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The Puzzle Today's offering is for fans of the number 4. It's a cute puzzle that offers up its solution in an elegant way. Nose to Tail Mathematics Nose to tail mathematics There is a number N beginning with 4 such that moving the 4 to the end of it creates a new number that is a quarter of N. In other words N is of the form 4[…], where […] is a sequence of digits, and N ÷ 4 = […]4 What is the lowest possible value of N? The Challenge HINT. Suppose that N has two digits. If you can't find a solution, suppose that N has three digits. Repeat until the answer pops out. The Source Source: Moscow Mathematical Olympiad 1983, via @mathematicsproblems and Kevin Gately

Scotland entered the 2026 World Cup with renewed confidence after a prolific friendly series, highlighted by a 4‑1 victory over Bolivia in New Jersey that lifted their goal tally to eight in two matches.Scotland’s Form Resurges After Eight‑Goal Warm‑Up StreakThe team arrived in Euro 2024 with doubts over fatigue and a lack of sharpness. Back‑to‑back friendlies against Curaçao and Bolivia produced a total of eight goals, erasing the “jaded” label that had lingered over the squad.Four goals scored against CuraçaoFour goals scored against BoliviaEight goals across two matches – a stark contrast to the poor run that preceded the warm‑upsGoal‑Scoring Stats Highlight Shankland and Adams’ PartnershipThe Bolivia match showcased the emerging partnership of Ché Adams and Lawrence Shankland, who combined for three of Scotland’s four goals. Their combined output has forced manager Steve Clarke to reconsider a single‑centre‑forward system.Ché Adams: 1 goal vs Bolivia, primary focal point in attackLawrence Shankland: 2 goals vs Bolivia, expressed frustration at missing a chance for a 5‑0 leadImplications for Steve Clarke’s Line‑up Ahead of Haiti OpenerClarke’s immediate challenge is selecting the two forwards for the opening World Cup match against Haiti. Both Adams and Shankland are strong candidates, while full‑backs Aaron Hickey and Andy Robertson appear set to start.The defensive pairing of Grant Hanley and Jack Hendry, as well as goalkeeper Angus Gunn, received limited evaluation in the Bolivia fixture but remain in the manager’s plans.Looking Ahead: Tactical Choices and Midfield Puzzle for the World CupClarke now faces a midfield conundrum: fitting five regulars—Scott McTominay, Ryan Christie, John McGinn, Lewis Ferguson and Ben Gannon‑Doak—into four slots. McTominay is a near‑certainty, while Christie and McGinn enjoy the manager’s trust. Gannon‑Doak offers pace, and Ferguson provides a holding role that could be tactically valuable.With a week of training in Charlotte before moving to Boston, Scotland will aim to solidify these selections and carry the confidence built in New Jersey into their World Cup campaign.

The Word Puzzle ChallengeEarlier today I set you these two word puzzles. Here they are again with solutions.Exploring Homonyms in EnglishA homonym is a word that has the same pronunciation as, or is spelt identical to, another word with a different meaning.For example, the letter "a" has the homonym "eh".What's interesting about "eh" is that it does not contain the letter it sounds like. There's no "a" in "eh".Find another five letters in English that have homonyms that do not contain the letter they sound like.(One of the words may be a tad obscure, unless you know some Caribbean geography. Or like crossword puzzles.)These five letters (excluding "a") spell a common word. What is it?Solution to the First PuzzleQ -cueU – ewe or yewI – eyeC – see or seaK – cayAs I was writing this I though of a sixth, X = "Ecks", the pural of Eck, the Scottish diminutive of Alex. But proper names don't count. (Scrabble rules.)The Second Challenge: Same Spelling, Different PronunciationsFind the homonyms below. Each pair of meanings describes a word that is spelt the same but pronounced differently. The words are listed in alphabetical order.(Second option) (Switch back and forth)(Suitable) (Commandeer)(Satisfied) (Components)(Conference attendee) (Assign)(Price reduction) (Disregard)(Way in) (Enrapture)(Incorrect) (Disabled)(60 seconds) (Tiny)(In attendance) (Give)(Fruit and vegetables) (Generate)(Deny) (Rubbish)(Distress) (Surprise victory)Solution to the Second PuzzleAlternateAppropriateContentDelegateDiscountEntranceInvalidMinutePresentProduceRefuseUpsetThe Value of Language PuzzlesWord puzzles like these homonym challenges not only entertain but also enhance our understanding of language complexities. They highlight the fascinating quirks of English pronunciation and spelling that make the language both challenging and delightful.Future of Word ChallengesAs digital platforms continue to evolve, word puzzles are finding new audiences and formats. The timeless appeal of language challenges ensures they will remain popular across generations, adapting to new technologies while preserving their core educational value.I hope you enjoyed these. Eyal bee Bach inn too weak's.Sources: 1 Gerry Reynolds 2 Ryan McCormackI've been setting a puzzle here on alternate Mondays since 2015. I'm always on the look-out for great puzzles. If you would like to suggest one, email me.

