Pascal’s Triangle and its Secrets

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Probability, statistics and Pascal’s other contributions

By David Benjamin. Posted March 4, 2022

This is the final part in the Pascal pentalogy, a series of guest posts by David Benjamin exploring the secrets of Pascal’s Triangle.

Probability and combinations

In Part 1 of this series we stated that Pascal is credited with being the founder of probability theory – but credit also needs to be given to other mathematicians, in particular the Italian polymath Girolamo Cardano.

The connection between probability and the numbers in Pascal’s triangle can be shown by looking at the outcomes when one or more coins are tossed. The table below, from row two, lists the outcomes for one, two and three unbiased coins.

				$1$
			$1$ H		$1$ T
		$1$ HH		$2$ HT, TH		$1$ TT
	$1$ HHH		$3$ HHT, HTH, THH		$3$ HTT, THT, TTH		$1$ TTT
$1$		$4$		$6$		$4$		$1$

Reading from the left: all possible outcomes, heads decreasing by one moving to the right.

For four coins there is $1$ outcome for four heads, $4$ outcomes for three heads and one tail, $6$ outcomes for two heads and two tails, $4$ outcomes for one head and three tails and one outcome for $4$ tails.

Row four shows us that when three unbiased coins are tossed, the probability they will land showing two heads and one tail in any order is $\frac{3}{1+3+3+1}=\frac{3}{8}$.

As the sum of the $n^{th}$ row is $2^{n}$, the number of possible outcomes for four coins is $2^4=16$, $32$ for five coins, $64$ for six coins, …

Quincunx

A Quincunx, or Galton Board, is named after the English explorer and anthropologist Francis Galton (1822-1911) – although this name is now less popular, because of Galton’s views on eugenics and racist attitudes.

The board is a triangular array of pegs. Balls are dropped onto the top peg and then bounce their way down to the bottom where they are collected in containers. Each time a ball hits one of the pegs, it bounces either left or right with an equal probability of $\frac{1}{2}$ and the balls collect in the containers to form the classic bell-shaped curve of the normal distribution.

Animation of a board of pegs with balls falling and bouncing to the left or right — Generated from Math Is Fun’s page on the Quincunx

The Quincunx is like Pascal’s triangle with pegs instead of numbers. The number on each peg represents the number of different paths a ball can take to reach that peg. If there are $10$ rows and the last row contains the containers, then the probability of landing in the third container from the right can be calculated by using the formula for the Binomial distribution.

The probability of landing in the third bin from the right is $120\times(\frac{1}{2})^3\times(\frac{1}{2})^7=\frac{15}{128}=0.1171875$, where $120$ is the number of different paths to that bin.

Statistics and permutations

The link between statistics and the triangle can be demonstrated using combinations. Consider these 5 mathematicians Euler, Pascal, Ramanujan, Hilbert and Conway and the possible teams for a three-legged race.

There are $10$ different teams of $3$:

EPR EPH EPC ERH ERC EHC PRH PRC PHC RHC

The formula to calculate the number of combinations is $_n{C}_r =\frac{n!}{r!(n-r)!}$ where $n$ represents the total we are choosing from, $r$ the number in the team and

\[ n!=n\times(n-1)\times(n-2)\times(n-3)\times…\times1\]

In our example $n=5$, $r=3$ and $\frac{5!}{3!(5-3)!}=\frac{120}{6\times2}=10$

$_n{C}_r$ can be used to calculate the rows of Pascal’s triangle as shown below for row $6$, where in the calculation of $_5{C}_0$, $0!=1$

$_5{C}_0$	$_5{C}_1$	$_5{C}_2$	$_5{C}_3$	$_5{C}_4$	$_5{C}_5$
$1$	$5$	$10$	$10$	$5$	$1$

The animation film Of Dice and Men by John Weldon is a lovely way to introduce students to probability and statistics.

Pascal the polymath: mathematics, inventor, science and religion

Pascal’s father was a tax collector and in 1642 Blaise invented a mechanical calculator to assist his father. It was called the Pascaline and had a wheel with eight movable parts for dialing. Each part corresponded to a particular digit in a number. Numbers could be added by turning the wheels located along the bottom of the machine. Subtraction was carried out by exploiting a method called nines’ complement representation, the use of which allows subtraction to be reduced to addition. Each digit in the answer was displayed in a separate window. The workings of the Pascaline are demonstrated here.

