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Introducing the "Nose to Tail" Number PuzzleToday’s offering is for fans of the number 4. The Guardian invites readers to solve a classic brain‑teaser: find the lowest integer N that starts with 4 such that moving the leading 4 to the end yields a number equal to one‑quarter of N.Mathematical Formulation of the ChallengeThe puzzle can be expressed as: N = 4[…], where the digits in brackets form a sequence, and N ÷ 4 = […]4. In other words, shifting the initial 4 to the terminal position produces a number exactly one‑quarter of the original.Guidance Provided to SolversA simple incremental hint is given: start by testing two‑digit numbers, then three‑digit numbers, and continue increasing the digit count until a solution is found.Community Interaction and Anticipated SolutionThe article asks readers to refrain from spoilers and instead discuss “snouts.” The author promises to return at 5 pm UK with the solution, linking to an update that will contain the answer.Source Attribution and Further EngagementThe puzzle originates from the Moscow Mathematical Olympiad 1983, shared via @mathematicsproblems and Kevin Gately. The author also invites suggestions for future puzzles via email.

The Lead The Venus flytrap, one of nature's most impressive predators, has long fascinated scientists with its ability to capture prey with lightning speed. Now, researchers have finally uncovered the surprising mechanism behind this phenomenon, resolving a problem that stumped Charles Darwin and many others. The Trigger Mechanism Scientists found that a hair-trigger detection causes the cells on the outer surface of the leaf to soften, prompting the flytrap to flip into a closed position within a second of a bug landing on the leaf. This is achieved through an intricate series of experiments, including the use of a device called a nanoindenter to measure the pressure on the leaf's outer surface. The Data Analysis The Venus flytrap's leaves have three trigger hairs on each lobe, which bend to prompt an electrical signal to spread across both sides of the trap within one-tenth of a second. The leaf's outer surface softens immediately after the trap is activated, allowing it to flip into a closed position. The Impact Analysis The discovery sheds light on the fascinating world of plant biology and the complex mechanisms that govern plant behavior. According to Dr. Yoël Forterre, a physicist at the French National Centre for Scientific Research (CNRS) and Aix-Marseille University, "Plants are just amazing. It makes you realise how all plants can sense their surroundings, transport information, react, defend themselves, feed." The Prediction This breakthrough has the potential to inspire new research into plant biology and biomimicry, with possible applications in fields such as robotics and materials science. As Dr. Forterre noted, "I'm not aware of any other plants with this kind of very rapid change of mechanical properties of the cells." The findings are published in the journal Science.

The Medical Breakthrough Scientists have achieved what was once considered impossible: developing a drug that targets pancreatic cancer, a disease long deemed 'undruggable' due to its complex biology and resistance to conventional treatments. The new daily pill, daraxonrasib, has shown remarkable efficacy in clinical trials, doubling survival time for patients diagnosed with this aggressive form of cancer. The Clinical Trial Results Recent clinical trials have demonstrated that daraxonrasib can significantly extend the lives of pancreatic cancer patients. The drug works by targeting specific genetic mutations that drive cancer growth, a approach that has eluded researchers for decades. Patients receiving the treatment showed a median survival rate that was double that of the control group, representing a unprecedented improvement in outcomes for this deadly disease. The Implications for Cancer Treatment This breakthrough marks a paradigm shift in how we approach pancreatic cancer treatment. Historically, patients have faced grim prognoses with limited therapeutic options. The success of daraxonrasib opens new avenues for targeting previously 'undruggable' cancers and may inspire similar approaches for other difficult-to-treat malignancies. The pharmaceutical industry is likely to accelerate investment in similar targeted therapies, potentially transforming the treatment landscape for multiple cancer types. The Future of Pancreatic Cancer Treatment With this promising development on the horizon, researchers are now focusing on optimizing daraxonrasib for broader patient populations and exploring combination therapies that could further improve outcomes. The drug's success may also lead to earlier detection methods and personalized treatment approaches based on individual genetic profiles. As this treatment moves toward regulatory approval and potential market availability, patients and healthcare providers alike are cautiously optimistic about a new era in pancreatic cancer care.

