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Tanya Khovanova has long been a leading voice in the world of recreational mathematics, authoring a popular blog and running the Number Gossip site where enthusiasts can inquire about any number.She now debuts her first book, Mathematical Puzzles and Curiosities, co‑written with fellow puzzle‑enthusiasts Ivo David and Yogev Shpilman. Published by World Scientific, the volume offers fresh challenges and inventive twists on classic problems.To give readers a taste, Khovanova selected three puzzles for the Guardian column:1. Battleships – As a naval admiral you must choose between sending a single vessel with a success probability of P % or two vessels each with a P/2 % chance, where the mission succeeds if at least one ship does. Which strategy maximises the odds?2. The Two Oracles – You face two respondents, Randie, who answers randomly, and Rando, who randomly decides to tell the truth or lie before each answer. Is there a method to distinguish them?3. Bad Maths – A student computes 5548 − 5489 as 59, claiming the “548” cancels out. He then experiments with a subtraction of the form XXYZ − XYZW (distinct digits) and observes the result is XW. How many digits in the new expression match those in the original calculation?Solutions to the trio were later posted online (update), but readers were urged to avoid spoilers until the scheduled reveal.Khovanova also reminds fans that she has been curating puzzles for the Guardian every other Monday since 2015 and welcomes new submissions via email.

Liverpool FC is gearing up for a crucial Champions League showdown against Paris Saint-Germain, with manager Arne Slot expressing confidence in his team's ability to turn the tide in their favor. The Reds are facing a daunting task, having lost 2-0 in the first leg at Parc des Princes.Slot emphasized that his team must deliver a great performance to overcome the European champions, stating that while the task is complicated, it is not impossible. He drew on his team's past successes, highlighting that in 36 of their 50 home games under his management, Liverpool has scored two or more goals.The Liverpool head coach is instilling belief in his players, citing examples of their ability to perform at a high level, even against strong opponents. He noted that his team has shown resilience in the face of adversity, such as coming back from 4-0 down against Manchester City in the FA Cup.Dominik Szoboszlai, Liverpool's midfielder, echoed Slot's sentiments, urging his teammates to go 'all in' from the first minute to the final whistle to knock out PSG. Szoboszlai expressed his desire to secure a win, stating that anything can happen at Anfield in a one-off game.Meanwhile, PSG's manager, Luis Enrique, warned his players not to underestimate Liverpool, cautioning that there could be pitfalls and that the game can change quickly. He emphasized the importance of taking care of details and not allowing Liverpool to benefit from their home crowd and noise.

For the first time, thousands of cancer patients from Black and minority ethnic backgrounds will benefit from an enhanced genetic test offered by the NHS. The new screening expands the panel of DPYD gene variants from four to five, directly addressing a long‑standing bias that left non‑white patients vulnerable to dangerous chemotherapy side‑effects. In England, patients slated for chemotherapy undergo a genetic check that can guide dose adjustments and mitigate adverse reactions such as mouth sores, hair loss, nausea, fatigue, and, in severe cases, death. Up to 40% of the 38,000 individuals receiving fluoropyrimidine‑based chemotherapy each year experience a harmful drug reaction. Previously, the test only targeted four DPYD variants common in people of European descent, meaning many Black patients received inaccurate “all‑clear” results. The addition of a fifth variant—more prevalent among African, Caribbean and other minority groups—means clinicians can now identify patients at risk who were previously missed. Since its rollout at Manchester University NHS Foundation Trust last September, three minority‑ethnic patients have had their initial chemotherapy doses adjusted, lowering their chance of a potentially fatal reaction. Dr Veline L’Esperance, senior clinical adviser at the NHS Race and Health Observatory, called the change “tangible results for patients who have historically been left behind.” She emphasized that the update shifts the discussion on ethnic health inequality from rhetoric to actionable care. Prof Habib Naqvi, chief executive of the NHS Race and Health Observatory, described the development as a “groundbreaking outcome” for chemotherapy safety, while noting that ethnic minorities remain under‑represented in genomic research and biobanks. He warned that broader inclusion is essential for the promised benefits of precision medicine to reach all communities. Prof Dame Sue Hill, chief scientific officer for NHS England, highlighted the significance of discovering the fifth variant: “Personalising chemotherapy based on genetics can save lives and reduce harmful side‑effects, especially for patients of African ancestry.” She added that the North West NHS Genomic Medicine Service has already demonstrated the practical impact of this approach. These steps come amid broader evidence that minority patients in the UK face longer diagnostic waits, more GP visits before a cancer diagnosis, and lower perceived support during treatment. The expanded DPYD test represents a concrete effort to close those gaps and ensure equitable, science‑driven care for all cancer patients.

