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Science

1,585 cards. Each one is a 40-second story with a five-minute read behind it. Page 14 of 27.

781. Most people alive today carry a little Neanderthal in their DNAFor decades the textbook story said modern humans left Africa and simply replaced the Neanderthals. Then, on 7 May 2010, a draft Neanderthal genome showed that the two had interbr…
782. Table salt holds together because sodium gives away an electronDrop sodium and chlorine together and each sodium atom hands one electron to a chlorine atom. The result is two ions of opposite charge, pulled together so strongly that they stac…
783. When blood stops reaching tissue, restoring it can cause fresh harmIschemia means a part of the body is not getting enough blood. Oxygen runs short, nutrients stop arriving and waste piles up. Strangely, when the flow comes back, the returning bl…
784. Jared Diamond won a Pulitzer and a long quarrel with historiansJared Diamond trained to study how the gallbladder absorbs salt. He then built a second career watching birds in New Guinea, and a third explaining world history. His book Guns, G…
785. A Dutch doctor went looking for the missing link and actually found a fossilNobody had ever found an early human fossil by setting out to look for one. In 1887 a young Dutch anatomist, Eugene Dubois, quit academia, joined the colonial army as a surgeon an…
786. Riemann set aside proving his famous hypothesis after a few quick attemptsIn his 1859 paper on counting primes, Bernhard Riemann remarked that a certain property of his zeta function was very probably true, admitted a rigorous proof would be welcome, an…
787. A tiny line segment holds exactly as many points as an endless lineTake a segment one centimetre long and a line stretching forever in both directions. Intuition says the line has more points. Set theory says they have exactly the same number. Ge…
788. Russell's paradox wrecked Frege's attempt to build arithmetic from pure logicGottlob Frege wanted to ground all of mathematics in logic, assuming any well-defined property picks out a set. Bertrand Russell then pictured a set gathering every set that fails…
789. In infinite sets, a part can be exactly as big as the wholeEvery natural number is also a rational number, but plenty of rationals, such as fractions, are not natural numbers. So the naturals form a strict part of the rationals. Yet the t…
790. Theoretical computer science began by proving what can never be provedSome of the deepest results about computing concern what cannot be done. In 1931 Kurt Gödel showed with his incompleteness theorem that some statements can be neither proved nor d…
791. No computer program can decide, in general, whether another program will haltIt sounds like a simple question: given a program and its input, will it eventually finish or run forever? Yet no Turing machine can answer it for every case. This halting problem…
792. Combine no sets at all and set theory says you get the empty setThe union of sets gathers every element that appears in any of them. Merge {1, 2, 3} with {2, 3, 4} and you get {1, 2, 3, 4}, since a set never lists anything twice. Push the idea…
793. John Venn never called his overlapping circles Venn diagramsJohn Venn introduced his diagrams in an 1880 paper but referred to them as Eulerian circles, crediting an older tradition. He had met Euler's diagrams in 1862 and only later hit o…
794. Checking a Sudoku is easy, but nobody knows whether solving one must be hardHand someone a filled-in Sudoku and they can confirm it in moments by scanning rows, columns and boxes. Whether a fast method exists to solve every enlarged version of the puzzle …
795. A ladder metaphor explains how finite reasoning proves infinitely many casesIf you can step onto the bottom rung of a ladder, and from any rung you can always reach the next, you can climb as high as you like. That is mathematical induction: two short arg…
796. Paradoxes and strange functions pushed nineteenth-century mathematicians to put logic under the microscopeKarl Weierstrass built continuous functions that have no slope anywhere, and flaws turned up in Euclid's supposedly airtight geometry. Unsettled, nineteenth-century mathematicians…
797. Paul Erdős imagined a heavenly book holding the most beautiful proof of every theoremPaul Erdős liked to say that an especially elegant proof came straight from The Book, an imaginary volume containing the perfect argument for each theorem. In 2003 a real volume, …
798. Why letting any property define a set leads straight into contradictionTry forming the set of all sets that are not members of themselves, and you hit Russell's paradox: it can neither contain itself nor fail to. That single puzzle showed that the in…
799. The purest branch of mathematics ended up guarding online secretsCarl Friedrich Gauss crowned number theory the queen of mathematics, and for generations it was prized precisely because it seemed useless outside the subject. Then in the 1970s p…
800. Babylonian astronomers tracked Jupiter with a rule computers still use for areasLong before calculus, Babylonian sky-watchers used the trapezoidal rule to work out how far Jupiter moved along the ecliptic. The same idea, adding up weighted samples of a functi…
