by DK
The triangle is most commonly named after French philosopher and mathematician Blaise Pascal, who explored it in detail in his Treatise on the Arithmetical Triangle in 1653. In Italy, however, it is known as Tartaglia’s triangle after mathematician Niccolò Tartaglia, who wrote about it in the 1400s. In fact, the origins of the triangle date back to ancient India in 450 BCE (see The ancient triangle).
There are two types of mind… the mathematical, and… the intuitive. The former arrives at its views slowly, but they are… rigid; the latter is endowed with greater flexibility.
Blaise Pascal
Pascal’s triangle is created by adding together two adjacent numbers (as shown by the arrows) to give the sum in the next row. Each row begins and ends with the number 1.
Probability theory
Pascal’s contribution to the triangle was notable because he set out a clear framework for exploring its properties. In particular, he used the triangle to help lay the foundations of probability theory in his correspondence with fellow French mathematician Pierre de Fermat. Before Pascal, mathematicians such as Luca Pacioli, Gerolamo Cardano, and Tartaglia had written about how to work out the chances of dice rolling particular numbers or hands of cards coming out a certain way. Their understanding was shaky at best, and it was Pascal’s work with the triangle that pulled the strands together.
Dividing stakes
Pascal was asked to look into probability in 1652 by a notorious French gambler. Antoine Gombaud, the Chevalier de Méré, wanted to know how to divide stakes fairly if a game of chance was suddenly broken off. If a game would normally end only when one player had won a certain number of rounds, for instance, de Méré wanted to know if the division of the stakes should reflect how many rounds each player had won. Pascal combined the numbers step by step to represent the rounds played. The natural consequence was an ever-widening triangle. As Pascal showed, the numbers in the triangle count the number of ways various occurrences can combine to produce a given result.
The probability of an event is defined as the proportion of times it will happen. A dice has six faces, so the probability of it landing on any particular face when you roll it is 1⁄6. In other words, it is a question of noting how many ways the event can occur, and dividing this by the total number of possibilities. While this is easy enough for a single dice, with multiple dice, or 52 playing cards, the calculations become complicated. However, Pascal found that the triangle could be used to find the number of possible combinations when you choose a number of objects from a particular number of available options.
BLAISE PASCAL
Born in Clermont-Ferrand, France, in 1623, Blaise Pascal was a mathematics prodigy. As a teenager, his father took him to Marin Mersenne’s mathematical salon in Paris. Around the age of 21, Pascal developed a mechanical adding and subtraction machine, the first ever marketed. As well as his mathematical contributions, Pascal played an important role in many scientific developments of the 1600s, including explorations of fluids and the nature of a vacuum, which contributed to the understanding of the idea of air pressure: the scientific unit of pressure is called the Pascal. In 1661, he launched what may have been the world’s first public transportation service in Paris, with linked five-person coaches. He died from unexplained causes in 1662, aged just 39.
Key works
1653 Traité du triangle arithmétique (Treatise on the Arithmetical Triangle)
1654 Potestatum Numericarum Summa (Sums of Powers of Numbers)
Binomial calculations
As Pascal realized, the answer lay in binomials—expressions with two terms, such as x + y. Each row of Pascal’s triangle gives the binomial coefficients for a particular power. The zeroth row (the top of the triangle) is used for the binomial to the power of 0: (x + y)0 = 1. For the binomial to the power of 1, (x + y)1 = 1x + 1y, so the coefficients (1 and 1) correspond to the first row of the triangle (the zeroth row is not counted as a row). The binomial (x + y)2 = 1x2 + 2xy + 1y2 has the coefficients 1, 2, and 1, as on the second row of Pascal’s triangle. As binomial expansion leads to ever longer expressions, the coefficients continue to match a corresponding line on the triangle. For example, in the binomial (x + y)3 = 1x3 + 3x2y + 3xy2 + 1y3, the coefficients match the third row of the triangle. The probabilities are calculated by dividing the number of possibilities by the total of all the coefficients in the row that reflects the total number of objects: for example, in a family of three children (the total number of objects), the probability of one girl and two boys is 3/8 (the sum of all the coefficients in the third row of the triangle is 8, and there are three ways of having one girl in a family of three children).
Pascal’s triangle made it simple to find probabilities. As Pascal’s triangle can continue forever, this works with any powers. The relationship between binomial coefficients and the numbers in Pascal’s triangle reveals a fundamental truth about numbers and probability.
The Bat Country, a jungle gym project by American artist Gwen Fisher, is a Sierpinski tetrahedron featuring softball bats and balls. This tetrahedron is a 3-D structure made of Sierpinski triangles.
Visual patterns
Pascal’s simple number pattern proved to be the launchpad, with Fermat’s work, for the mathematics of probability, but its relevance does not stop there. For a start, it provides a quick way of multiplying out binomial expressions to high powers, which would otherwise take a very long time. Mathematicians are continually finding new surprises in it. Some of the patterns in Pascal’s triangle are extremely simple. The outside edge is entirely made up of the number 1, and the next set of numbers, in the first diagonal, is a simple number line of 1, 2, 3, 4, 5, and so on.
