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The Perfect World Cup Send-offIn a last outing before a first World Cup appearance in 28 years, Scotland dismantled Bolivia with a commanding 4-0 victory. The performance, featuring goals from Lawrence Shankland, Scott McTominay and a brace from Ché Adams, suddenly worries over a potentially tournament defining joust with Haiti next weekend evaporated. If Steve Clarke's men are this ruthless and efficient when the proper stuff starts, they have a serious chance of emerging from the group phase for the first time in Scotland's international history.First Half DominationIn sweltering New Jersey heat, the Scots made a fine start. Andy Robertson completed a one-two with Ryan Christie before chipping a cross to the back post. The Bolivia goalkeeper, Guillermo Viscarra, should have done better with Shankland's header but Scotland's No 20 had no cause to care. This marked his third goal in two games. Adams, who was lively, tested Viscarra before firing a shot only narrowly wide.A second goal was soon forthcoming anyway. Shankland nudged the ball back to McTominay, whose low drive from 18 yards beat the highly unconvincing Viscarra. Scotland's third was a fine goal too, with Aaron Hickey feeding the marauding Ben Gannon-Doak, who put in a perfect cross to the feet of Adams. The Torino man could not miss.Scotland's Attacking ProwessScotland were to match a first-half scoreline margin feat first achieved against England in 1878 and, until now, last versus the Faroe Islands 20 years ago. Gannon-Doak, staking a huge claim for a World Cup start, drove from midfield. The Bournemouth man smoothly found Adams, who beat Viscarra at the second attempt. The Scots were 4-0 to the good at the break. This also meant they had scored a quartet of goals in successive games.Unlike Ally MacLeod's infamous hubris of 1978, Steve Clarke has never been prone to bold or rash predictions. Excitement will be left to everyone else. Who can reasonably deny them that? This friendly, in theory an exercise in box ticking, instead gave reasons for huge Scottish confidence. Scotland will remember their first ever game against Bolivia with great fondness.Tactical ConsiderationsShankland can now be considered a guaranteed Scotland starter for the Haiti clash. He joins Robertson, Hickey, McTominay and John McGinn in that bracket. The likelihood is that Clarke would be keen to deploy Adams in attack alongside Shankland, but the impact made by Lyndon Dykes during last weekend's win over Curaçao inserted doubt to that scenario.Clarke has tough calls over who starts in goal and at centre half in Boston. Christie excelled against Bolivia, which should enhance his Haiti prospects. Scott McKenna, Dykes and Nathan Patterson were the only outfield Scots not to feature at all. This was now a game they only had to manage adequately, including without the injury distress that ended Billy Gilmour's World Cup dream seven days earlier.World Cup OutlookBolivia are ranked above Haiti, whom Scotland will surely have to defeat for the opportunity to reach the World Cup's last 32. The performance against Bolivia suggests Clarke's team has the quality to achieve this objective. While it is fair to point towards limited opposition, that is precisely what Scotland will face in game one of Group C.Foot was removed from gas in the second period but there was still something hugely admirable about Scotland's approach. Now for the key question; will it be alright on the night? This time, there was no sour note. Clarke will have no desire to summon his inner or outer MacLeod. Nonetheless, Scotland head to the World Cup with collective spring in step.