801. A 1677 bell-ringing manual worked out factorials before most mathematicians named themFabian Stedman wanted to know how many orders a set of church bells could ring in. Two bells give two sequences, three give six, and by five he had tabulated 120 before giving up …
802. A prime is any number of dots that refuses to form a rectangleLine up six dots and they fit neatly into two rows of three. Try the same with five or seven and nothing works except a single line. That is the essence of a prime: a whole number…
803. Primes thin out at a precise rate that a teenage Gauss first guessedAmong whole numbers up to 1000 digits long, roughly one in 2300 is prime; double the length to 2000 digits and the odds fall to about one in 4600. The prime number theorem pins do…
804. Nobody can say for certain who invented the truth tableThe truth table is among the most familiar tools in logic, yet its origin is murky. Ludwig Wittgenstein and Emil Post usually get credit for the grid layout, perhaps independently…
805. Jewish scribes invented error checking over a thousand years before computersBetween the seventh and tenth centuries, Jewish scribes counted every word and letter of the Hebrew Bible, even marking each book's middle verse, so copies could be checked. A sin…
806. Euclid's famous proof of infinite primes is not actually a proof by contradictionTextbooks often say Euclid assumed a largest prime and derived an absurdity. He did nothing of the sort. His argument in the Elements works directly on any finite list of primes, …
807. The language mathematicians use for foundations cannot pin down the ordinary counting numbersFirst-order logic became the standard language for writing mathematics as axioms by the 1940s. It is powerful enough that every statement true in all its models can be proved. Yet…
808. A notation for describing machines, drafted in 1907, anticipated programming languagesIn Vienna in 1907, Leonardo Torres Quevedo presented a system of symbols for describing machines. Heinz Zemanek later judged it equivalent to a programming language for controllin…
809. The reason 1 is not a prime number is a theorem about uniquenessEvery whole number above 1 breaks into primes in exactly one way, apart from the order. 1200 always contains four 2s, one 3 and two 5s. If 1 counted as prime, you could tack on as…
810. Arithmetic with addition alone is complete; add multiplication and Gödel's limits biteIn 1931 Kurt Gödel showed that any consistent, algorithmically listable set of axioms rich enough for ordinary arithmetic leaves some true statements unprovable, and cannot prove …
811. Handshakes and debts show the difference between two basic kinds of graphDraw dots for guests at a party and a line whenever two shake hands, and you have an undirected graph, since a handshake always goes both ways. Draw an arrow when one guest owes a…
812. Proving four colours suffice for any map took a century and a computerIn 1852 Francis Guthrie asked whether four colours are always enough to paint a map so that neighbouring regions differ. Mathematicians offered wrong proofs for over a hundred yea…
813. Why is approximately equal not a proper kind of sameness in mathematics?Nudge a number by a tiny amount, then again, then again, and each step looks almost equal to the last while the ends drift far apart. That failure of transitivity is exactly why m…
814. Equations that demand whole-number answers, and why no machine can settle them allAllow fractions and most equations are easy to satisfy. Insist that every answer be a whole number and the problems can become fiendish. These are Diophantine equations, named aft…
815. The mathematics of countable things grew up alongside the digital computerCalculus deals in smooth, unbroken change. Discrete mathematics deals in separate, countable things: whole numbers, networks, logical statements. It barely existed as a university…
816. The set with nothing in it has a Scandinavian symbol and odd powersThe familiar slashed-circle sign for the empty set was chosen in 1939 by André Weil of the Bourbaki group, borrowing the letter Ø from the Danish and Norwegian alphabets. That bor…
817. Zhang Heng's earthquake detector was mocked until a messenger galloped inWhen Zhang Heng's new seismoscope signalled a quake to the northwest, nobody in the Han capital of Luoyang had felt a thing, and his rivals at court enjoyed the apparent failure. …
818. Class field theory: the century-long project that grew from Gauss's reciprocity lawDavid Hilbert usually gets credit for class fields, but Leopold Kronecker already knew the idea and Heinrich Weber coined the name before Hilbert's key papers appeared. What follo…
819. Why a scratched CD still plays: the mathematics of deliberate redundancyA music CD survives dust and scratches because its data is written twice over in a clever way. A Reed–Solomon code adds carefully chosen extra information, and the bits are shuffl…
820. Exactly 2,598,960 different poker hands, and how to count them fastDeal five cards from a standard 52-card deck and the odds of getting any one particular hand are 1 in 2,598,960. Nobody listed them all to find that out. The figure comes from a s…