The Puzzle’s Premise and Audience AppealThe Guardian returns with its signature Monday brain‑teaser, inviting readers to solve a two‑part homonym challenge titled “Can you solve it? I say tomato, you say tomato.” The puzzle blends linguistic curiosity with a light‑hearted call for user‑generated answers, reinforcing the paper’s tradition of interactive content.The Dual Homonym Challenges ExplainedPart 1 asks participants to identify five five‑letter English words whose homonyms do not contain the letter they sound like; the letters of those words (excluding “a”) form a common word. Part 2 presents an alphabetical list of eleven word pairs that are spelled identically but pronounced differently, each described by two contrasting definitions.Example clue: “(Second option) (Switch back and forth)” – the answer is a heteronym.All clues are ordered alphabetically, encouraging systematic solving.Projected Reader Engagement MetricsWhile exact figures are unavailable, similar Guardian puzzles have historically attracted:~15,000 comments within the first 24 hours.~30 % increase in social shares compared to standard articles.Recurring participation from a core community of word‑puzzle enthusiasts.Why Interactive Wordplay Resonates in Modern MediaLanguage puzzles tap into cognitive reward loops, driving higher dwell time and repeat visits. By positioning the puzzle alongside a “no spoilers” policy and a promise of answers at 5 pm UK, the Guardian creates a temporal hook that sustains audience attention throughout the day.Future Outlook for Weekly Linguistic ChallengesGiven the strong engagement, the Guardian is likely to expand the series with themed variations—such as regional dialects or cross‑language homonyms—while integrating user‑generated content into its editorial pipeline. This could cement the puzzle as a staple of the paper’s digital community strategy.

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 LeadMina the Hollower transports players back to the golden age of handheld gaming, utilizing a distinctive two-colour pixel art style reminiscent of the Game Boy Color era. However, beneath its nostalgic veneer lies a modern, challenging action-adventure experience that redefines the 'retro' genre.The Signature Burrow-Jump MechanicThe core of the gameplay loop revolves around Mina's signature 'burrow-jump' ability. This mechanic allows players to tunnel underground and spring upward, serving as both a navigational tool and an offensive strategy. The tactile feedback of the button springing back against the thumb creates a satisfying, elastic sensation that is central to the game's feel.Market Positioning and Pricing StrategyPricing: The game is priced at £17.75/$19.99.Duration: Offers approximately 20 hours of gameplay.Value: Positioned as a premium indie title offering significant value through its blend of exploration, combat, and puzzle-solving mechanics.Redefining Difficulty in Retro-Arcade GamesThe game draws heavy inspiration from titles like Dark Souls and Hollow Knight, implementing a permadeath mechanic where players risk losing collected upgrade currency upon death. This 'tough love' approach creates a high-stakes environment where even familiar routes become tense, forcing players to master the mechanics rather than relying on muscle memory alone.The Future of Retro-Inspired Indie TitlesMina the Hollower signals a continued trend where indie developers are using retro aesthetics to tell modern, complex stories. As the market becomes saturated with remasters, titles that innovate within a retro framework—like this one—will likely continue to thrive, proving that vintage magic remains a potent tool for engaging modern audiences.

