Probability

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 8, 2026 Read

Will the Sun rise tomorrow? The rule of succession

Classical probability goes from causes to effects: knowing the coin is fair, what is the chance of heads? Often the problem is the reverse. We see the effects and want to know about the cause: after seeing certain results, what can we say about the coin? The first to treat this probability of causes mathematically was the English clergyman Thomas Bayes, in a paper published after his death, in 1763. Laplace arrived at the same ideas independently in 1774, without knowing of Bayes, and it was he who developed and applied them.

October 30, 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 11, 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 8, 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 8, 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 8, 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 7, 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 8, 2026 Read

ITA 2025 – 2nd phase: Mathematics solved

Source: ITA – 2nd phase Mathematics exam. Files on the ITA past exams page. The 2nd phase Mathematics exam of ITA 2025 has 10 written questions. ITA does not publish official solutions for the written part (the 2nd phase answer key only covers the multiple-choice Portuguese questions); the solutions below are ours, and each answer was checked by a second route (substitution, another formula or a computer check). Answers Question Topic Answer 1 Remainder of the division by $x^2 - x + 1$ $a = -1$, $b = 3$ 2 Ellipse: maximum of $y/x$ $r = \sqrt{3}$ 3 Trigonometric system $\sin(\alpha + \beta) = \frac{5 + \sqrt{7}}{8}$ 4 Excircle $15\sqrt{3}$ 5 First digit of $3^{100}$ 5 6 Probability: fourth head on toss $n$ $n = 6$ and $n = 7$ 7 Polynomial with a double root and a right triangle $a = \pm\frac{3\sqrt[4]{2}}{2}$, $b = \mp\frac{1}{\sqrt[4]{2}}$ 8 Sum of determinants 1013 9 Integer matrices with $A^{-1} = A^T$ 3840 10 Projection of the cube perpendicular to the diagonal $4\sqrt{3}$ Question 1: The remainder of the division by x² − x + 1 Statement Find the real values $a$ and $b$ such that the polynomial $p(x) = x^{57} + ax^{14} + bx^7 + 1$, when divided by $x^2 - x + 1$, leaves remainder $2x + 1$.

October 28, 2026 Read

ITA 2023 – 1st phase: Mathematics solved

Source: ITA – 1st phase exam and official answer key. Files on the ITA past exams page. The 1st phase of ITA 2023 had 60 multiple-choice questions, 12 per subject (Physics, Portuguese, English, Mathematics and Chemistry). This post covers the mathematics questions, numbered 37 to 48 as in the exam. Exam conventions. $\mathbb{N} = \{1, 2, 3, \dots\}$; $i^2 = -1$; $M_n(\mathbb{R})$ is the set of $n \times n$ real matrices; $\overline{AB}$ is the segment and $AB$ the line through $A$ and $B$; $m(\overline{AB})$ is the length of $\overline{AB}$; $A\hat{O}B$ is the angle with vertex $O$; $\det A$ is the determinant of $A$.

October 28, 2026 Read

ITA 2022 – 1st phase: Mathematics solved

Source: ITA – 1st phase exam and official answer key. Files on the ITA past exams page. The 1st phase of ITA 2022 had 70 multiple-choice questions: 15 in Physics, 15 in Portuguese, 10 in English, 15 in Mathematics and 15 in Chemistry. This post covers the mathematics questions, numbered 41 to 55 as in the exam. Exam conventions. $\mathbb{N} = \{1, 2, 3, \dots\}$; $i$ is the imaginary unit ($i^2 = -1$); $M_n(\mathbb{R})$ is the set of $n \times n$ real matrices; $\overline{AB}$ is the segment with endpoints $A$ and $B$, and $m(\overline{AB})$ is its length; $A\hat{O}B$ is the angle with vertex $O$ formed by the rays $\overrightarrow{OA}$ and $\overrightarrow{OB}$.

October 28, 2026 Read

FUVEST 2026 – 2nd phase: Mathematics solved

Source: FUVEST – test on the 2nd day of the 2nd phase (specific tests; Mathematics has questions M01 to M06) and expected Mathematics answers. Files in the FUVEST collection 2026. In the 2nd phase of FUVEST 2026, the 2nd day consisted of specific essay tests for each area; The Mathematics test has six questions, M01 to M06, each with items a), b) and c). For this test, FUVEST published the expected answers; the resolutions below are ours, we checked each one in a second way and all match the official expected answers.

September 28, 2026 Read