A photo of a gold metal box, with a set of dials on the front and numbers showing through holes across the top. — A Pascaline. An original is displayed in The Musée des Arts et Métiers in Paris, France

Close-up photo of the dials on the box, which have the digits 1-9 engraved and some French script text above — A close-up of the dials which are rotated by inserting a spoke

The Musée des Arts et Métiers in Paris has one of the original Pascalines. The invention was not a commercial success – it was very expensive and often only purchased as a novelty rather than for use. Essentially, it was an adding machine. Subtraction was turned into a form of addition, as was multiplication. Division was done by repeated subtraction. Nines’ complement representation is still used in modern digital computers by a similar technique called ones’ complement which is used to represent negative numbers and hence perform subtraction in the same way as addition. Pascal did not discover this method but his calculator is the earliest known device to employ it. He continued to make improvements to his design until 1652.

Conic sections – normally just called conics – are obtained when a mathematical cone is sliced by a plane. Depending on the angle of the slice, the intersections create a circle, an ellipse, a parabola and a hyperbola. Conics have many applications including the wheel of course, ophthalmic, parabolic mirrors and reflectors, telescopes, searchlights and projectile motion.

Pascal wrote a short treatise, Essai pour les coniques (Essay on Conics) when only 16. In it he included what is known as Pascal’s Theorem which states that if a hexagon is inscribed in a conic section then the three intersection points of opposite sides lie on a straight line – the Pascal line. The theorem [also referred to as Pascal’s Hexagrammum Mysticum Theorem] was his first important mathematical discovery and a breakthrough in the field of projective geometry.

Photo of a page of the essay, with some geometric diagrams at the top and the title 'Essay Pour Les Coniques' above French text — A rare copy of the *Essay pour les coniques* which is kept in the National Library of France

In 1647 Pascal expanded on the work of the Italian physicist Evangelista Torricelli, the inventor of the barometer by writing Experiences nouvelles touchant le vide (New experiments with the vacuum) in which Pascal gave detailed rules to describe to what degree various liquids could be supported by air pressure. In 1971 the SI unit for pressure [equal to one newton per square metre] was named the pascal.

Photograph of a circular gauge with values from 0 to 4000 psi and 0 to 28000 kPa, with an arrow that rotates and the text 'U.S. Gauge / MADE IN USA' across the bottom. Text in the centre reads 'USE NO OIL'. — A pressure gauge reading in psi (red scale) and kPa (black scale)

Also in 1647 he discovered Pascal’s Law of hydrostatics allowing for the development of the hydraulic press. Pascal himself used the principle to invent the syringe.

A diagram of a car on a wide platform, joined via a hydraulic setup to a hand plunger, which has a much narrower area of fluid to push down on — Pascal’s Law is the principle behind hydraulic lifting and pressing devices

Pascal wrote an extremely influential theological work which was unfinished at the time of his death. It was posthumously called Pensées (Thoughts) and contained a detailed and coherent examination and defence of the Christian faith.

Image of a book page in French; it's a title page that reads PENSÉES DE M.PASCAL SUR LA RELIGION ET SUR QUELQUES AUTRES SUJETS, with publication details below and a fancy decorative flourish in the middle — Pascal – Pensées, édition de Port-Royal, 1670

In 1655 Pascal was trying to invent a perpetual motion machine, a machine that continues to operate without drawing energy from an external source. The laws of physics now say this is impossible. Naturally he failed but he ended up inventing a basic roulette wheel, now upgraded and used in casinos as a game of chance.