The Discovery of Ancient Plague Outbreak The earliest evidence for an outbreak of plague has been uncovered at late stone age cemeteries in south-eastern Siberia where dozens of hunter-gatherers and their children were buried. Ancient DNA collected from the remains suggests the disease tore through the sparse communities in devastating waves that began about 5,500 years ago, at least two centuries after the bacterium responsible, Yersinia pestis, first emerged. The Spread of Disease Among Hunter-Gatherers The hunter-gatherers probably became infected after butchering or eating raw marmots, a risky practice that still causes plague deaths today. After spilling over from the chunky ground squirrels, the primary animal reservoir in the area, the disease spread from person to person, decimating families and others in close contact. The Impact on Children and Communities While older hunter-gatherers might have survived past brushes with the disease and gained some immunity, young children were exceptionally vulnerable. At least two-thirds of the dead at two of the cemeteries were under 15 years old. Many who died shared graves with siblings or other family members. The Analysis of Ancient DNA The international team, including researchers in Copenhagen, Alberta, Cambridge and London, analysed dental pulp in the teeth of skeletons excavated from the cemeteries. Tests on 42 hunter-gatherers buried at four cemeteries on the Angara river found that 18 of them (39%) contained Y. pestis DNA, a higher proportion than is seen in some medieval plague pits. The Future of Plague Research The work resolves a longstanding mystery of why so many children were among the dead at one cemetery in particular, named Ust-Ida, on the bank of the Angara River north-west of Lake Baikal, the oldest and deepest lake in the world. The findings provide new insights into the history of plague and its impact on ancient human populations.

France's Shift Away from Palantir's AI Tools France's domestic intelligence service is to ditch AI data tools from the US tech company Palantir in favour of a domestic provider in an effort to avoid 'strategic dependency', the prime minister, Sébastien Lecornu, has said. The Drive for Domestic AI Solutions 'We must use our own AI models; we cannot accept new strategic dependencies in ‌the digital sphere,' Lecornu posted on social media. 'We cannot rely on tools developed by foreign powers. France must have its own tools.' Concerns Over Foreign Technologies There is increasing concern among European governments at their reliance on US-controlled technologies. Washington decided last week to restrict foreign nationals' access to Anthropic's latest AI model. The Data Analysis ChapsVision, which was founded in 2019 and made €200m (£173m) in revenue in 2025 against Palantir's $4.5bn (£3.3bn), said it would become the 'technological foundation' for 'many public agencies for their critical data processing needs'. The French DGSI intelligence agency would replace Palantir's tools with those from ChapsVision, although since the US company's long-term contract was renewed in 2025, the process is likely to take several years. The Impact Analysis France must 'build real autonomy' and 'not depend on the goodwill of certain partners, who are capable of turning off the access tap' for artificial intelligence, the prime minister said. Campaign groups have long warned that the US company's products pose risks relating to surveillance, infringements on individual freedoms and data protection. The Prediction France planned to invest €655m ​in artificial intelligence and set up ​a shared chatbot for ​all state services. It will also create a public health chatbot for the state-owned health insurance agency Ameli. The money would fund 'infrastructure, computing capacity, research, companies and industrial sectors', he said.

