Mathematics keeps better stories than its reputation suggests. Here are people who worked under false names, numbers that sit at the center of arguments and jokes, ideas that arrived in the wrong century, and puzzles whose famous inventors did not invent them. Every one of them is checkable, and most of them are not what the usual telling says.
Every tile in a Mathematics game carries one of these. Open one to read it now, or leave them for the board to hand you.
Mathematicians
nine stories
1She borrowed a dropout's name to do mathematics
On the tile: M. LeBlanc1804
The École Polytechnique did not take women, so Sophie Germain got hold of the lecture notes instead, including those from Lagrange's course on analysis, and sent in written work as M. LeBlanc. The name was not invented. Antoine-Auguste LeBlanc was a real student who had quickly given up his studies, and course material still went out to his name by mail. Lagrange went looking for the author, found a woman, and became her sponsor anyway. Between 1804 and 1809 she wrote a dozen letters to Gauss, signing as LeBlanc at first.
In the spring of 1976 George Andrews went to the Wren Library at Trinity College, Cambridge, to look through the papers of the late G. N. Watson. In one of the boxes he found loose sheets in the handwriting of Srinivasa Ramanujan, who had died in 1920: 138 written sides, holding the work of his last year. Robert Rankin had sent them to the library in December 1968, where they sat for seven years. The manuscript was soon named the lost notebook, and the find has been compared to turning up a tenth Beethoven symphony.
Women in Russia could not live apart from their families without written permission from a father or husband, and Sofia Kovalevskaya's father would not let her leave home for a university. At eighteen, after great difficulty getting his consent, she married the young paleontologist Vladimir Kovalevsky and went abroad. Heidelberg would not matriculate women either, so she talked her way into lectures with each professor's permission. In June 1889 she became the first woman since Laura Bassi and Maria Gaetana Agnesi to hold a chair at a European university.
4Almost half his work came after he lost his sight
On the tile: Euler1771
Leonhard Euler went nearly blind after an illness in the 1760s. In 1771 a fire destroyed his house and he saved only himself and his manuscripts; a cataract operation later that year gave him a few days of sight before he lost it for good. He worked on regardless, holding calculations in his head and dictating to his sons and assistants. Almost half of his total output came after his return to St Petersburg at the age of fifty-nine, in spite of the blindness.
Giuseppe Piazzi spotted a new small planet, Ceres, on the first day of 1801, and tracked it across only nine degrees of its orbit before it slipped behind the Sun. Astronomers published rival guesses about where it would come out. One of them, from Carl Friedrich Gauss, then in his twenties, differed greatly from the rest. When Franz Xaver von Zach found Ceres again on 7 December 1801, it was almost exactly where Gauss had said. He had used least squares, and did not explain his method at the time.
Young French mathematicians began meeting in a Paris café in December 1934 to write a better analysis textbook. By the summer of 1935 they had agreed to publish as one man: Nicolas Bourbaki, a name taken from a Franco-Prussian war general and first heard in a student hoax, a lecture of invented theorems given to first-year students by a senior in a false beard. The first volume appeared in 1939. When R. P. Boas of Mathematical Reviews called Bourbaki a pseudonym, the publisher received an indignant letter from Bourbaki himself, objecting that his right to exist had been questioned.
Paul Erdős put his name on more than 1,500 papers and spent his life moving between the homes and offices of the people he wrote them with. Ronald Graham handled his money and kept an Erdős room in his own house. When Erdős died in 1996 Graham counted 458 collaborators, and mathematicians had taken to measuring themselves by their distance from him: a coauthor has Erdős number 1, a coauthor's coauthor has 2. On a count of the Mathematical Reviews database made in 2004, 504 people had number 1 and about 6,600 had number 2.
The Fields Medal has been given out since 1936, and for seventy-eight years it went only to men. In 2014 Maryam Mirzakhani won it for her work on the geometry of curved surfaces. She had grown up in Tehran wanting to be a writer, and mathematics took her over at school; she was teaching at Stanford when the medal came, and she died in 2017 at the age of forty. In 2022 Maryna Viazovska became the second woman to receive it.