821. An ancient Indian physician counted 63 ways to mix six tastesThe Sushruta Samhita, a classic of Indian medicine, notes that six basic tastes can be combined in 63 different ways, singly, in pairs, in threes and so on. That tally, every poss…
822. Almost every collection of numbers is beyond the reach of any computerThere are only countably many possible Turing machines, so only countably many sets of whole numbers can ever be computed. Cantor showed the sets of whole numbers themselves are u…
823. A function can be computable even if no real computer could ever finish itIn mathematical logic, computable does not mean practical. A function counts as computable if some finite set of instructions produces its answer eventually, with unlimited time a…
824. Complexity theory asks how hard a problem is for every possible algorithmAsking whether a tour of Germany's 14 biggest cities can be done in under 2,000 kilometres is one question. Asking how hard that kind of question is in general, no matter which me…
825. Predicting the next symbol well is the same skill as compressing dataA language model built for text, DeepMind's Chinchilla 70B, squeezed images to 43.4 percent and audio to 16.4 percent of their original size, beating the PNG and FLAC formats. Tha…
826. Rob Pike's rule: pick the right data layout and the algorithm writes itselfProgrammer Rob Pike has argued that how you arrange your data almost always matters more for speed than which algorithm you choose, because once the layout is right the method ten…
827. The imaginary computers that let programs run anywhereLong before Java promised write once, run anywhere, computer scientists in the late 1950s dreamed of a universal in-between language that every machine could understand. That drea…
828. The famous number theory book Dedekind wrote and credited to his teacherIn 1863 Richard Dedekind published Lectures on Number Theory under Peter Gustav Lejeune Dirichlet's name and called it Dirichlet's book for the rest of his life. Yet he wrote near…
829. A teenage Gauss scribbled the rule for primes in his logarithm tableAround 1792, aged 15 or 16, Carl Friedrich Gauss jotted a short note in his book of logarithms guessing how many primes lie below any large number. He never published it and only …
830. Mathematicians stopped asking whether axioms are true and started asking what followsFor Euclid, a starting assumption in geometry was a plain fact about the world, like the ability to draw a straight line between any two points. Modern mathematicians treat axioms…
831. You can tell two piles are equal without counting either oneLay out a heap of apples and a heap of oranges, then match them off one apple to one orange. If nothing is left over, the heaps are the same size, and you never needed a single nu…
832. Wolfgang Pauli, the physicist whose mere presence supposedly broke equipmentColleagues joked that laboratory apparatus failed whenever Wolfgang Pauli walked nearby, and he loved the reputation. The Austrian theorist was also the sharpest critic in physics…
833. The Wright brothers solved flight by thinking like cyclists, not engineersRivals of Wilbur and Orville Wright bolted powerful engines onto airframes and hoped to stay aloft. The Dayton bicycle makers believed the real puzzle was control. Their first pat…
834. Yang Chen-Ning showed nature is not perfectly mirror-symmetricPhysicists long assumed that any process and its mirror image obey the same laws. In 1956 Yang Chen-Ning and Tsung-Dao Lee proposed that the weak nuclear force breaks that rule. C…
835. W. D. Hamilton turned kindness toward relatives into a simple equationWhy would an animal sacrifice itself for another? W. D. Hamilton's answer, published in 1964, was that helping a relative can spread your own genes, provided the benefit multiplie…
836. Hay fever sent Heisenberg to a rocky island where quantum mechanics took shapeIn June 1925, Werner Heisenberg was so miserable with hay fever that neither aspirin nor cocaine helped. He fled to Helgoland, a nearly pollen-free island in the North Sea, and wo…
837. Wernher von Braun's rockets reached space for Hitler, then the Moon for AmericaIn 1944 a V-2 rocket co-designed by Wernher von Braun became the first human-made object to cross into space. Twenty-five years later, the Saturn V he architected sent Apollo astr…
838. Queen Elizabeth's doctor worked out that Earth itself is a magnetWhy does a compass point north? Before 1600, some blamed the pole star or a giant magnetic island at the top of the world. William Gilbert, physician to Elizabeth I, built a small…
839. William Harvey mapped blood's circuit and exposed a witch's ordinary toadIn 1628 William Harvey published the first complete description of blood being pumped by the heart around the body and through the lungs. Patients deserted him over it. The same s…
840. William Henry Bragg shared a Nobel Prize with his own sonIn November 1915, William Henry Bragg and his son Lawrence jointly won the Nobel Prize in Physics for using X-rays to reveal how atoms sit inside crystals, the only father and son…

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