40 Fun Facts About Pascal's Triangle

Which casino game is Blaise Pascal, of Pascal's triangle fame, said to have devised while chasing perpetual motion?

Roulette, many historians believe, began with a primitive wheel Pascal introduced in the 17th century during his hunt for a machine that never stops. The name means 'little wheel'.

Which chip-dropping peg-board game on TV's The Price Is Right spreads its chips by the odds in Pascal's triangle?

Plinko chips bounce left or right at every peg, so the routes to each slot follow the numbers of Pascal's triangle. That is why the middle slots catch far more chips than the edges.

Which carol's gift totals, running 1, 4, 10, 20 up to 364, trace a diagonal of Pascal's triangle?

Twelve Days of Christmas comes to 364 presents if every verse is delivered in full, one short of a gift for each day of the year. The running totals are the tetrahedral numbers.

Entry 5 of row 52 in Pascal's triangle is 2,598,960, the number of possible hands in which card game?

Poker deals five cards from a deck of 52, and row 52 of the triangle lists how many ways each size of hand can be dealt. Only 4 of those 2,598,960 hands are a royal flush.

Which sports item names the Pascal's triangle pattern where a diagonal run sums to the number below its end?

Hockey stick is the nickname because the summed diagonal forms the long shaft and the total, one step off to the side, forms the blade. For example, 1 + 3 + 6 + 10 lands on 20.

Summing the shallow diagonals of Pascal's triangle gives 1, 1, 2, 3, 5, 8, the start of which famous sequence?

Fibonacci numbers, where each term is the sum of the two before, hide in those slanted totals. They are named after a medieval Italian who used them in 1202 to model a growing population of rabbits.

Blaise Pascal, the namesake of Pascal's triangle, also gives his name to the SI unit of which physical quantity?

Pressure is measured in pascals because he had a barometer carried up the Puy de Dôme mountain, where the mercury fell as the air thinned. Poor health kept him from making the climb himself.

Each inner number in Pascal's triangle comes from doing what to the two numbers directly above it?

Adding them is the whole recipe: the 1 and 5 in one row make the 6 beneath, and the 5 and 10 beside them make 15. Every pattern in the triangle grows from that single rule.

Which religious symbol names a Pascal's triangle theorem about the six numbers surrounding any entry?

Star of David is the shape traced by those six neighbours, and the theorem says its two triangles of three numbers share the same greatest common divisor. Their products match too.

Computer scientist Niklaus Wirth honoured the man behind Pascal's triangle in 1970 by naming what after him?

A programming language called Pascal was Wirth's tribute, built as a small, tidy tool for teaching good habits. It became a standard language for teaching programming around the world.

Pascal's triangle takes its Western name from Blaise Pascal, a 17th-century thinker from which country?

France was home to Pascal, but the triangle was no discovery of his: scholars in India, Persia, China, Germany and Italy had all studied it before him.

In China, Pascal's triangle is named after which 13th-century mathematician, who drew it in a book of 1261?

Yang Hui gave the credit to an earlier scholar, Jia Xian, who had used the triangle in the 11th century to extract square and cube roots, some 600 years before Pascal.

In Iran, Pascal's triangle is named after which poet-astronomer who helped reform the Persian calendar?

Omar Khayyam used the triangle's numbers to extract roots of any degree, and his fame carried the pattern across the Islamic world. Western readers know him better for the Rubaiyat.

In Italy, Pascal's triangle is named after which 16th-century algebraist, whose nickname means 'the stammerer'?

Tartaglia got his nickname after a soldier's sabre slashed his jaw when he was a boy, leaving him with a stammer. He printed six rows of the triangle in 1556, a century before Pascal's treatise.

Which ancient Indian scholar's study of poetic metres, over 2,000 years old, holds the seed of Pascal's triangle?

Pingala was tallying the ways long and short syllables can fill a line of Sanskrit verse, and a 10th-century commentator laid his method out as the triangle. Poetry got there before algebra.

Medieval Indian scholars called the pattern now known as Pascal's triangle the staircase to which sacred mountain?

Mount Meru is the five-peaked centre of the universe in Hindu, Jain and Buddhist tradition. The 10th-century writer Halayudha called the stepped number pattern Meru-prastara.

Which English scientist extended the expansion rule behind Pascal's triangle to fractional and negative powers?

Isaac Newton found in 1665 that the pattern carries on as an endless series when the power is not a whole number. He was in his early twenties and about to begin developing calculus.

Blaise Pascal, of Pascal's triangle, solved a famous gambling puzzle in 1654 letters with which Toulouse lawyer?

Pierre de Fermat and Pascal worked out how to split the stakes fairly when a game is cut short. Pascal went on to solve the general case with the numbers of his triangle.

Pascal's triangle helped Blaise Pascal found which branch of mathematics, sparked by a gambler's question in 1654?

Probability took shape as Pascal used the triangle's rows to tally how an interrupted game might have finished. His treatise on the triangle applied the same numbers to games of chance.

Years before his work on Pascal's triangle, the teenage Blaise Pascal built what to help his tax-collector father?

A mechanical calculator, later called the Pascaline, handled sums and subtractions with a turn of its dials, sparing his father endless tax arithmetic. Pascal went through some fifty prototypes to perfect it.

