Probability

Fermat and Pascal: fairly dividing an interrupted game

Two players bet the same amount on a game of several rounds: whoever first wins a certain number of rounds takes everything. Midway through, the game has to be stopped. How should the money be divided fairly? This is the problem of points. It looks simple, but it defeated mathematicians for more than 150 years. The wrong answer In 1494 the Italian friar Luca Pacioli proposed dividing the prize in proportion to the rounds already won. If A has won 2 rounds and B has won 1, A takes $\frac23$ of the prize.

October 11, 2026 Read

Banburismus: weighing evidence in decibans

The bombe was fast, but not fast enough to test every rotor order of the naval Enigma. In 1940 Turing invented a pencil-and-paper method to rule out most of them before switching on the machines. Because the sheets used in the method were printed in the town of Banbury, his colleagues named it Banburismus. The method is statistical. And to devise it, Turing reinvented one of the central ideas of modern statistics.

October 16, 2026 Read

The Bernoulli numbers: adding a thousand powers in seven minutes

With the help of this table it took me less than half of a quarter of an hour to find that the tenth powers of the first thousand numbers, added together, give 91,409,924,241,424,243,424,241,924,242,500. — Jakob Bernoulli, Ars conjectandi (1713), part II The story goes that young Gauss added the numbers from 1 to 100 in seconds, pairing the first with the last: $1+2+\cdots+n=\frac{n(n+1)}2$. For squares and cubes there are formulas too, known since Archimedes and the Arab mathematicians:

October 14, 2026 Read

The cycloid: the toothache curve

Paint a dot on a bicycle tyre and push the bike in a straight line. The dot rises, makes an arch and comes back down to touch the ground, once for every turn of the wheel. The curve it traces is called the cycloid. In the 17th century it was the fashionable curve. The French called it the roulette. So many mathematicians quarrelled over discoveries about it that it earned a nickname: the Helen of geometers, after Helen of Troy, who started a war.

October 11, 2026 Read

The legacy of James Bernoulli

James Bernoulli lived only fifty years and published little in his lifetime, almost all of it in short papers in the Acta Eruditorum. Even so, few mathematicians left so many words, symbols and ideas that we still use every day: integral, Bernoulli trial, binomial distribution, Bernoulli numbers, law of large numbers, stochastic. The word integral Leibniz called the operation inverse to the derivative calculus summatorius, the calculus of sums, and wrote it with $\int$, an elongated S for summa. In May 1690, solving the isochrone problem, Jakob wrote that the two expressions had “equal integrals”. It was the first time the word integral appeared in print in a mathematical text. Johann adopted the term, and in 1696 Leibniz agreed to swap his “summatory calculus” for the Bernoullis’ integral calculus. The symbol stayed Leibniz’s; the name, Jakob’s.

October 14, 2026 Read

Pascal's principle: multiplying forces with water

An elephant and a woman in high heels: who sinks deeper into soft ground? It may be the woman. The elephant weighs fifty times more, but its weight spreads over four broad feet. Hers is concentrated on the thin tip of a heel. What sinks into the ground is not force but pressure: the force divided by the area it rests on. $$p=\frac FA.$$The unit of pressure is the pascal: $1\ \text{Pa}=1\ \text{N/m}^2$, a force of one newton spread over one square metre. It is a tiny pressure; at sea level the atmosphere exerts about $100{,}000$ Pa.

October 11, 2026 Read

Image credits

Historical images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Jakob Bernoulli, painted in 1687 by his brother Niklaus Bernoulli: source page · public domain. Title page of the Ars conjectandi, Basel, 1713: source page · public domain. Jakob Bernoulli’s tombstone in the cloister of Basel Minster, photo by Wladyslaw Sojka: source page · Free Art License. Cross-section of a nautilus shell, photo by Florian Elias Rieser: source page · CC BY-SA 4.0. Blog figures The cover, the drawing of the two spirals and the graph of the proportion of heads in simulated coin tosses are original illustrations, with versions in each of the site’s languages. The table of notations for the integral, in the post on the legacy, was redone from a table in this site’s old calculus course.

October 14, 2026 Read

The weight of air: the Puy de Dôme experiment

For two thousand years scholars repeated a saying of Aristotle’s: nature abhors a vacuum. If you suck the air out of a straw, the liquid rises because nature will not let the space stay empty. It seemed to explain everything. But the well-diggers of Florence had a problem. Suction pumps could not raise water more than about 10 metres, however hard they tried. Why would nature’s horror of the vacuum stop at 10 metres?

October 11, 2026 Read

Pascal's wager: a calculation, not a proof

Many people have heard that Pascal “proved that God exists”. He did not, and he said plainly that it could not be done. What he proposed is more curious: since reason cannot settle the question, change the question. Instead of “does God exist?”, ask “what is the better bet?”. And answer this second question with the mathematics he had just invented with Fermat, the mathematics of fair division. A sheet full of scribbles Autograph sheets of the Pensées, pasted into a large album after Pascal’s death, at an exhibition at the National Library of France in 2016. Pascal wrote on scraps of paper, in several directions, and crossed out a great deal.ActuaLitté · Wikimedia Commons · CC BY-SA 2.0 In the last years of his life, Pascal gathered notes for a book in defence of Christianity. He never wrote it. After his death, in 1662, his friends found hundreds of fragments, some cut out with scissors and strung together on threads, and published a selection in 1670 under the title Pensées (Thoughts).

October 11, 2026 Read

Pascal's legacy

The heart has its reasons, which reason does not know. — Blaise Pascal, Pensées Pascal left few finished books. Almost everything he wrote on mathematics came out as loose sheets, in letters or after his death. Even so, few authors turn up in so many different places: in the triangle at school, in the weather forecast, in a car’s brakes, in an insurance calculation and even in the name of a programming language.

October 11, 2026 Read

Image credits

Historical portrait The portrait attributed to Rolland Lefebvre comes from Wikimedia Commons and is in the public domain. We use the thumbnail provided by Commons, without editing, and the caption keeps the attribution given by the collection. Source page. Public domain mark. Blog figures The cover and the refraction sketch are original illustrations. The interactive figures for remainders of powers, the area of rectangles and the division of the interrupted game were created for this trail. They are teaching aids and do not reproduce Fermat’s manuscripts. The division figure, for example, assumes independent rounds and players with equal chances.

October 11, 2026 Read

Pascal image credits

Historical images All of them come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Blaise Pascal, unknown artist, around 1690, Versailles: source page · public domain. Yang Hui’s triangle, in Zhu Shijie’s book, 1303: source page · public domain. First page of the Triangulus arithmeticus, copy held by Clermont Auvergne Métropole: source page · CC BY-SA 4.0. Pascaline at the Musée des Arts et Métiers, photo by David Monniaux, 2005: source page · CC BY-SA 3.0. Marker of the experiment on the Puy de Dôme, Agence Rol, 1923, Bibliothèque nationale de France: source page · public domain. Plaque of the grave in Saint-Étienne-du-Mont, Agence Meurisse, 1923, Bibliothèque nationale de France: source page · public domain. Autograph manuscript of the Pensées, exhibition at the Bibliothèque nationale de France, photo by ActuaLitté, 2016: source page · CC BY-SA 2.0. First page of the Pensées, counterfeit of the 1670 edition, copy held by Clermont Auvergne Métropole: source page · CC BY-SA 4.0. Blog figures The cover, the fountain of bowls, the carry diagram and the cycloid arch are original illustrations. The controls for the triangle, the split of the stakes and the hydraulic press were made for this trail. They are teaching aids, not reproductions of manuscripts or of the internal mechanism of the Pascaline.

October 11, 2026 Read