At 19,000 miles above the lunar surface, the Artemis II astronauts heard the voice of Apollo‑13 commander Jim Lovell greeting them from beyond the grave, a message recorded shortly before his death at 97. Mission control transmitted the tribute on the morning of 6 April, when astronaut Reid Wiseman held a silk square bearing the original Apollo 8 patch—handed to him by Lovell’s son—before the crew’s launch. Launch director Charlie Blackwell‑Thompson secured a unanimous "go" from every console, a moment the author describes as a snapshot of humanity’s capacity to honor the past while reaching for the future. The ceremony stood in stark contrast to the Pentagon’s agenda. Two decades after a 2006 Iraqi kill‑board note reading “Let the bodies hit the floor,” the current defense chief pledged “death and destruction from the sky all day long” against Iran, a stance legal scholars warn could constitute a war crime. That rhetoric accompanies a proposed $1.5 trillion defense budget, including an additional $200 billion earmarked for operations against Iran—figures that dwarf NASA’s entire annual budget of $24.4 billion. Amid this fiscal disparity, crew members reflected on their place in history. Victor Glover, the first Black astronaut to travel deep space, described the view of Earth as an "oasis" in the void, while Christina Koch, the first woman to orbit the Moon, pressed her face to the window and formed a heart with her hands, calling the mission’s spirit "humility." Reid Wiseman watched the planet rotate beneath him, noting the sight of Africa, Europe, and the aurora‑lit north, saying it "paused all four of us in our tracks." Canadian astronaut Jeremy Hansen broke Jim Lovell’s 56‑year distance record and asked mission control to name a lunar feature after his late wife, Carroll. The crew agreed, designating a bright spot on the Moon as "Carroll," a tribute that will endure beyond any single lifetime. When Koch announced the spacecraft’s descent, she radioed, "We are now falling to the moon rather than rising away from Earth," a poetic reversal that underscored the mission’s symbolic return. The article recalls the author’s 2022 reflection on the James Webb Space Telescope, noting how humanity’s finest inventions now aim outward, seeking answers to our origins rather than turning inward toward conflict. Artemis II’s vessel, named Integrity, carried four breathing humans who, in a moment of collective grief, pressed their faces to the glass and imagined the faces of every loved one ever lived, captured in a single frame. Beyond the spacecraft, cultural threads wove through the mission: Hansen’s patch displayed the Seven Sacred Teachings of the Anishinaabe people, and the Chinese myth of Chang’e reminded viewers of the Moon’s timeless allure. As Carl Sagan famously wrote, "We are made of star‑stuff," the Artemis crew embodied that sentiment—not as passive observers, but as active participants in a fragile, hopeful narrative that rises above the relentless tally of bodies on Earth’s battlefields.

England and Spain meet again on Tuesday night in front of more than 70,000 at Wembley in the Women's World Cup 2027 qualifiers. This match comes eight-and-a-half months after they faced off in the Euro 2025 final, which England won.Spain has undergone significant changes since then, with a new head coach, Sonia Bermúdez, and a crop of emerging young players. Despite these changes, they remain unbeaten since the Euros with five wins and a draw in six matches. Bermúdez has already led them to the Nations League title.The visitors have a number of talented young players, including Fiamma Benítez, the 21-year-old Atlético Madrid midfielder, who has made a strong impact with seven goal contributions in the Champions League this season. Another player to watch is Vicky López, the Barcelona winger, who has become a regular starter.England head coach Sarina Wiegman believes that despite the changes, Spain's DNA remains the same: they want to have the ball and play dynamically. Wiegman noted that there might be some tweaks but the principle will be very similar.For England, Leah Williamson is an injury doubt due to a hamstring injury, but she has started training with her international teammates. Keira Walsh is set to win her 100th cap and Wiegman praised her vision and game understanding.England will have to make at least one change from the starting XI that beat Spain in Switzerland, with Ella Toone out injured. Jess Park is likely to start in her place.

