Pascal's Triangle
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40 free Pascal's Triangle trivia questions with answers — science & nature quiz, new questions added Oct 2026.
Pascal's triangle looks like a classroom exercise: start with a 1, then build each new row from the one above. Yet this simple stack of numbers hides a remarkable amount of mathematics, from well-known sequences to a famous fractal, and it turns up in places as unlikely as game shows, card tables and a seasonal song. This quiz turns the best fun facts about Pascal's triangle into a game. Expect questions on how the rows are built, the patterns that run along the diagonals, the puzzles you can solve by reading off a single entry, and the scholars across Asia and Europe who studied the triangle long before the man it is named after. It suits students meeting the triangle for the first time, teachers who want a ready-made class activity, and anyone who enjoys recreational mathematics. Every question comes with its answer and a short explanation, so you will leave knowing why each pattern works.
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Q 01Which casino game is Blaise Pascal, of Pascal's triangle fame, said to have devised while chasing perpetual motion?
Roulette
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'.
Q 02Which chip-dropping peg-board game on TV's The Price Is Right spreads its chips by the odds in Pascal's triangle?
Plinko
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.
Q 03Which carol's gift totals, running 1, 4, 10, 20 up to 364, trace a diagonal of Pascal's triangle?
Twelve Days of Christmas
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.
Q 04Entry 5 of row 52 in Pascal's triangle is 2,598,960, the number of possible hands in which card game?
Poker
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.
Q 05Which sports item names the Pascal's triangle pattern where a diagonal run sums to the number below its end?
Hockey stick
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.
Q 06Summing the shallow diagonals of Pascal's triangle gives 1, 1, 2, 3, 5, 8, the start of which famous sequence?
Fibonacci numbers
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.
Q 07Blaise Pascal, the namesake of Pascal's triangle, also gives his name to the SI unit of which physical quantity?
Pressure
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.
Q 08Each inner number in Pascal's triangle comes from doing what to the two numbers directly above it?
Adding them
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.
Q 09Which religious symbol names a Pascal's triangle theorem about the six numbers surrounding any entry?
Star of David
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.
Q 10Computer scientist Niklaus Wirth honoured the man behind Pascal's triangle in 1970 by naming what after him?
A programming language
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.
Q 11Pascal's triangle takes its Western name from Blaise Pascal, a 17th-century thinker from which country?
France
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.
Q 12In China, Pascal's triangle is named after which 13th-century mathematician, who drew it in a book of 1261?
Yang Hui
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.
Q 13In Iran, Pascal's triangle is named after which poet-astronomer who helped reform the Persian calendar?
Omar Khayyam
Q 21The third diagonal of Pascal's triangle runs 1, 3, 6, 10, like racks of bowling pins, listing which family of numbers?
Triangular
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.
Q 22Just inside the border of 1s, the second diagonal of Pascal's triangle lists which familiar set in order?
Counting numbers
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.
Q 23Alternating plus and minus signs along any row of Pascal's triangle below the top gives which total?
0
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.
Q 14In Italy, Pascal's triangle is named after which 16th-century algebraist, whose nickname means 'the stammerer'?
Tartaglia
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.
Q 15Which ancient Indian scholar's study of poetic metres, over 2,000 years old, holds the seed of Pascal's triangle?
Pingala
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.
Q 16Medieval Indian scholars called the pattern now known as Pascal's triangle the staircase to which sacred mountain?
Mount Meru
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.
Q 17Which English scientist extended the expansion rule behind Pascal's triangle to fractional and negative powers?
Isaac Newton
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.
Q 18Blaise Pascal, of Pascal's triangle, solved a famous gambling puzzle in 1654 letters with which Toulouse lawyer?
Pierre de Fermat
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.
Q 19Pascal's triangle helped Blaise Pascal found which branch of mathematics, sparked by a gambler's question in 1654?
Probability
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.
Q 20Years before his work on Pascal's triangle, the teenage Blaise Pascal built what to help his tax-collector father?
A mechanical calculator
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.
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.
Q 24Moving down Pascal's triangle one row at a time, what happens to the total of the numbers in each row?
It doubles
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.
Q 25Read as a single number, the Pascal's triangle row 1, 3, 3, 1 becomes 1,331, which is the cube of what number?
11
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.
Q 26Like every row of Pascal's triangle, the row 1, 6, 15, 20, 15, 6, 1 is which type of sequence?
Palindromic
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.
Q 27In Pascal's triangle, row 7's inner numbers (7, 21, 35) are all multiples of 7 because 7 is what?
Prime
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.
Q 28Pascal's triangle shows there are 10 ways to pick 2 friends from 5, a tally where order is ignored, called what?
Combinations
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'.
Q 29In Pascal's triangle, which number fills the gap in the row that runs 1, 4, blank, 4, 1?
6
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.
Q 30Pascal's triangle supplies the coefficients for expanding (x + y) to any whole power, a result known as which theorem?
Binomial
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.