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 Budget iPhone Gets a Major UpgradeThe cheapest new iPhone has been upgraded for this year with a faster chip, double the storage, automatic portraits and MagSafe, providing even more of the core Apple smartphone experience for less. The iPhone 17e is an upgraded version of the mid-range "e" line launched last year with the first iPhone 16e and is the latest member of the iPhone 17 family. It starts at £599 (€699/$599/A$999), undercutting the iPhone 17 and iPhone 16 by £200 and £100 respectively to be the cheapest new iPhone sold by Apple.Design and Build QualityThe new 17e is the spitting image of the model it replaces, giving it the older iPhone 14-like design with a large notch at the top of the screen and a slower 6.1in OLED screen. The aluminium sides feel great and the screen glass has been upgraded to the latest Ceramic Shield 2, which is tougher and includes an extremely effective anti-glare treatment that makes it a lot easier to see outdoors. The 17e has MagSafe built into the back for magnetic accessories, such as Popsockets, wallets and chargers, which have been a key part of the iPhone experience since 2020.Key SpecificationsScreen: 6.1in Super Retina XDR (OLED) (460ppi)Processor: Apple A19 (4-core GPU)RAM: 8GBStorage: 256 or 512GBOperating system: iOS 26Camera: 48MP rear; 12MP front-facingConnectivity: 5G, wifi 6, NFC, Bluetooth 5.3, USB-C, Satellite and GNSSWater resistance: IP68 (6 metres for 30 mins)Dimensions: 146.7 x 71.5 x 7.8mmWeight: 170gPerformance and Battery LifeThe 17e has the A19 chip from the regular iPhone 17 but with one less GPU core, which reduces graphics performance slightly. Not that anyone will probably notice, as the phone is very fast and still capable of handling top-spec games. It also has a decent 256GB of storage as standard, which should be enough space for most with additional cloud backup. The battery life is great, too, lasting a good 52 hours between charges with general usage across 5G and wifi, meaning most will need to charge it every other night.The 17e lacks a few of the more advanced hardware features common to Apple's other phones, including wifi 7, Thread and Ultra Wideband (UWB), the latter of which is used for the precision finding tool and for some digital car keys, among other features.Sustainability and RepairabilityThe battery should last in excess of 1,000 full-charge cycles, with at least 80% of its original capacity, and can be replaced for £95. Out-of-warranty screen repairs cost £225. The 17e has repair guides available and was awarded seven out of 10 for repairability by the specialists iFixit.It contains more than 30% recycled material including aluminium, cobalt, copper, glass, gold, lithium, plastic, rare earth elements, steel, tin and tungsten. The company breaks down the phone's environmental impact in its report, and offers trade-in and free recycling schemes including for non-Apple products.Camera CapabilitiesThe single camera on the back may be a deal killer for some. The iPhone 17e features automatic portrait mode functionality, which was previously reserved for more expensive models in Apple's lineup. This allows users to create professional-looking portrait shots with depth effects even with the single rear camera setup.Market Position and Value PropositionWith the iPhone 17e, Apple is clearly targeting budget-conscious consumers who want to enter the iOS ecosystem without paying premium prices. The inclusion of features like MagSafe, the A19 chip, and 256GB of storage at this price point represents a significant value proposition compared to previous generations. This strategy helps Apple capture market share from Android manufacturers in the mid-range segment while maintaining brand loyalty.Future Outlook for Apple's Budget LineThe iPhone 17e sets a new standard for Apple's budget lineup, suggesting that future "e" models will continue to incorporate more premium features at lower price points. As Apple faces increasing competition in the smartphone market, particularly in the mid-range segment, we can expect continued innovation in this product category. The success of the iPhone 17e may influence Apple's entire product strategy, potentially leading to more aggressive pricing and feature inclusion across all iPhone tiers.

The Revival of Motion Gaming For a wonderful moment in the noughties, video games became a truly universal pursuit. The Nintendo Wii flew off the shelves, inspiring a wave of competitors such as the Xbox Kinect camera that encouraged people to play games by moving their bodies. But the tide turned: outside of still-niche VR gaming and the odd controller-waggler on the Switch, motion-controlled gaming has barely been seen for more than a decade. Nex Playground Enters the Scene Now, 20 years later, a new console is aiming to get the whole family flailing in front of the TV once again: the Nex Playground. Launching in the UK later this month, the first thing that struck me about this family-friendly device is just how tiny it is. The size of two and a half Rubik’s Cubes taped together, this impressively unintrusive device swaps cumbersome controllers for camera-controlled minigames, putting you and your family directly in the game. The Technology Behind Nex Playground Using a wide-angle lens and AI-powered tracking tech, the Nex Playground offers over 50 games that track players’ bodies as they leap, flail and dance about the living room. It’s not hard to see the appeal. Physically leaping through puddles in Peppa Pig: Jump and Jiggle, dancing in time to Rick Astley on Starri and slicing up watermelons with my hands in the perennial hit Fruit Ninja, I’m impressed by how seamlessly – and accurately – the tech works. Market Performance and Pricing The Playground retails at £269 ($299) – significantly less than any other games console at the moment. But it comes with just five free games. The rest of its library is locked behind an eye-watering £90 annual subscription. In the US, where it launched in 2023, the Playground has sold over a million units, even outselling Microsoft’s Xbox consoles during 2025’s Black Friday week. Safety and Future Plans Nex appears to be taking great care to earn families’ trust. None of the camera data from Nex play sessions is saved – either offline or online – meaning that families can happily embarrass themselves without worrying that an omniscient tech firm is tracking their every movement. Online multiplayer is coming to Playground soon, via parent-controlled “playdates”, and Lee hopes that this will also help older relatives stay connected with their families. The Future of Family Gaming Game publishers who’ve previously made games for Kinect and VR are already coming to Nex, Kang says. Child-focused brands such as Hasbro, DreamWorks and Mattel have already licensed games for it, perhaps seeing it as a safer alternative to social media and smartphone platforms – a view that most parents are likely to share. The most family-friendly dedicated games console currently available, Nintendo’s Switch 2, recently raised its price to £395.99, with new games at £50+ each; a lot of families are looking for a more affordable option. Nex Playground launches in the UK on 22 June.

