and UN WarningThe United Nations has warned that the world must prepare for the imminent return of El Niño, a powerful weather pattern that raises global temperatures and drives extreme weather. The World Meteorological Organization (WMO) reported an 80% chance of El Niño forming before September and a 90% chance before November.Projected Climate Impacts Across Key Asian RegionsIndia: Expected below‑average monsoon rainfall, extending the current heatwave and threatening wheat and mustard crops.China: Anticipated 20% higher rainfall in southern regions, with some areas forecast to receive over 200 mm of rain; the Qinghai‑Tibetan plateau warned of “unpredictable and extreme” conditions.General: Intensifying heat and drought could stress agriculture, power grids, and water supplies across the continent.Sectoral Vulnerabilities: Agriculture, Power Grids, and Water SuppliesExperts highlight a “deadly combination” for India, where delayed monsoon rains could exacerbate the ongoing energy crisis and jeopardise food security. In Mumbai, the city’s seven rain‑fed lakes hold only enough water for 45 days, raising the risk of a severe water shortage if rains are delayed. In China, flood‑prone regions face heightened storm risk, while drought‑sensitive areas worry about power‑grid strain.Looking Ahead: Preparedness Measures and Uncertain OutlookNational climate agencies in both India and China are urging stockpiling of emergency supplies and issuing weather warnings. The UN stresses that El Niño’s impacts will be “super‑charged” by human‑driven climate change, making the upcoming summer and autumn seasons especially unpredictable for the region.

The Climate Policy Reversal in Blue States Democratic-led states are eroding their climate policies, as red states are scaling up their clean energy deployment. California on Friday scaled back its cap-and-invest program, offering more than $3bn in free pollution allowances to polluting companies. Earlier the same week, New York weakened its groundbreaking climate law, delaying a plan to regulate carbon from 2024 until 2028 and reducing emissions-slashing targets. Rhode Island's governor, meanwhile, is attempting to roll back aggressive clean-energy programs. The Economic Justification vs. Climate Imperative The moves come as Donald Trump's administration withdraws clean energy incentives and energy savings programs, and as energy prices spike across the country amid trade disruptions stemming from the US-Israeli war on Iran. Proponents have said the changes are necessary to suppress electricity costs, but climate advocates say that view is short-sighted and misguided. "Using affordability as a cudgel to weaken climate policy is a major error that will not solve either crisis, ultimately amplifying both," said Johanna Bozuwa, executive director of the Climate and Community Institute, a left-leaning thinktank. "Extreme weather and fossil-fuel dependency directly inflate costs – for food, energy, transportation, housing, and health – across the economy for working people." American Public Opinion on Climate Change Polls show most Americans are concerned about the climate crisis. An annual poll from Gallup, published in April, shows that 44% of American adults say they worry "a great deal" about global warming – one of the highest levels of concern since 1989, when the poll was first conducted, behind only 2020 and 2017. About 65% of registered voters in the US also think global heating is driving up the cost of living, according to a report published in December by Yale University and George Mason University. Red States Lead Clean Energy Buildout In contrast to many Democratic-led jurisdictions, red states have tended to dominate renewable energy deployment in recent years. In terms of growth of utility-scale renewables, states that voted for Donald Trump in the 2024 presidential election made up eight of the top 10 in the year to March, according to Energy Information Administration data. Indiana tops the list of states with the most clean energy capacity growth in that timeframe, followed by Kentucky and Utah. More broadly, though, it is Texas that has emerged as the country's leading clean energy superpower, despite its strong ties to the oil and gas industry and unsuccessful attempts within the Republican-led legislature to curb the growth of wind and solar. Texas leads the country in wind energy production, followed by fellow red states Iowa, Oklahoma and Kansas, and in March overtook California in utility-scale solar, too. The Paradox of Climate Leadership Meanwhile, the states scaling back their emissions-cutting policies have long called themselves climate leaders. When Governor Gavin Newsom of California extended his state's cap-and-invest program last year, he said: "We're doubling down on our best tool to combat Trump's assaults on clean air … by making polluters pay for projects that support our most impacted communities." The changes could end up giving more money to the fossil fuel producers and distributors who have been increasing consumers' energy prices amid the Iran war, said Bahram Fazeli, Policy Director with Communities for a Better Environment, a grassroots organization in California. "There's no reason to think that giving them more free allowances will actually help motivate them to lower gas prices more," he said. Long-Term Economic Implications New York advocates are also skeptical about whether the weakening of the 2019 Climate Leadership and Community Protection Act – which the state touted as among the strongest climate laws the country – will deliver long-term benefits. The state legislature last week reached a deal with Governor Kathy Hochul to remove a 2030 mandate to cut planet-warming pollution by 40% from 1990 levels, instead including language to aim for a 60% by 2040 if it is "feasible and cost effective" to do so. "Even though you might see bill savings initially, that's going to come at the cost of locked-in, higher energy costs in the future, as the grid has to procure more energy that would otherwise have been saved," Anna Johnson, a senior policy manager State at American Council for an Energy-Efficient Economy, told Baltimore's NPR affiliate WYPR; she estimates that the moves could ultimately increase households' electricity costs by $592m. The True Cost of Inaction The climate crisis itself also costs for working people, said Mar Zepeda Salazar, legislative director of the national environmental justice coalition Climate Justice Alliance. "You can lower costs on paper by weakening protections, but the bill still comes due," she said. "It just shows up in emergency rooms, insurance premiums, utility bills, lost wages, and disaster recovery – that families pay, not industry."