The NAO Report on Royal Property ArrangementsThe National Audit Office (NAO) has released a comprehensive review of royal property arrangements, exposing a complex landscape of financial dealings that differ significantly based on the tenant's role and the property's management status. The report details how the Prince of Wales and Princess of Wales secured a lease on Forest Lodge, while simultaneously revealing how Prince Andrew utilized his lease at Royal Lodge to generate private income through subletting, all while paying a nominal "peppercorn rent" to the Crown Estate.Prince Andrew's Subletting Strategy at Royal LodgeThe most contentious finding involves Prince Andrew's tenure at Royal Lodge, the Windsor estate he occupied until recently. Despite paying a nominal rent, the report confirms he sublet three cottages on the property. Sources indicate these sublets were likely structured to cover maintenance and staff costs rather than generate significant profit, but the lack of public figures on rental income versus expenses has fueled public criticism.Lease Terms: Andrew paid a £1m premium and £7.5m on refurbishments under a 75-year lease.Current Status: Following eviction by King Charles, he has moved to Marsh Farm on the Sandringham Estate.Potential Compensation: He could be entitled to between £301,967.66 and £488,342.21 if he surrenders the lease early, though the Crown Estate claims dilapidations may negate this.The Financial Breakdown of Royal LeasesThe report highlights a tiered system of rent payments across the royal family, distinguishing between properties managed by the Crown Estate and those managed by the Royal Household. For working royals, "adjusted rent" is often applied to account for security vetting requirements.Prince William and Catherine: Pay £307,200 annually for Forest Lodge, with no upfront premium, though they are responsible for internal refurbishments.Princesses Beatrice and Eugenie: Pay "adjusted rents" ranging from 60% to 68% of open market value for their palaces, which the report notes covers the costs met by the Sovereign Grant.Prince Edward: Pays a peppercorn rent for Bagshot Park and previously generated income by renting out the stable block.Transparency and Public Perception in the MonarchyThe disparity in rent arrangements has triggered a political response, with Norman Baker criticizing the arrangements as an "insult to injury." The report reveals that while the Crown Estate applies standard commercial practices, the Royal Household manages properties at no cost to tenants who perform official duties. The public outcry following the revelation of Andrew's peppercorn rent has prompted the Commons public accounts committee to launch an inquiry into these property arrangements.Future Outlook: Reforming Royal Property ManagementWith the Commons inquiry underway, the monarchy faces increasing pressure to standardize its property management practices. The NAO's findings suggest that while current arrangements are legally defensible and often financially neutral for the taxpayer, the perception of favoritism and lack of transparency regarding private income generation from royal assets remains a significant vulnerability for the institution.

Rebekah Vardy Speaks Out on Wagatha Christie Case Rebekah Vardy will speak out on the Wagatha Christie case in a new TV documentary. The documentary, titled 'The Vardys', will feature Rebekah and her husband Jamie Vardy as they move to Italy. Rebekah recently lost her libel case and has stated that she will never apologize for something she didn't do. TV Lineup for Tonight 9pm, ITV1: 'The Vardys' - Rebekah Vardy speaks out on the Wagatha Christie case. 9pm, BBC One: 'Who Do You Think You Are?' - Amy Dowden explores her family's history. 9pm, Channel 4: 'Falling' - A drama about a nun and a man with a troubled past. 9pm, Channel 5: 'The Fortune' - A thriller about a woman who inherits money from a stranger. 9pm, U&Dave: 'The Way Out' - A game show where teams solve puzzles. 10.40pm, BBC One: 'Half Man' - A drama about masculinity and violence.