The Lead As the 10th anniversary of the Brexit vote approaches, the verdict on Britain’s economic performance is clear: voting to leave has resulted in severe costs for households and businesses. The Pound's Value Has Not Recovered The value of the pound swung wildly after the polls closed on 23 June 2016. The collapse in the pound drove up the cost of importing goods, triggering an inflation shock that damaged the public finances and inflicted financial pain on households across the country. A decade later, the pound has never returned above its pre-Brexit level. UK Growth Has Slowed According to the Office for Budget Responsibility, the independent Treasury watchdog, the UK is on track to suffer a 4% hit to national income over a 15-year period. UK GDP per head is between 6% and 8% lower than it would have been without Brexit. Trade Has Suffered from More Border Friction Brexit involved erecting trade barriers, which has hit goods exports. The EU is still the UK’s largest trading partner: in 2025, exports to the bloc were worth £385bn (41% of all UK exports) and imports £474bn (49% of the total). Since the end of the EU transition period on 31 December 2020, growth in UK goods exports has slowed relative to the G7. Uncertainty Sapped Business Investment After a shock result, no clear plan from the government or leave campaigners led to years of infighting over just what Brexit – never properly defined, and often subjective – should be in practice. Amid that political turmoil businesses froze their investment plans. Investment is estimated to be close to 18% lower than it would have been under remain and productivity up to 4% lower. Employment Has Suffered Unemployment in the UK fell after the Brexit referendum to among the lowest rates since the 1970s, before rising sharply during the pandemic. However, experts say this obscured underlying challenges. Wage growth has stagnated, and Britain emerged as the worst-performing country in the G7 for the pace of its recovery in workforce participation after the easing of pandemic restrictions. Brexit Support Has Faded Public support for Brexit has steadily fallen since the 52%-48% leave vote. Polling last month by YouGov shows 70% of Britons support a closer relationship with the EU without rejoining the bloc, its single market or customs union. More than two-thirds think looser ties would be a mistake. A majority – 56% – would back rejoining the bloc outright.

Rebalance Earth is providing a multi‑million‑pound injection to the Broughton Sanctuary estate in North Yorkshire, unlocking the next stage of a five‑year rewilding programme that will transform roughly 700 hectares of the 1,100‑hectare property.The Funding Boost for Broughton Sanctuary’s RewildingThe fund, which describes itself as a “natural capital asset manager”, aims to treat ecosystems as critical infrastructure that can generate both environmental and economic returns. The latest capital injection follows an earlier phase that saw the planting of 330,000 native trees and the reintroduction of beavers, otters and curlews.Scale of Investment and Ecological TargetsFunding amount: described as “a few million” pounds by CEO Rob Gardner.Land area: 700 ha (about two‑thirds of the estate) slated for intensive habitat restoration.Previous achievements: 330,000 trees planted over five years; beaver pair introduced in April produced a second litter.Current land use: 60 cattle, orchards, allotments and a spa‑based tourism operation employing ~700 people across 50 companies.Backer investment: West Yorkshire Pension Fund has placed £25 m in Rebalance Earth.Ecological and Economic Implications for North YorkshireThe rewilding effort is already reshaping the landscape: spruce stands are being felled and replaced with native species, while Iron Age pigs and Dales ponies will graze former sheep fields to break compacted soils. Wildlife returns—including otters, curlews and beavers—signal improving biodiversity, and the natural‑capital model promises reduced flood, drought and coastal‑erosion risks for surrounding communities.Future Outlook: Expanding Natural Capital on the EstateWith the new capital, Rebalance Earth plans to “unlock the next level of rewilding”, extending habitat diversity and demonstrating a scalable model where private investment funds ecosystem restoration. Success could encourage further pension‑fund backing and inspire other historic estates to adopt similar natural‑capital strategies, positioning the Yorkshire Dales as a benchmark for climate‑resilient land management.

The Unexpected Stabilization of InflationFollowing the initial shock of Iran choking off oil supplies in March, the UK economy has demonstrated surprising resilience. While dire warnings predicted a surge in inflation that would force the Bank of England to implement three quarter-point interest rate hikes, the latest data paints a different picture. In May, inflation remained steady at 2.8%, a figure that surprised economists who had expected a rise to 3%.The Discrepancy Between Fuel and Food CostsThe data reveals a complex landscape where energy costs are rising, but broader consumption is cooling. The Office for National Statistics reported that motor fuels were up 25% year-on-year, yet food prices actually fell by 0.1% month-on-month. This divergence suggests that while the immediate impact of the Middle East conflict is being felt at the pump, it has not yet fully permeated the wider economy.UK Inflation: Steady at 2.8% in May (vs. expected 3%)Fuel Prices: Up 25% year-on-yearFood Prices: Down 0.1% month-on-monthUS Inflation: Surged to 4.2% (record high)Why Firms Lack Pricing PowerThe resilience of the UK economy can be attributed to a lack of "pricing power" among businesses. Bank of England Governor Andrew Bailey noted that firms do not believe cash-strapped shoppers would tolerate higher prices. This is a stark contrast to the 2022 peak of 11.1% inflation, which was driven by strong consumer demand coinciding with the Russia-Ukraine war. Currently, the fear of a downturn in the jobs market is becoming a more pressing concern for the central bank than rising prices.The Path Forward for Interest RatesThe announcement of a US-Iran peace deal, which has already pushed oil prices below $80 a barrel, has eliminated the Bank’s worst-case scenario. Consequently, the Monetary Policy Committee is expected to keep interest rates on hold at 3.75% at their upcoming meeting. While analysts still expect at least one rate hike this year, markets are now betting on a move in November rather than September. The next major shift may not be a hike, but a potential cut, contingent on the health of the jobs market.