One particularly appealing property of Pascal’s triangle is the “hockey stick” pattern, which can be used for addition. If you take a diagonal down from any of the outer 1s, then stop anywhere, you can then find the total sum of all of the numbers in the diagonal by taking one step further in the opposite direction. For example, starting at the fourth 1 on the left edge and going down diagonally to the right, if you stop at the number 10, then the total of the numbers passed so far (1 + 4 + 10) can be found by going one diagonal step down to the left: 15.
Coloring in all of the numbers divisible by a particular number creates a fractal pattern, while coloring all of the even numbers creates a pattern of triangles identified by Polish mathematician Wacław Sierpinski in 1915. This pattern can be made without Pascal’s triangle by breaking an equilateral triangle into ever smaller triangles by connecting the midpoints of each of the triangles’ three sides. The division can continue indefinitely. Today, Sierpinski triangles are popular for use in knitting patterns and in origami, where a Sierpinksi triangle is converted into three dimensions to create a Sierpinski tetrahedron.
I cannot judge my work while I am doing it. I have to do as painters do, stand back and view it from a distance, but not too great a distance.
Blaise Pascal
Number theory
There are also many more complex patterns hidden within the triangle. One of the patterns found in Pascal’s triangle is the Fibonacci sequence, which lies on a shallow diagonal (see below). Another link to number theory is the discovery that the sum of all the numbers in the rows above a given row is always one less than the sum of the numbers in the given row. When the sum of all the numbers above a given row is a prime, it is a Mersenne prime—a prime number that is one less than a power of 2, such as 3 (22 - 1), 7 (23 - 1), and 31 (25 - 1). The first list of these primes was made by Pascal’s contemporary, Marin Mersenne. Today, the largest known Mersenne prime is 282,589,933 -1. If Pascal’s triangle were drawn at a sufficiently large scale, this number would be found there.
The numbers on the left form the Fibonacci sequence, which can be calculated by adding the numbers on the shallow diagonals (indicated here by the color shading) of Pascal’s triangle.
The ancient triangle
Myanmar’s Hsinbyume pagoda represents the mythical Mount Meru, whose st
aircase inspired another name for Pascal’s triangle.
Mathematicians knew about Pascal’s triangle long before the 1600s. In Iran, it is known as Khayyam’s triangle after Omar Khayyam, but he was just one of many Islamic mathematicians to have studied it between the 7th and 13th century—a golden age for learning. In China, too, c. 1050, Jia Xian created a similar triangle to show coefficients. His triangle was taken up and popularized by Yang Hui in the 1200s, which is why it is known in China as Yang Hui’s triangle. It is illustrated in the 1303 book by Zhu Shijie entitled Precious Mirror of the Four Elements.
The most ancient references to Pascal’s triangle, however, come from India. It appears in Indian texts from 450 BCE as a guide to poetic metre, by the name of “The Staircase of Mount Meru.” The mathematicians of ancient India also realized that the shallow diagonal lines of numbers in the triangle showed what are now known as Fibonacci numbers.
See also: Quadratic equations • The binomial theorem • Cubic equations • The Fibonacci sequence • Mersenne primes • Probability • Fractals
IN CONTEXT
KEY FIGURES
Blaise Pascal (1623–62), Pierre de Fermat (1601–65)
FIELD
Probability
BEFORE
1620 Galileo publishes Sopra le Scoperte dei Dadi (On the Outcomes of Dice), calculating the chances of certain totals when throwing dice.
AFTER
1657 Christiaan Huygens writes a treatise on probability theory and its applications to games of chance.
1718 Abraham de Moivre publishes The Doctrine of Chances.
1812 Pierre-Simon Laplace applies probability theory to scientific problems in Théorie analytique des probabilités (Theory of Probabilities).
Before the 1500s, predicting the outcome of a future event with any degree of accuracy was thought to be impossible. However, in Renaissance Italy, scholar Gerolamo Cardano produced in-depth analyses of outcomes involving dice. In the 1600s, such problems attracted the attention of French mathematicians Blaise Pascal and Pierre de Fermat. More renowned for findings such as Pascal’s triangle and Fermat’s last theorem, the two men took the mathematics of probability to a new level, laying the foundations for probability theory.
Forecasting the outcomes of games of chance proved a useful way of approaching probability, which, by definition, measures the likelihood of something occurring. For example, the chances of throwing a six with a die can be estimated by throwing the die a given number of times and dividing the amount of sixes thrown by the total number of throws. The result, called relative frequency, gives the probability of throwing a six, which can be expressed as a fraction, a decimal, or a percentage. This, however, is an observed finding, based on actual experiments. Theoretical probability of any single event is calculated by dividing the number of desired outcomes by the total number of possible outcomes. With one roll of a six-sided die, the probability of throwing a six is 1⁄6; the probability of throwing any other number is 5⁄6.
Probability theory is nothing but common sense reduced to calculation.
Pierre-Simon Laplace
Estimating the odds
One popular game in 17th-century France involved two players taking turns to throw four dice in a bid to obtain at least one “ace,” or six. The players contributed equal stakes and agreed, in advance, that the first one to win a certain number of rounds would take the whole stake. Writer and amateur mathematician Antoine Gombaud, who styled himself Chevalier de Méré, understood the 1⁄6 odds of an ace with one throw of a die, and sought to calculate the odds of throwing a double ace using a pair of dice.