Hypatia taught mathematics and astronomy in Alexandria, and every commentator who mentions her calls her a magnetic teacher. Not one of her own works survives. A later Greek reference book credits her with commentaries on Diophantus's Arithmetica and on Apollonius's Conics, and she worked alongside her father Theon on Ptolemy. What has come down to us instead is her mail: letters from a former student, Synesius of Cyrene, asking her advice on how to build an astrolabe and a hydroscope.
In 628 the Indian astronomer Brahmagupta set down what zero actually does. He defined it as what is left when a number is subtracted from itself, and said that adding or subtracting zero leaves a number unchanged while multiplying by zero gives zero. Then he pushed into territory nobody had mapped: division. A number divided by zero, he wrote, is a fraction with zero underneath. Zero divided by zero is zero. That last claim is simply false, and the attempt was still remarkable.
Visiting Ramanujan at his nursing home in Putney, in southwest London, G. H. Hardy mentioned that his taxi had carried the number 1729, which struck him as rather a dull one. Ramanujan told him it was the smallest number that can be written as two cubes added together in two different ways: 1 and 12 cubed, or 9 and 10 cubed. Nearly a century later Ken Ono and Sarah Trebat-Leder found that Ramanujan's notes on 1729 held a kind of surface mathematicians did not name until the 1950s.
Edward J. Goodwin, an Indiana physician, believed he had squared the circle, and in 1897 he offered his proof to his home state royalty free. Among its errors it made pi equal 3.2. Lawmakers passed the bill along from the Committee on Canals to the Committee on Education, then approved it without a single dissenting vote. Clarence A. Waldo, who led mathematics at Purdue, happened to be at the statehouse lobbying for his budget. He stayed to coach the senators, who postponed the bill indefinitely.
The American mathematician Edward Kasner wanted a name for 1 followed by a hundred zeros, so he asked his nine-year-old nephew. In Mathematics and the Imagination, published in 1940, Kasner reported that the boy invented the word googol, being very certain the number was not infinite and therefore equally certain it had to have a name. The child also offered googolplex. Much later, a graduate student in Stanford's computer science building misspelled googol while checking whether a domain name was available.
Mersenne primes are one less than a power of two, and they are scarce. On October 12, 2024, Luke Durant showed that 2 raised to the 136,279,841st power, minus 1, is prime. It runs to 41,024,320 digits, more than sixteen million longer than the record it broke, and it is the fifty-second Mersenne prime known. He did not find it on a desktop. His search ran on thousands of datacenter graphics cards across seventeen countries, ending a twenty-eight-year run of ordinary computers holding the record.
6Four digits, one subtraction, always the same end
On the tile: 61741949
Take any four-digit number whose digits are not all the same. Rearrange the digits into the largest and the smallest numbers they can make, keeping four digits and any leading zeros, subtract the smaller from the larger, and repeat with the answer. You will arrive at 6174, and nowhere else, in at most seven steps. Starting from 2005 it takes all seven. Once 6174 turns up the operation hands it back forever. D. R. Kaprekar, a schoolteacher in Devlali, India, hit on it in the 1940s and announced it in 1949.
The number is real mathematics: about 1.618, the way to cut a line so that the whole is to the longer part as the longer part is to the shorter. The famous claims about it are another matter. The idea that the Parthenon and the Egyptian pyramids were built to it dates only to the mid-nineteenth century, appears in no classical manuscript, and does not survive measurement; every diagram fitting a golden rectangle to the Parthenon leaves out steps or includes empty air. Nothing in Leonardo's writings connects it to the Mona Lisa.
One word, two numbers. For generations a billion meant a million million in Britain and a thousand million in the United States. In December 1974 the member of Parliament Robin Maxwell-Hyslop asked the Prime Minister to require ministers to use only the British sense. Harold Wilson answered no. The word was by then used internationally to mean a thousand million, he said, and it would be confusing for British ministers to use it any other way, though he would ask colleagues to leave no ambiguity.