More than 1,000 film and television professionals have signed an open letter opposing Paramount’s pending acquisition of Warner Bros Discovery, a deal valued at $111 billion. The signatories include high‑profile names such as Joaquin Phoenix, Ben Stiller, Mark Ruffalo, Yorgos Lanthimos, Kristen Stewart, Jane Fonda, and Emma Thompson.The letter, published on BlocktheMerger.com, warns that the merger would undermine the integrity, independence and diversity of the U.S. media sector, consolidating the number of major studios to just four and jeopardising a "vibrant future" for what it calls America’s "single most significant export" – its cultural content.Signatories argue that media consolidation already weakens competition, leading to fewer mid‑budget films, reduced independent distribution, higher production costs and fewer jobs across the ecosystem. They stress that competition is essential for both a healthy economy and a healthy democracy.Among the notable supporters are directors Denis Villeneuve, Boots Riley, Mimi Leder and Nicole Holofcener, as well as TV veterans David Chase, Noah Wyle, Ramy Youssef, Rob Delaney, Jason Bateman and Ted Danson. The letter also praises California Attorney General Rob Bonta and other state officials for scrutinising the deal.Paramount CEO David Ellison, who outbid Netflix for Warner Bros, claims the merger will boost creative output, pledging to release 30 theatrical titles annually and invest in both studios. Critics, however, remain skeptical, pointing to the Ellisons’ political ties and the risk of fewer politically‑engaged films.Recent accolades underscore the stakes: Warner Bros productions captured a record 11 Oscars in March, while Paramount films earned no nominations. The industry fears that the combined entity could further diminish quality and lead to significant job losses.Paramount has responded with a statement emphasizing that the transaction will “create a company that can greenlight more projects, back bold ideas, support talent across multiple stages of their careers, and bring stories to audiences at a truly global scale—while strengthening competition.” The letter’s authors remain unconvinced, urging regulators to block the merger to preserve competition, protect jobs, and safeguard the cultural export that defines American cinema.

CF Montreal, formerly known as Montreal Impact, is facing a tough time in Major League Soccer (MLS). The team's rebranding in 2021 was met with criticism from fans, who felt it was a step backward. The team's performance on the field has been equally disappointing, with five defeats in their opening six games, including a 5-0 drubbing and a pair of 3-0 humblings. The team's owner, Joey Saputo, has invested heavily in Bologna FC, an Italian Serie A club where he has been majority shareholder and chair since October 2014. Despite the close ties between the two clubs, CF Montreal are not benefiting much from their association. The team's academy is underdeveloped, with just three homegrown products having logged a combined 1,737 MLS minutes. In contrast, Bologna FC had a successful season, playing Champions League football and winning the Coppa Italia. The stark difference in performance between the two clubs has raised concerns about the future of CF Montreal. The team's latest struggle led to the departure of head coach Marco Donadel after a dismal start to the season. His replacement has not been announced yet. The team's poor performance has made them a worryingly neglected holdover in the league.