The Swiss computer scientist Niklaus Emil Wirth, born in 1934, named one of his programming languages Pascal in honour of Blaise. Wirth along with Helmut Weber also designed the programming language named after another mathematician, Euler. [Recommended read: Euler: The Master of Us All ]

Pascal died in extreme pain at the young age of 39. He had a malignant growth in his stomach which had spread to his brain. Like many others, such as Évariste Galois and Franz Schubert, we are left wondering what else Pascal could have achieved had he lived longer. His work with Fermat into the calculus of probabilities helped the German mathematician Gottfried Leibniz [1646-1716] develop the infinitesimal calculus. Pascal is buried in the Saint-Étienne-du-Mont church in Paris and his death mask is held at the J. Pau l Getty museum in Los Angeles, California.

Old photograph of a death mask, which looks like a plaster case of a sleeping man's face, seen from the side — The death mask of Blaise Pascal

Numbers and number patterns in Pascal’s triangle

By David Benjamin. Posted February 4, 2022

This is the fourth in a series of guest posts by David Benjamin, exploring the secrets of Pascal’s Triangle.

Triangles and fractals

If we highlight the multiples of any of the Natural numbers $\geq 2$ in Pascal’s triangle then they create a pattern of inverted triangles.

Diagrams showing Pascal's triangle, with numbers in hexagonal cells, with cells coloured as per their factors; in each of six diagrams, the multiples of 2, 3, 6, 7, 10 and 11 are shaded, and each gives a different pattern - all containing smaller triangles shaded the same colour

The images above are evocative of the Sierpinski sieve (also known as the Sierpinski gasket or Sierpinski’s triangle), a fractal described in 1915 by the Polish mathematician Waclaw Sierpiński (1882-1969).

Image of the Sierpinski sieve/triangle, in blue on a white background — The Sierpiński Sieve

Medal of Waclaw Sierpiński in the Yale University art gallery

Fractals are beautiful geometric shapes. Small, even down to (theoretically) infinitesimal areas of a fractal are identical to the entire shape. The Koch snowflake, generated geometrically by successive iterations on an equilateral triangle, is an example of a fractal. Julia sets and Mandelbrot sets are examples of fractals generated using recursion on complex functions. Many examples of fractals appear in nature, and the Polish-born French-American polymath Benoit Mandelbrot (1924-2010) suggested that fully developed turbulent flows are fractals.

It is a lovely surprise to discover that a simple fractal can be found inside Pascal’s triangle. It is achieved by considering all the numbers in the triangle modulo 2 – equivalent to colouring in only the multiples of 2, as in the first diagram at the top of the post. In this version, every odd number becomes $1$ and every even number becomes $0$, and by considering sufficiently many lines of the triangle, the Sierpinski pattern emerges.

A red Sierpinski triangle with the gaps filled in black — The areas containing the odd numbers have been shaded red and the areas containing the even numbers, black.

Pascal's triangle with numbers modulo 2; cells containing 1s are red and 0s are black, and the pattern matches that of the Sierpinski triangle — Pascal’s triangle modulo 2. The entries when concatenated can be read as binary numbers which are palindromic.

Number patterns in the triangle

If we consider the first 32 rows of the mod$(2)$ version of the triangle as binary numbers: $1, 11, 101, 1111, 10001,…$ and convert them into decimal numbers, we obtain the sequence:

\[1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, \]

\[196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, \]

\[50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295\]

Interestingly, all members of this sequence are factors of the final term, $4294967295 = 2^{32} – 1$. Since this is one less than a power of two, it’s a Mersenne number. Why the first $31$ terms are all factors of the 32nd term is difficult to summarise here but there is a thread on StackExchange discussing what happens to the pattern after the $32nd$ term.

$4294967295$ has prime factorisation $3 \times 5 \times 17 \times 257 \times 65537$. These five prime factors are Fermat numbers – numbers of the form $2^{2^{n}}+1$ – in this case with $n = 0, 1, 2, 3$ and $4$. As of the time of writing these are the only known Fermat numbers which are also prime.

These patterns in the rows of the triangle are intriguing, and my own efforts to understand them have uncovered a few other interesting discoveries – notably, that while the 32nd term is not divisible by the 33rd, the 34th term is exactly 3 times the 33rd. The pairs of terms after that seem to alternate, as they do from the start of the sequence, between a non-integer ratio and a ratio of exactly 3, which I conjecture is a pattern that will continue.