9In real data, 1 leads far more often than it should
On the tile: Leading digit1881
If the first digits of real-world numbers were evenly spread, each would lead about one time in nine. They are not. Across a great many data sets, 30.1 percent of the entries begin with 1 and 17.6 percent begin with 2, and the pattern holds however you change the units. The astronomer Simon Newcomb noticed it in 1881 because the early pages of logarithm tables, the ones for numbers starting with 1, were grubbier than the rest. Frank Benford found it again in 1938, and the law took his name.
In October 1852 a student named Francis Guthrie asked his brother to put a question to Augustus De Morgan: can any map be colored with four colors so that no two regions sharing a border match? De Morgan could not say. Nobody could, for more than a century. In 1976 Kenneth Appel and Wolfgang Haken settled it by boiling every possible map down to about 1,500 configurations and checking them, which took 1,200 hours of computer time. It was the first major theorem with a proof no mathematician could verify directly.
On 23 June 1993, at the end of three lectures at the Isaac Newton Institute in Cambridge, Andrew Wiles said, "I think I'll stop here." He had just presented a result that carried Fermat's Last Theorem with it, after more than 350 years. Witnesses say the applause was thunderous. Then the checking began, and the proof turned out to contain a hole. Wiles and Richard Taylor, his former student, spent close to a year closing it, and in 1994 the theorem was finally proved.
3Arithmetic cannot prove everything true about itself
On the tile: Gödel1931
David Hilbert wanted mathematics resting on foundations secure enough to settle every question in principle. In 1931 Kurt Gödel, then in his mid-twenties, published a paper showing that any consistent system rich enough to describe arithmetic contains true statements it cannot prove, and cannot establish its own consistency. Mathematics, it turned out, was not a finished object. One consequence: no computer can ever be given a procedure that answers every mathematical question.
For two thousand years geometers tried to derive Euclid's parallel postulate from his others. János Bolyai instead assumed it was false and worked out the geometry that follows. In 1823 he wrote to his father that he had discovered things so wonderful he was astounded, and that out of nothing he had created a strange new world. He published it as a 24-page appendix to his father's book. Gauss said he had reached the same results earlier without publishing. Lobachevsky had published his own version in 1829, in Russian, in a journal neither man saw.
In the summer of 1654 Blaise Pascal and Pierre de Fermat exchanged five letters and, between them, laid the foundation of probability. Two questions drove the correspondence: how many throws of a pair of dice you should expect before a double six, and how to split the stakes when a game of dice is broken off partway. They solved the second for two players and could not get it for three or more. Pascal wrote one of the letters from bed, unwell, having just been handed Fermat's.
A medieval scribe scraped the ink from seven older manuscripts and wrote a prayer book across the reused pages. In 1906 the Danish scholar Johan Ludvig Heiberg studied that prayer book and identified the erased writing underneath as Archimedes: seven treatises, two of which, The Method and The Stomachion, exist nowhere else. The book then dropped out of sight for most of a century. It sold at Christie's in New York in October 1998 for two million dollars, and the buyer deposited it at the Walters Art Museum in Baltimore.
Counting is really matching things up, and Georg Cantor pushed that idea further than anyone. In a paper of 1874 he showed that the counting numbers can be matched one for one with all the solutions of polynomial equations with whole-number coefficients, but not with all the real numbers, which form a strictly larger infinity. A consequence is that almost every number is transcendental. In 1877 he proved that a square holds no more points than one of its sides, and wrote to Dedekind that he could see it and did not believe it.
William Sealy Gosset joined Arthur Guinness Son and Company in Dublin as a chemist in 1899 and hit a problem. Brewing quality control handed him tiny samples, and the statistics of the day assumed large ones. So he worked out the distribution that handles small samples. Guinness had a firm rule against employees publishing, imposed after one of them gave away trade secrets, so his 1908 paper in Biometrika went out under the name Student. Statistics classes have called it Student's t-test ever since.
9Binary arithmetic before Leibniz, and no one knew
On the tile: Harriot1679
Leibniz gets the credit for binary arithmetic, base two, which he had worked out by 1679 and did not publish until 1701. Laplace wrote that Leibniz saw creation in it, with unity for God and zero for the void, and hoped the emblem might convert the Emperor of China. He was not first, though. The English mathematician Thomas Harriot had already done arithmetic in base two, on pages that stayed in the papers he left unpublished, and their contents were not pointed out until a paper of 1951.