Messi’s Quest for a Second World Cup: A Narrative OverviewThe article reflects on Lionel Messi’s 2022 World Cup performance in Qatar, framing it as a possible final chapter in his international career and questioning whether the 2026 edition could rewrite his legacy.Why Qatar 2022 Felt Like a Climactic FinaleMessi entered Qatar with a recent Copa América win, a reputation for quiet leadership, and a memorable post‑match outburst that suggested a shift from his reserved persona. The narrative positions the tournament as a culmination of decades of success, yet leaves open the possibility of an encore.Statistical Snapshot: Age, Appearances, and Recent Club FormAge: 39 during the 2026 World Cup, making him the oldest Argentine to play at a World Cup.Pre‑2022 World Cup club activity: 13 Ligue 1 matches and 5 Champions League appearances.2024‑25 season: 14 MLS games and 2 CONCACAF Champions League matches.International output: continued productivity in Copa América, qualifiers, and friendlies.Implications for Argentina’s Football Identity and Global PerceptionMessi’s potential second triumph would shift the long‑standing comparison with Diego Maradona, who won a single World Cup in 1986. A repeat victory could redefine Argentine football’s narrative from “Maradona’s shadow” to “Messi’s era,” influencing fan culture, media framing, and future player development.What the 2026 World Cup Could Mean for Messi’s LegacyIf Messi lifts the trophy again, he would join an elite group of players with multiple World Cup wins, cementing a legacy that transcends club achievements. Conversely, an early exit could cast his extended career as a cautionary tale about longevity and the risks of playing beyond peak performance. The article suggests that the 2026 tournament will be the decisive test of whether Messi’s story ends in a historic climax or an anticlimactic denouement.

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!

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.

The Lead: Spurs lose 105‑104 in Game 2 despite a 14‑0 rallyThe San Antonio Spurs fought back from a 14‑point fourth‑quarter deficit, briefly taking a one‑point lead before a series of late miscues handed the New York Knicks a 105‑104 victory. The Missed Game‑Winning ShotWith the score tied at 104‑104 and 9.5 seconds remaining, Wembanyama attempted a potential game‑winner that rimmed out. He later said, “I liked the shot, but I need more poise and control.” The Turnover That Sealed the GameMoments earlier, Wembanyama threw a pass intended for teammate Stephon Castle, but Castle never saw it, resulting in a turnover that halted the Spurs’ momentum. Score tied at 104‑104 with 9.5 seconds left. Wembanyama’s pass to Castle turned over. Final jump shot missed the rim. Series Implications: A Hole Too Deep?The loss puts the Spurs in an unprecedented position, needing a comeback as the series shifts to New York for Games 3 and 4. No NBA team has ever lost the first two finals games on its home floor and still captured the title. Looking Ahead: Turning Regret into FuelWembanyama acknowledged his error, saying, “I threw that one away. I messed up,” but added he will use the disappointment to motivate himself and the team for the next game.