Unexpectedly, three mathematicians have now shown that a well-known hypothesis in probability theory called the bunkbed conjecture falls into this category. The conjecture—which is about the different ways you can navigate the mathematical mazes called graphs when they’re stacked on top of each other like bunk beds—seemed natural, even self-evident. “Anything our brain tells us suggests the conjecture should be true,” said Maria Chudnovsky, a graph theorist at Princeton University who was not involved in the new work.

But they were wrong. Last month, a trio of mathematicians announced a counterexample, disproving the conjecture. The result offers fresh guidance on how to approach related problems in physics about properties of solid materials. But it also taps into deeper questions about how mathematics works. A lot of mathematical effort is spent trying to prove conjectures true. It’s lonelier to try to pull them apart. The team behind the new work failed many times before they finally found their counterexample. Their story suggests that mathematicians may need to question their assumptions more often.

Football pitches around Sudan’s capital, Khartoum, and in the neighbouring city of Omdurman have been used as burial grounds for the dead rather than games. The 19-month conflict has caused what is, according to the United Nations, “the worst humanitarian crisis on earth”.

“The numbers are so large that you can’t even get your head around the scale of human suffering,” the United States’ special envoy for Sudan, Tom Perriello, told reporters this week. “The numbers are astronomical…(and) the death toll is probably more than anything that’s been estimated.”

People from Sudan have found themselves fighting for peace but also for attention, as conflicts in the Middle East and Ukraine dominate headlines.

Sudan’s football team have been forced into a nomadic existence, playing “home” games in South Sudan (which became its own nation in 2011), Mauritania, Saudi Arabia and Libya. But they have achieved remarkable results: Sudan have qualified for the Africa Cup of Nations (AFCON) starting in Morocco in December 2025 and are top of their group competing to reach the World Cup, a tournament they have never played in before, in the United States, Canada and Mexico in 2026.