1Euler said the bridge puzzle was barely mathematics
On the tile: Königsberg1730s
Königsberg sat on a river with two islands and seven bridges, and nobody could find a walk that crossed every bridge exactly once. Townspeople wrote to Leonhard Euler asking him to settle it. He replied dismissively that the question had little relationship to mathematics, and in a sense he was right: the mathematics it needed did not exist yet. He solved it anyway, by throwing away distances and shapes and keeping only which piece of land touched which. That is graph theory.
The Tower of Hanoi reached Paris in 1883 with a story attached. In a temple at Benares, it said, priests were moving a tower of sixty-four golden discs between diamond needles, and when they finished the world would end. The French mathematician Édouard Lucas made up the tale along with the puzzle. The arithmetic is reassuring. Moving sixty-four discs one at a time, never setting a larger disc on a smaller, takes 2 to the sixty-fourth power minus 1 moves, a twenty-digit number.
3A thousand doctorates wrote in to say she was wrong
On the tile: Monty Hall1990
Three doors, a car behind one of them. You choose a door, the host opens a different one to reveal a goat, and you are offered the switch. In September 1990 Marilyn vos Savant answered in her Parade column that you should switch, because switching wins two times in three. About a thousand people with doctorates wrote in to tell her she was wrong. Paul Erdős was among the unconvinced and came around only after watching a computer simulation. She was right.
The sliding 15 puzzle set off a craze that ran from January 1880 in the United States through that July, and it is still commonly credited to the American puzzle maker Sam Loyd. He began claiming it in 1891 and kept the claim up for twenty years, until his death. Research by Slocum and Sonneveld, published in 2006, showed that Loyd neither invented the puzzle nor helped popularize it. The inventor was Noyes Chapman, a postmaster in Canastota, New York, who applied for a patent in March 1880.
Nikoli, the Japanese publisher that made Sudoku famous, says plainly that it is not their puzzle. They found it in an American magazine under the name Number Place and brought it to Japanese readers in 1984, first with a long title meaning the digits must be single, then shortened to Sudoku: su for number, doku for single. Nikoli registered that name as a trademark in Japan, which is why rival Japanese magazines print the same puzzle under its original American title, Number Place.
In the upper right of Albrecht Dürer's engraving Melencolia I hangs a four-by-four grid holding the numbers 1 through 16. Harvard Art Museums, describing its own impression, notes a magic square with numbers adding up to 34 in all directions. Rows, columns, diagonals, each quadrant and the middle four squares all reach that total. At the bottom center sit 15 and 14, side by side: the year the engraving was made, tucked into the arithmetic where it balances.
7Euler's impossible parade and the square he missed
On the tile: 36 officers1782
Euler asked whether 36 officers, six ranks drawn from six regiments, could stand in a six-by-six square with no rank and no regiment repeated in any row or column. In 1782 he admitted he believed it impossible but could not prove it. Gaston Tarry supplied the proof in papers of 1900 and 1901. Euler also guessed that such squares fail whenever the side is 6, 10, 14 and so on. In 1960 Bose, Shrikhande and Parker, with computers to help, found them for every size except 2 and 6.
In a letter to Euler in June 1742, Christian Goldbach floated an idea: every number greater than 2 is the sum of three primes, counting 1 as a prime. Euler recast it in the form that stuck, that every even number greater than 2 is the sum of two primes, and nobody has proved it. In 2000, promoting a novel about the problem, the publisher Faber and Faber offered a million dollars for a proof produced within two years and published in a respectable journal within two more. Faber insured the offer for a five-figure sum. The prize expired unclaimed.
Pick a number. If it is even, halve it. If it is odd, triple it and add 1. Repeat. Every number anyone has tried eventually falls to 1, and no one can prove they all must. Lothar Collatz probably posed the question in the 1930s, and mathematicians treat it as a quagmire and warn each other away from it. In September 2019 Terence Tao got further than anyone had in decades, proving the conjecture almost true for almost all numbers. The question itself is still open.