The World Cup's Most Prolific Goal ContributorsIn the history of the FIFA World Cup, certain players have stood out not just for their individual brilliance, but for their extraordinary contribution to their team's goalscoring output. A detailed analysis reveals that Gary Lineker holds the remarkable record of scoring 85.71% of England's goals during the 1986 tournament in Mexico, making him the most dominant goalscorer relative to his team's total in World Cup history.Lineker's Golden Boot AchievementLineker's performance at Mexico 86 was exceptional, scoring 6 of England's 7 goals (with Peter Beardsley scoring the other one). This remarkable feat earned him the Golden Boot as the tournament's top scorer. The English striker's clinical finishing and positioning made him virtually unstoppable throughout the competition, despite England ultimately being eliminated in the quarter-finals.Statistical Breakdown of World Cup Goal DominanceThe statistical analysis reveals several players who came close to matching Lineker's achievement:85.71% Gary Lineker (England, 1986)6 out of 7 goals83.33% Peter McParland (Northern Ireland, 1958)5 out of 680% Marcelo Salas (Chile, 1998); Jon Dahl Tomasson (Denmark, 2002); Christian Vieri (Italy, 2002); Robert Vittek (Slovakia, 2010)All 4 out of 575% Luis Artime (Argentina, 1966)3 out of 471.43% Teófilo Cubillas (Peru, 1978)5 out of 766.67% Anatoliy Byshovets (USSR, 1970); Michel (Spain, 1990) Both 4 out of 662.50% Roberto Baggio (Italy, 1994)5 out of 8, including 5 out of 6 in the knockout roundsThe Significance of Individual Brilliance in Team SportsThese statistics highlight a fascinating aspect of football history - the rare instances when a single player's performance becomes almost synonymous with their team's entire goalscoring output. McParland, a winger rather than a traditional striker, scored 5 of Northern Ireland's 6 goals in 1958, including crucial goals against West Germany and Czechoslovakia that took his team to the quarter-finals.Similarly, Peru's Teófilo Cubillas scored 71.43% of his team's goals in 1978, while Chile's Marcelo Salas contributed 80% of his nation's goals in 1998, including a memorable brace against Italy. These performances demonstrate how individual excellence can sometimes carry a team beyond expectations.The Evolution of Team Dependence on Star PlayersThe analysis reveals an interesting pattern across different eras of the World Cup. In earlier tournaments, it was more common for teams to rely heavily on one or two star players for their goalscoring. As the game has evolved tactically and become more specialized, such extreme dependencies have become less frequent.Modern football tends to distribute goalscoring responsibilities more evenly across the squad, making Lineker's 1986 achievement even more remarkable in today's context. The data suggests that while individual brilliance remains crucial, team-oriented approaches have become more prevalent in recent World Cups.Future Prospects for Breaking Goal RecordsAs the World Cup continues to evolve, breaking records like Lineker's 85.71% contribution becomes increasingly difficult. The tactical sophistication of modern defenses, coupled with the emphasis on balanced team performances, makes it challenging for any single player to dominate their team's goalscoring to such an extent.However, with the emergence of new talents and the unpredictable nature of tournament football, there remains the possibility that a future star could emerge to challenge these historic records. The 2026 World Cup and beyond will undoubtedly produce new statistical anomalies that captivate football fans and analysts alike.