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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.

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.

The highly anticipated third and final season of Euphoria premieres on Monday at 9pm on Sky Atlantic. The series, created by Sam Levinson, follows the lives of a group of high school students as they navigate love, friendship, and identity.The new season stars Zendaya, Jacob Elordi, and Sydney Sweeney, who have all become major Hollywood stars since the show's debut. The story picks up five years after high school, with the characters now in their early 20s. Previews are scarce, but it's known that Cassie and Nate are married, Jules is in art school and is a 'sugar baby', and Rue is involved in some shady business in Mexico.Other TV shows airing tonight include Miriam Margolyes Made Me Me on BBC Two, Great Japanese Railway Journeys on BBC Two, Missed Call on Channel 5, Charmain and the Prophet on BBC Two, and DTF St Louis on Sky Atlantic.

Ukraine and Russia have exchanged accusations of violating the Orthodox Easter ceasefire, as the conflict between the two nations continues to escalate. The truce, which was agreed upon for 32 hours, was intended to provide a temporary reprieve from hostilities during the religious holiday.According to the Ukrainian military's general staff, 2,299 ceasefire violations were recorded, including 28 enemy assault actions, 479 enemy shellings, and 747 strikes by attack drones. In response, Russia's Ministry of Defence accused Kyiv of nearly 2,000 breaches, including 258 artillery or tank attacks and 1,329 FPV drone strikes.The ceasefire violations have raised concerns about the prospects of a lasting peace agreement. The Kremlin has stated that it will not extend the truce unless Kyiv accepts its terms, which include control of 17-18 percent of Ukraine's disputed Donetsk region. Ukraine has proposed freezing the conflict along the current front lines, but Russia has rejected this, seeking control of the entire Donetsk region.Despite the challenges, the truce did bring some relative calm to the region, with the Ukrainian army reporting no long-range Shahed drone attacks, guided aerial bombings, or missile strikes during the ceasefire period. However, the conflict has continued to take a toll on civilians, with Governor Alexander Khinshtein of Russia's Kursk region accusing Kyiv of breaking the ceasefire by attacking a petrol station, injuring three people, including a baby.

Tehran – Iranian officials warned that the United States must increase pressure for any peace agreement while urging their supporters to keep the streets under control.Parliament speaker Mohammad Bagher Ghalibaf said the US delegation in Islamabad “failed to gain the trust of the Iranian delegation” during the marathon talks.US President Donald Trump announced an immediate naval blockade of the Strait of Hormuz and declared that US forces are “locked and loaded” to finish Iran at the “appropriate moment.”Iran hailed the delegation’s refusal to accept Washington’s core demands – ending nuclear enrichment and relinquishing control of the Strait – as a sign of defiant resolve.Judiciary chief Gholam‑Hossein Mohseni‑Ejei thanked the Islamabad team for “guarding the rights” of government supporters, including paramilitary forces that have been gathering in Tehran’s main squares for more than six weeks.State television aired an IRGC aerospace officer urging flag‑waving crowds not to worry, stating, “If the enemy does not understand, we will make them understand,” which was met with cheers demanding more missile and drone attacks.Iranian officials blamed Trump’s “excessive demands” for the talks’ failure, noting the Ministry of Foreign Affairs did not expect an agreement after a single day of negotiations.Hard‑line parliamentarians expressed satisfaction with the stalemate, viewing Iran as holding the upper hand. Deputy speaker Hamidreza Haji‑Babaei said only a UN Security Council resolution signalling US surrender would lift sanctions on Iran.Lawmaker Amir Hossein Sabeti of the Paydari faction thanked the negotiating team for standing by red lines and called for “resistance in the field against these evildoers.”The sudden announcement of a two‑week cease‑fire left pro‑state voices disappointed; local media reported the Iranian delegation numbered over 85 members, including state‑affiliated journalists and factional analysts.The talks confirmed that a diplomatic breakthrough was not imminent and hinted at further escalation, though not necessarily a return to full‑scale fighting.National security commission head Ebrahim Azizi dismissed Trump’s blockade threats as “excessive talk.”The IRGC warned it would meet any vessel passing the Strait of Hormuz with “full force” and rejected US claims that two warships had already transited the waterway.Russian President Vladimir Putin told Iranian President Masoud Pezeshkian he remains ready to facilitate a Middle‑East peace settlement.Pezeshkian, focused on domestic affairs, reaffirmed support for Supreme Leader Ayatollah Ali Khamenei and announced that schools and universities will continue online via a limited intranet.Iran’s economy continues to suffer from chronic inflation, rising unemployment and a near‑total internet shutdown, deepening public hardship.