Conversations about mundane topics are often dreaded, but research suggests that people enjoy them more than they anticipate. A study involving 1,800 volunteers found that participants enjoyed chatting about dull subjects, such as the stock market or vegan diets, more than they expected. The findings indicate that by avoiding potentially dull exchanges, individuals may miss out on the mood boost and health benefits that come with connecting with others. Elizabeth Trinh, a PhD candidate in management and organisations at the University of Michigan, noted that people tend to underestimate how interesting and enjoyable conversations about boring topics can be. Trinh and her colleagues conducted a series of experiments to gauge how people responded to conversations on dull topics. Participants were asked to predict how much they would enjoy discussing topics they considered boring, and then engage in brief conversations about those topics. The results, published in the Journal of Personality and Social Psychology, showed that people consistently found the conversations more enjoyable than they expected. The surprise enjoyment arises because individuals base their expectations on static components, such as the topic and the other person, rather than dynamic components, like the conversation itself. Trinh suggests that people may benefit from lowering the bar for what makes a conversation worth having and reframing conversational goals. Instead of focusing on enjoyment, individuals could think about what they might learn from the conversation. Nicholas Epley, a professor of behavioural science at the University of Chicago, added that the fear of a boring conversation shouldn’t prevent people from starting one. “If a conversation is boring once you’re in it, you also have a surprising amount of power to make it better!”

While the two‑week pause in the US‑Israel campaign against Iran remains uncertain, one constant is clear: Donald Trump lacks a concrete plan, but Benjamin Netanyahu does. The war’s stated aim – to cripple Iran’s state capacity – is only a stepping stone toward a larger vision of a Greater Israel. For Israel’s right‑wing, the phrase often evokes a purely territorial ambition: enlarging the land Israel claims. History shows this expansionist drive has repeatedly displaced Palestinians, a process that has accelerated dramatically in recent years. Since the war began, Israel has flattened Gaza, killing tens of thousands and reducing the civilian‑inhabitable area to roughly 12 % of its pre‑war size. In the West Bank, a wave of settlement expansion and property destruction rivals the scale of the 1967 conflict. Beyond the occupied territories, Israel has seized parts of Syria and is forging a de‑facto occupation zone in southern Lebanon, with ministers from Religious Zionism, Jewish Power and Likud openly demanding Israeli sovereignty there. Finance minister Bezalel Smotrich even called for an expansion “to Damascus,” and Netanyahu has publicly expressed a deep personal connection to this territorial vision. However, Greater Israel is as much a geopolitical and strategic construct as a land‑grab. Netanyahu’s ambition extends beyond occupying borders; he seeks a regional dominion built on new alliances and hard‑power dependencies. After the October 7 attacks and the ensuing Gaza devastation, Israel’s prospects for Arab‑state normalization stalled. Faced with a choice between a conciliatory approach and a zero‑sum rejection of a Palestinian future, Netanyahu chose the latter, aiming to eliminate Iran as a regional counterweight – a move that inevitably required massive US military involvement. Former Israeli security analysts note that, from the perspective of Sunni Gulf states, a weakened Iran would elevate Israel to the role of “dominant regional power.” Achieving this, according to the article, also means softening the Gulf Cooperation Council (Bahrain, Kuwait, Oman, Qatar, Saudi Arabia, UAE) and making them dependent on Israel for security and energy routes. The spill‑over of Iranian drone and missile attacks on GCC infrastructure is portrayed not as an accident but as a calculated element of Israel’s strategy. When the US‑Israel coalition struck Iranian energy sites, Iran retaliated against the Gulf, disrupting global oil flows through the Strait of Hormuz. Netanyahu seized the moment to propose “alternative routes” – oil and gas pipelines that would bypass Hormuz and Bab‑al‑Mandab, ending at Israeli Mediterranean ports. In a meeting with Indian Prime Minister Narendra Modi, Netanyahu outlined a “hexagon of alliances” linking India, Arab nations, African states, Greece, Cyprus and other Asian partners, positioning Israel as the central hub. Recent IDF strategy papers echo this, suggesting Israel could achieve “operational control” far beyond its borders without permanent occupation, likening the Middle East to a “jungle” where Israel would become the “queen.” Netanyahu now describes Israel not merely as a “regional superpower” but, in some contexts, as a “global superpower.” He promises the hexagonal alliance will confront a “radical Shia axis” and an “emerging radical Sunni axis,” with Turkey singled out as the next strategic threat. Dismissal of the Greater Israel rhetoric as wartime hyperbole would be misleading. The article warns that a permanent war‑oriented mindset permeates Israel’s political elite, security establishment and media, posing a risk of overreach and regional blowback. Containing this expansive vision may become one of the most pressing post‑war challenges for the Middle East.