Two welcome appearances

$e$ and $\pi$ are two of the most used transcendental numbers. The Swiss mathematician Leonhard Euler (1707-1783) connected them with the most beautiful equation, called Euler’s identity:

\[e^{i\pi}+1=0\]

There are many approximations connecting $e$, $\pi$ and other irrational numbers to be found here.

In 2012 Harlan J. Brothers proved that

\[\displaystyle\lim_{n\to \infty} \frac{\ \displaystyle\frac{s_{n+1}}{s_n}\ }{\displaystyle\frac{s_n}{s_{n-1}}}=e\]

where $s_n$ is the product of the numbers on row $n$ of Pascal’s triangle. The proof can be found on Cut the Knot, part of the wonderful website of Dr Ron Knott.

In 2007 Jonas Castillo Toloza discovered a connection between $\pi$ and the reciprocals of the triangular numbers (which can be found on one of the diagonals of Pascal’s triangle) by proving

\[\pi= 2 + \frac{1}{1} + \frac{1}{3} – \frac{1}{6} – \frac{1}{10} + \frac{1}{15} + \frac{1}{21} – \frac{1}{28} – \frac{1}{36} + \frac{1}{45} + \frac{1}{55} – \ldots\]

Three proofs are given on Cut the Knot.

Harmony in the triangle

The infinite sum of the reciprocals of the Natural numbers is called the harmonic series, $H_n$, where

$H_n = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} \ldots$

The series is divergent, but it crawls its way towards infinity, and takes $15092688622113788323693563264538101449859497$ terms just to pass a total of $100$.

The harmonic series can be used to create a version of Pascal’s triangle – the series itself is placed along the two leading diagonals, and the entries are then related by each being the difference of the fraction to its left, and the one diagonally above it and to its left. For example, $\frac{1}{30} = \frac{1}{5}-\frac{1}{6}$.

A triangle like Pascal's but with fractions in place of numbers; the rows read 1/1; 1/2 1/2; 1/3 1/6 1/3; 1/4 1/12 1/12 1/4; 1/5 1/20 1/30 1/20 1/5; 1/6 1/30 1/60 1/60 1/30 1/6

Dividing the first term in the $n^{th}$ row by every other term in that row creates the $n^{th}$ row of Pascal’s triangle. The table below shows the calculations for the $5^{th}$ row:

$\frac{1}{5}$	$\frac{1}{20}$	$\frac{1}{30}$	$\frac{1}{20}$	$\frac{1}{5}$
$\frac{1}{5}\div \frac{1}{5} =1$	$\frac{1}{5}\div \frac{1}{20} =4$	$\frac{1}{5}\div \frac{1}{30} =6$	$\frac{1}{5}\div \frac{1}{20} =4$	$\frac{1}{5}\div \frac{1}{5} =1$

In our next post, we’ll talk about probability and statistics in Pascal’s triangle, and consider some of Pascal’s other contributions.

Fibonacci, Lucas and the Golden Ratio in Pascal’s Triangle

By David Benjamin. Posted January 14, 2022

This is the third in a series of guest posts by David Benjamin, exploring the secrets of Pascal’s Triangle.

Leonardo Pisano (1170-1250), now universally known as Fibonacci, was born in Pisa, Italy, where he was also living at the time of his death. He was educated in north Africa as his father worked there, representing the merchants of the Republic of Pisa when they were trading in Bugia, now called Béjaïa, a Mediterranean port in Algeria.

Photograph of a Fibonacci statue. The statue stands on a cuboid of marble and depicts a man in a cape holding a piece of paper. — The Fibonacci statue by Giovanni Paganucci preserved in the monumental Cemetery of Pisa

Fibonacci returned to Pisa in about 1200 where he wrote a number of important books. His book Liber abaci introduced the Hindu-Arabic place-valued decimal system and the Arabic numerals we now use. Books and any copies had to be handwritten, as it predated the printing press. Fibonacci is now mostly remembered for introducing the Fibonacci numbers and sequence which appeared in the third section of Liber abaci as a problem about rabbits:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

The resulting sequence is $1, 1, 2, 3, 5, 8, 13, 21, 34, 55…$ (although Fibonacci did not include the first term in the book).