The third diagonal of Pascal's triangle runs 1, 3, 6, 10, like racks of bowling pins, listing which family of numbers?

Triangular numbers are the totals you get by stacking rows of 1, 2, 3, 4 and so on, like the ten pins at the end of a bowling lane. The next in line, 15, is a full rack of pool balls.

Just inside the border of 1s, the second diagonal of Pascal's triangle lists which familiar set in order?

Counting numbers 1, 2, 3, 4 and so on run down that diagonal, so the second entry of any row is simply the row's own number. Row 9, for instance, begins 1, 9.

Alternating plus and minus signs along any row of Pascal's triangle below the top gives which total?

0 is the result every time: 1 − 6 + 15 − 20 + 15 − 6 + 1, for example, cancels exactly. The positive and negative halves of each row always balance.

Moving down Pascal's triangle one row at a time, what happens to the total of the numbers in each row?

It doubles every time, so the row totals run 1, 2, 4, 8, 16, because each number is used twice to build the row below: once to its left and once to its right.

Read as a single number, the Pascal's triangle row 1, 3, 3, 1 becomes 1,331, which is the cube of what number?

11 is the base: the rows 1, then 1 1, then 1 2 1 and 1 3 3 1 spell out 1, 11, 121 and 1,331. After the first five rows the trick needs carrying, because entries reach two digits.

Like every row of Pascal's triangle, the row 1, 6, 15, 20, 15, 6, 1 is which type of sequence?

Palindromic rows read the same backwards as forwards, because choosing 2 items from 6 leaves 4 behind, so both choices can be made in the same 15 ways.

In Pascal's triangle, row 7's inner numbers (7, 21, 35) are all multiples of 7 because 7 is what?

Prime row numbers always behave this way, since nothing in the formula's denominator can cancel the row number on top. Row 6 fails the test: its middle entry, 20, is no multiple of 6.

Pascal's triangle shows there are 10 ways to pick 2 friends from 5, a tally where order is ignored, called what?

Combinations ignore order, so picking Ann then Ben is the same as Ben then Ann. Entry k of row n gives the tally, which is why mathematicians read it aloud as 'n choose k'.

In Pascal's triangle, which number fills the gap in the row that runs 1, 4, blank, 4, 1?

6 fills the gap, the sum of the two 3s sitting above it in the row 1, 3, 3, 1. The finished row gives the coefficients of (x + y) raised to the fourth power.

Pascal's triangle supplies the coefficients for expanding (x + y) to any whole power, a result known as which theorem?

Binomial means 'two terms', and the theorem turns a slog of multiplying brackets into reading off a row: (x + y) cubed is x³ + 3x²y + 3xy² + y³, straight from the row 1, 3, 3, 1.

Row 49 of Pascal's triangle contains 13,983,816, the ways to pick six numbers from 49 that set the odds in what?

A lottery draw of six balls from 49 has exactly that many possible tickets, so a single ticket's chance of the jackpot is about one in 14 million.

The three-dimensional cousin of Pascal's triangle, which stacks its numbers in layers, is known as Pascal's what?

Pyramid is the name because each three-sided layer rests on a larger one. Its numbers give the coefficients for expanding three terms at once, such as (x + y + z) raised to a power.

Shading only the odd numbers in Pascal's triangle reveals a famous fractal named after which Polish mathematician?

Wacław Sierpiński described the triangle-within-triangle pattern in 1915, and the odd entries of Pascal's triangle trace it ever more exactly as the rows pile up. The even entries form the holes.

Which Victorian scientist invented the bean machine, a peg board whose falling beads follow Pascal's triangle?

Francis Galton, a half-cousin of Charles Darwin, unveiled the board at an 1874 lecture to show how many small random bounces pile up into a bell curve. The device is also called a quincunx.

Charted as bars, a long row of Pascal's triangle approaches which bell-shaped distribution from statistics?

Normal curves emerge because each entry tallies the routes to one outcome of many 50-50 steps, and middling outcomes have the most routes. The longer the row, the smoother the bell.

Ratios built from the products of each row of Pascal's triangle close in on which famous constant, roughly 2.718?

Euler's number, written e, appears when the product of each row is compared with those of its neighbours, a link set out in a 2012 paper by H. J. Brothers. The same constant governs compound interest.

Every whole number above 1 appears in Pascal's triangle at least twice, except which one, which appears only once?

2 sits alone in the middle of the third row and never returns. Every larger number n shows up at least twice, because it is the second and the second-to-last entry of row n.

Apart from 1, which number is the only one known to appear eight times in Pascal's triangle?

3,003 turns up eight times, in rows 14, 15, 78 and 3,003 itself, twice in each. Whether any other number manages eight appearances is still an open question.

Which Rubik's Cube pioneer proposed in 1971 that numbers above 1 can repeat only so often in Pascal's triangle?

Singmaster wrote the notation still used to record Cube moves and one of the first published solutions. His conjecture about the triangle remains unproven after more than fifty years.

Which German astronomer printed Pascal's triangle at the front of his 1527 book on business calculations?

Petrus Apianus, also known as Peter Apian, put the triangle on show 96 years before Pascal was born. He was better known for his books on the heavens, and a crater on the Moon carries his name.