De Méré suggested that the chance of getting two aces from two throws of a dice was 1⁄36, that is, 1⁄6 as likely as getting an ace with one die in one roll. To make these odds the same, he argued that a pair of dice should be rolled six times for each roll of the single die. To have the same chance of rolling a double ace as you would from getting one ace when four dice are thrown, the pair should be thrown 6 × 4 = 24 times. De Méré consistently lost the bet and was compelled to deduce that a double ace from 24 throws of a pair of dice was less likely than one ace from four throws of a single die.
In 1654, de Méré consulted his friend Pascal about this problem, and about the further question of how a stake should be divided between the players when a game was interrupted before completion. This was known as the “problem of points,” and it had a long history. In 1494, Italian mathematician Luca Pacioli had suggested that the stakes should be divided in proportion to the number of rounds already won by each player.
In the mid-1500s, Niccolò Tartaglia, another prominent mathematician, had noted that such a division would be unfair if the game was interrupted, say, after only one round. His solution was to base the division of the pot on the ratio between the size of the lead and the length of the game, but this also gave unsatisfactory results for games with many rounds. Tartaglia remained unsure whether the problem was solvable in a way that would convince all players of its fairness.
Probability is easily measured in the cases shown here. It is zero if the element in question (blue candies) is absent, and 0.5 (or 1⁄2, or 50 percent) if half of all candies are blue. When events are certain, probability = 1 (or 100 percent).
PIERRE DE FERMAT
Born in in 1601 in Beaumont-de-Lomagne in France, Pierre de Fermat moved to Orléans in 1623 to study law and soon began to pursue his interest in mathematics. Like other scholars of his day, he studied geometry problems from the ancient world and applied algebraic methods to try to solve them. In 1631, Fermat moved to Toulouse and worked as a lawyer.
In his spare time, Fermat continued his mathematical investigations, circulating his ideas in letters to friends, such as Blaise Pascal. In 1653, he was struck down by plague but survived to do some of his best work. As well as his ideas on probability, Fermat pioneered differential calculus, but is best remembered for his contribution to number theory and Fermat’s last theorem. He died in Castres in 1665.
Key works
1629 De tangentibus linearum curvarum (Tangents of Curves)
1637 Methodus ad disquirendam maximam et minimam (Methods of Investigating Maxima and Minima)
The Pascal–Fermat letters
During the 1600s, it was common for mathematicians to meet at academies—scientific societies. In France, the leading academy was that of the Abbé Marin Mersenne, a Jesuit priest and mathematician who held weekly meetings at his Paris home. Pascal attended these meetings, but he and Fermat had never met. Nonetheless, having pondered de Méré’s problems, Pascal chose to write to Fermat, communicating his thoughts on these and related issues and asking for Fermat’s own views. This was the first of the letters between Pascal and Fermat in which the mathematical theory of probability was developed.
On a standard roulette wheel, there is a 1⁄37 chance of the ball landing on any given number for a single spin of the wheel. This number gets closer to 1 the greater the number of spins.
Player versus banker
The Pascal–Fermat letters were sent via Pierre de Carcavi, a mutual friend. Seven letters exchanged in 1654 reveal the two men’s thoughts on the points problem, which they examined in different scenarios. They discuss a game between a player attempting to throw at least one ace in eight throws and a “banker” who takes the pot if the player is unsuccessful. If the game is interrupted before an ace has been thrown, Pascal seems to suggest that the stakes should be allocated according to the players’ expectations of winning. At the start of the game, the probability of eight rolls of the die without success is (5⁄6)8 ≈ 0.233, and the probability of throwing at least one ace is (1—0.233), or 0.7677. The game clearly favors the one who makes the throws, rather than the “banker.”
Choice means probability, and probability means mathematicians can get to work.
Hannah Fry
British mathematician
Laying down the theory
In other letters, Pascal and Fermat discuss ot
her cases of interrupted games, such as when the play alternates between two players until one is successful. Fermat notes that what matters is the number of throws remaining when the game stops. He points out that a player with a 7–5 lead in a game to 10 aces has the same chance of eventually winning as a player with a 17–15 lead in a game to 20.
Pascal gives an example with two opponents playing a sequence of games, each with an equal chance of winning, where the first to win three games wins the stake. Each player has staked 32 pistoles, so the stake is 64 pistoles. Over the course of three games, the first player wins twice and the other once. If they now play a fourth game and the first player wins, then he will take the 64 pistoles; if the other wins it, they will have each won two games and are equally likely to win the final game. If they stop at this point, each should take back his stake of 32 pistoles.
Pascal’s step-by-step methods and Fermat’s considered replies provide some of the earliest examples of using expectations when reasoning about probability. The correspondence between the two laid down basic principles of probability theory, and games of chance would continue to prove fertile ground for early theorists. Dutch physicist and mathematician Christiaan Huygens wrote a treatise translated as “On reasoning in games of chance,” which was the first book on probability theory.