A diagram showing the growth of a rabbit population - one pair of rabbits in the first row, the same (mature) pair in the second row, and successive generations with 2, 3 and 5 pairs of rabbits. — Fibonacci’s rabbits

The ratio of successive terms converges on the Golden Ratio, $\phi$.

$\phi = \displaystyle\frac{1 + \sqrt5}{2} \approx 1.618033988749. . .$

$\phi$ is an irrational number and is the positive solution of the quadratic equation $x^2 – x – 1 = 0$ Hence, since $\phi$ is the root of an integer polynomial, it is not transcendental, unlike $\pi$.

\[ \frac{1}{1} = 1 \qquad \frac{2}{1} = 2 \qquad \frac{3}{2} = 1.5 \qquad \frac{5}{3} = 1.666 \ldots \qquad \frac{8}{5} = 1.6\]

\[ \frac{13}{8} = 1.625 \qquad \frac{21}{13} \approx 1.615384 \qquad \frac{34}{21} \approx 1.619047 \qquad \frac{55}{34} \approx 1.617647 \qquad \ldots \]

Indeed, convergence to $\phi$ remains true if we start with any pair of Natural numbers and follow the same pattern where any term after the second is the sum of the previous two terms.

Terms	Ratio
3	2.33333…
7	1.428571…
10	1.7
17	1.58823…
27	1.62962…
44	1.61363…
71	1.61971…
115	1.61739…
186	1.61827…
301	1.61794…
487	1.61806…
788	1.61802…
1275	1.61803…

Convergence when the first term is smaller than the second term

Terms	Ratio
5	0.6
3	2.66666…
8	1.375
11	1.72727…
19	1.57894…
30	1.63333…
49	1.61224…
79	1.62025…
128	1.61718…
207	1.61835…
335	1.61791…
542	1.61808…
877	1.61801…

Convergence when the first term is larger than the second term

Terms	Ratio
2	0.5
1	3
3	1.33333…
4	1.75
7	1.57142…
11	1.36363…
18	1.61111…
29	1.62068…
47	1.61702…
76	1.61842…
123	1.61788…
199	1.61809…
322	1.61801…

This is called the Lucas Sequence.

In Liber abaci, Fibonacci included other numeracy problems – on perfect numbers, the Chinese remainder theorem and on the sum of arithmetic and geometric series. He wrote a book on geometry, Practica geometriae, and perhaps his most impressive work was Liber quadratorum in which he included methods for finding Pythagorean triples. But it is for his sequence for which he is mainly remembered.

The Fibonacci Sequence in Pascal’s triangle

Finding out that the Fibonacci sequence can be found in Pascal’s triangle was a delight for me and I find it hard to think it is just a coincidence. To view Fibonacci’s sequence we can display the triangle as a right-angled triangle.

Pascal's triangle with columns left aligned, with arrows pointing diagonally up and to the right showing that these lines of numbers each sum to the Fibonacci numbers — Fibonacci’s sequence is hidden in the triangle

The Golden ratio in art, music and architecture

My interest in mathematics began when the film Donald Duck in Mathmagic Land was shown to our class in my first year at secondary school in Burnage, Manchester, England and as a teacher of mathematics I showed it in the lesson before Christmas to many year 7 groups.

The film illustrates how the Golden Rectangle has been used by artists and architects throughout history as well as connections between the golden ratio and music. The film mimics some of the novel Alice in Wonderland by Lewis Carroll, the pseudonym of the mathematician Charles Lutwidge Dodgson.

Further connections between the golden ratio and music can be found here and between the ratio and a Stradivarius violin here:

Image of a violin with sections labelled - the whole violin's height split at the top of the body (labelled a1 and a2), the height from the top of the next to the bottom of the side curve, split at the top of the body (labelled b1 and b2) and the distance from the top of the body to the bottom of the side curve, split at the top of the side curve (labelled c1 and c2).