Tyson Fury, the Gypsy King of boxing, has once again called out Anthony Joshua for a highly anticipated superfight, but the question remains: will it ever happen? The two British boxing legends have been talking about facing each other for over 10 years, but the bout has been plagued by delays, postponements, and cancellations. Following his dominant victory over Arslanbek Makhmudov at Tottenham Hotspur Stadium, Fury expressed his desire to fight Joshua next, stating, "I've no idea" if the fight will happen. Despite Joshua's presence ringside, he refused to give a definitive answer, citing the need for time to recover from a tragic car crash in Nigeria that claimed the lives of two of his close friends. Fury, who has been chasing Joshua for a decade, sounded perplexed by Joshua's evasive response. "He didn't give an answer," Fury said. "In my opinion, he didn't want no smoke. He didn't look like he wanted it. He was shell-shocked." Fury emphasized that he has given Joshua the respect and space he deserves, especially after his recent personal tragedy. Joshua, on the other hand, appeared dismissive, stating "Contracts are being sent over and you’ll probably see us in the ring next, more than likely." When asked if he needed a warm-up fight, Joshua quipped, "Good question. That [Fury] could be a warmup fight, based on what I saw tonight." Fury dismissed the suggestion that Joshua needed a proper test against a real boxer before stepping into the ring with him. "It’s another stumbling block. He’s just knocked a man spark out," Fury said. "I’ve just come off a win. Let’s get it on. He’s 36. I’m 37. What’s the holdup?" The two fighters have had a long and complicated history, with Fury noting, "If it ain’t Anthony Joshua next I’m not interested in boxing. I’ll eat a thousand Easter eggs, go up to 35 stone. I’m out. It’s either him or I’m gone again. I’m not interested in up-and-comers. I’m not interested in someone trying to prove a point over me. I don’t care about rankings or belts. I only care now about AJ. That’s the defining fight for British boxing."

Djibouti's longtime President Ismail Omar Guelleh has claimed a landslide victory in the country's latest elections, ushering in his sixth consecutive term in the Horn of Africa country.Guelleh secured 97.81 percent of the votes cast on Friday, according to official results published by Djibouti's state-run news agency.His sole opponent, Mohamed Farah Samatar, earned just 2.19 percent of votes. Samatar's Unified Democratic Centre (CDU) opposition party has no seats in parliament, and he struggled to gain recognition ahead of the polls.Guelleh, 78, wrote on X as early results arrived: “Reelected”. Politicians last year removed presidential age limits, allowing him to seek another five years in power.Voter turnout was 80.4 percent on Friday, according to Djibouti media outlets. Roughly a quarter of the population – about 256,000 people – were registered to vote.Guelleh has ruled Djibouti's population of roughly one million since 1999. Neighbouring Eritrea, Ethiopia and Somalia, the country is strategically located at the Bab al-Mandeb strait, which provides access to the Red Sea from the Gulf of Aden.Even as voters turned in their ballots on Friday, few doubted who would win. Thousands had gathered at Guelleh's campaign rallies ahead of the election, while his posters could be seen plastered across the capital.

It's been a year since the Trump administration's sweeping cuts to the federal workforce, and the effects are still being felt. Tens of thousands of employees were offered buyouts or faced termination, leaving many to navigate a difficult job market. Maggie, a former employee of the US Office of Personnel Management, took a buyout offer in May 2025. She has since applied to over 250 jobs but is still waiting for an ethics letter to start work elsewhere. 'I couldn't be without health insurance through the delivery of my baby,' she said, highlighting the challenges faced by those who lost their jobs. The federal workforce has declined by about 355,000 employees since Trump took office, with 18,000 workers leaving in March 2026 alone. The cuts have left remaining government workers overwhelmed, trying to keep essential public services afloat. Charles Melton, a 20-year veteran of the US Department of Agriculture, took early retirement but still helps former colleagues with job applications. 'I'm still mad as hell,' he said. 'We just got thrown away like garbage.' The impact on public services has been significant, with customer service at the Social Security Administration worsening and healthcare workers at the Department of Veterans Affairs reporting ongoing staffing issues. The shutdown of USAID has resulted in hundreds of thousands of deaths worldwide due to the spread of infectious diseases and malnutrition. The White House has declined to comment, but Scott Kupor, OPM's director, stated that 'reshaping the federal workforce is essential to building a government that works for the American people, not the bureaucracy.'

The European Parliament's decision to block the extension of a law that permits big tech firms to scan for child sexual exploitation on their platforms has created a legal gap that child safety experts say will lead to crimes going undetected. The law, which was a temporary measure allowing companies to use automated detection technologies to scan messages for harms, including child sexual abuse material (CSAM), grooming, and sextortion, expired on April 3.Google, Meta, Snap, and Microsoft have expressed disappointment over the EU's decision, stating that they will continue to voluntarily scan their platforms for CSAM. In a joint statement posted on a Google blog, the companies said, 'We are disappointed by this irresponsible failure to reach an agreement to maintain established efforts to protect children online.'Child protection advocates had warned that allowing the legislation to lapse would probably trigger a steep fall in reports of child sexual abuse. They point to a similar legal gap that occurred in 2021, when reports of such material from EU-based accounts to the National Center for Missing and Exploited Children (NCMEC) fell by 58% over a period of 18 weeks.The EU's decision to prohibit scanning will have ripple effects in other regions around the world, child safety experts said. Many internet crimes are cross-border, with perpetrators sending illegal images to people or targeting children in other countries. 'The offender can be anywhere in the world, but they could have unfettered access to minors in Europe now that there's legal uncertainty around those safeguards and protections to identify when a child is being groomed,' said John Shehan, vice-president at NCMEC.In 2025, NCMEC received 21.3m reports that included more than 61.8m images, videos, and other files suspected of being related to child abuse, from around the world. About 90% of these reports are related to countries outside the US.