The Lady Blunt shown above shows the measurements connected to the golden ratio:

\[ \frac{a_1 +a_2}{a_2}=\frac{a_2}{a_1}=\frac{b_2}{b_1}=\frac{b_2}{c_2}=\frac{c_2}{c_1}=\phi \]

Below is a geometric interpretation of the golden ratio and the golden rectangle:

An interval split into two parts in the Golden ratio, labelled 'a' and 'b', with the equation a/b = (a+b)/a = phi

A rectangle split into two parts - a square piece with sides length a, and a rectangle b by a (a and b are in the Golden ratio)

The Lucas numbers in Pascal’s triangle

Black and white photograph of Edouard Lucas, a very French and rectangular looking white man with short dark hair and an excellent moustache, wearing a white shirt and coat — François Édouard Anatole Lucas

The French mathematician François Édouard Anatole Lucas (1842-1891) served as an artillery officer in the Franco-Prussian War, and subsequently became professor of mathematics at the Lycée Saint Louis and then professor of mathematics at the Lycée Charlemagne, both in Paris. Lucas did a lot of work on number theory and was particularly interested in the Fibonacci sequence and devised the test for Mersenne primes which is still used today.

Lucas died of erysipelas (a bacterial skin infection) a few days after a freak accident. He was at a banquet when a fragment of a dropped plate flew up and cut his cheek.

His sequence, the Lucas sequence, begins with the pair of numbers $2$ and $1$ and its terms are generated in the same way as for the Fibonacci sequence.

$2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521…$

There are a number of connections between the Fibonacci sequence and the Lucas sequence. The $3^{rd}$ Lucas number is the sum of the $1^{st}$ and $3^{rd}$ Fibonacci number, the $4^{th}$ is the sum of the $2^{nd}$ and $4^{th}$, the $5^{th}$ is the sum of the $3^{rd}$ and $5^{th}$, the $6^{th}$ is the sum of the $4^{th}$ and $6^{th}$,…

Division of the Fibonacci terms $2n$ and $n$ beginning with the $2^{nd}$ term yields the Lucas terms

$2^{nd} \div 1^{st} = 1 \div 1 = 1$

$4^{th} \div 2^{nd} = 3 \div 1 = 3$

$6^{th} \div 3^{rd} = 8 \div 2 = 4$

$8^{th} \div 4^{th} = 21 \div 3 = 7$

$10^{th} \div 5^{th} = 55 \div 5 = 11$,..

With some manipulation of Pascal’s triangle and some basic arithmetic, we can find the Lucas numbers in the triangle. We begin by setting out the triangle as below and sum the columns to obtain the Fibonacci sequence

The Fibonacci numbers revealed as the column sums

We now multiply each Pascal number by its column number and divide by its row number, starting with row $1$ column $1$ and then sum the new entries in each column. The first few calculations are shown below:

The Lucas numbers revealed as the column sums

Generally, $\displaystyle\frac{\phi^n -(\frac{1}{\phi})^n}{\phi -(\frac{1}{\phi})}$ is the formula for the $n^{th}$ Fibonacci number, $\displaystyle\frac{\phi^n +(\frac{1}{\phi})^n}{\phi +(\frac{1}{\phi})}$ is the formula for the $n^{th}$ Lucas number and $\phi^n =\displaystyle \frac{L_n+ \sqrt5 \times F_n}{2}$, where $L_n$ and $F_n$ represent the $n^{th}$ Lucas and Fibonacci numbers respectively.

In the next part, we’ll consider some more connections between the triangle and particular numbers, and types of numbers.

Sequences in the triangle and the fourth dimension

By David Benjamin. Posted December 19, 2021

This is the second in a series of guest posts by David Benjamin, exploring the secrets of Pascal’s Triangle.

Sequences in the diagonals

There are many sequences of numbers to be found in Pascal’s triangle. The Natural numbers occur in the second diagonal, running in either direction, and the next two diagonals after that contain other important sequences:

Sequences in the diagonals

Pascal’s Triangle and its Secrets – Introduction

By David Benjamin. Posted December 9, 2021

This is the first in a series of guest posts by David Benjamin, exploring the secrets of Pascal’s Triangle.

The triangle of Natural numbers below contains the first seven rows of what is called Pascal’s triangle. Each row begins and ends with the number 1, and each of the remaining numbers, from the third row onwards, is the sum of the two numbers ‘above’: