What is Pascal’s triangle?

How to make it

How to build Pascal’s triangle

Start with a 1 at the top. Every row after that starts and ends with 1, and each number in between is the two numbers above it added together. MathWorld puts it the same way: each row is “obtained by adding the two entries diagonally above.” Press Build row by row above and watch it happen.

1. Row 0 1
2. Row 1 1 1
3. Row 2 1 2 1
4. Row 3 1 3 3 1
5. Row 4 1 4 6 4 1
6. Row 5 1 5 10 10 5 1
7. Row 6 1 6 15 20 15 6 1

The top row is row 0, and the first number in each row is place 0. Counting from 0 is what makes the formula work. In row 6, the 20 sits at place 3 because 10 + 10 = 20 from the two numbers above it.

You do not need the rows above to find a number. The number at row n, place r is written C(n, r), said “n choose r,” and MathWorld identifies the entries as exactly these binomial coefficients. The formula is C(n, r) = n! / (r! × (n − r)!), where n! means 1 × 2 × 3 × … × n. Check it on the 20: 6! = 720, 3! = 6, and 720 / (6 × 6) = 20. Tap any number in the triangle and the readout does this sum for you.

Why the word binomial? Multiply out (x + y) to any power and the numbers in front of each term are a row of the triangle. (x + y)² = 1x² + 2xy + 1y², which is row 2. (x + y)³ = 1x³ + 3x²y + 3xy² + 1y³, which is row 3.

The patterns

What hides inside the triangle

Each button above lights up one of these. All of them are proven, not just noticed, and you can check every one by adding, like the other math curiosities on this site.

Odd and even: the Sierpinski triangle

Colour the odd numbers and leave the even ones dark, and a fractal appears: triangles inside triangles, forever. MathWorld: “Pascal’s triangle (mod 2) turns out to be equivalent to the Sierpiński sieve.” Push the slider to 64 rows and the picture repeats itself at 2, 4, 8, 16 and 32 rows.

Remainders: a new pattern for every number

Odd and even is just the remainder after dividing by 2. Pick any other number from 2 to 9 and each remainder gets its own colour. The zeros, the numbers that divide exactly, always gather into upside-down triangles. For 3, 5 and 7 the picture repeats itself at 3, 9 and 27 rows, at 5 and 25, and at 7 and 49. That repeat is our own check of the math, which you can see by sliding to 64 rows.

Fibonacci on the shallow diagonals

Add up the numbers along a shallow diagonal, going up one row and over one place each time, and you get the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13. MathWorld: “The shallow diagonals of Pascal’s triangle sum to Fibonacci numbers.” For example 1 + 3 + 1 = 5 and 1 + 4 + 3 = 8. Tap any number in Fibonacci mode to light its diagonal. And their last digits repeat every 60 numbers, as the Fibonacci repeat page shows.

Row sums: powers of 2

Every row adds up to double the row above: 1, 2, 4, 8, 16. Row n adds up to 2ⁿ. That is MathWorld’s rule, and it makes sense: each number is passed down twice, once to the left and once to the right.

The hockey stick

Start at a 1 on the edge and add down a diagonal. The total is the number one step down and to the side of the last one, and the line plus that turn makes the shape of a hockey stick. 1 + 3 + 6 + 10 = 20, and the 20 sits just below and beside the 10. MathWorld calls this “the Christmas stocking theorem, also known as the hockey stick theorem.” Tap any number in hockey stick mode and the page draws the stick that adds up to it.

Triangular numbers

The third diagonal runs 1, 3, 6, 10, 15, 21: the triangular numbers, the counts of dots you can stack into a triangle. MathWorld: “Pascal’s triangle contains the figurate numbers along its diagonals.” The nth triangular number is n(n + 1) / 2, which is C(n + 1, 2). The diagonal next to it is just 1, 2, 3, 4, the counting numbers.

Powers of 11

Read rows 0 to 4 as one number each and you get 1, 11, 121, 1331, 14641: the powers of 11. Harvey Mudd College’s Math Fun Facts explains why: “you can use them as a quick way to calculate the powers of 11, since 11=10+1.” From row 5 the numbers have two digits and you have to carry: 1 5 10 10 5 1 becomes 161051, which is 11⁵.

What it is for

Counting choices and coin flips

Pascal’s triangle answers “how many ways?” questions. C(n, r) is the number of ways to choose r things from n when the order does not matter.

Choosing. Five friends, and you can take two to the cinema. How many different pairs could you take? Look at row 5, place 2: 10. Write them out (AB, AC, AD, AE, BC, BD, BE, CD, CE, DE) and there are exactly 10.

Coin flips. Flip 4 coins. Row 4 is 1 4 6 4 1: there is 1 way to get no heads, 4 ways to get one head, 6 ways to get two, 4 ways to get three and 1 way to get four. That is 16 outcomes in all, which is 2⁴, the row sum. So the chance of exactly two heads is 6 out of 16, or 37.5%.

That is also why Pascal wrote it up. Cambridge University Library notes that he wrote his treatise after his 1654 letters “with Pierre de Fermat of Toulouse about some problems in calculating the odds in games of chance.”

History

Pascal was not first

The triangle has many names. MathWorld: it is “known as the Yang Hui triangle in China, the Khayyam triangle in Persia, and Tartaglia’s triangle in Italy.”

China. Jia Xian, who lived from about 1010 to about 1070, gave “a table of the resulting binomial coefficients in the form of Pascal’s triangle” up to row 6, according to the MacTutor history archive. His work is known to us through Yang Hui. “In 1261 Yang wrote the Xiangjie jiuzhang suanfa,” with the triangle up to the sixth row, “saying that he learnt it from Jia Xian’s treatise.” Zhu Shijie’s “Jade mirror of the four unknowns,” published in 1303, takes it to the eighth power.

Persia. MacTutor, writing about Omar Khayyam: “In the lost work Khayyam discusses the Pascal triangle but he was not the first to do so since al-Karaji discussed the Pascal triangle before this date.”

India, and a disputed reading. The oldest claim goes back to Piṅgala, whom Jayant Shah of Northeastern University dates to the 2nd century BCE. Shah: “It is almost universally accepted on the authority of Halāyudha (10th century, C.E.) that Piṅgala’s last sūtra, ‘pare pūrṅaṃ iti’, implies the construction of meru prastāra (what is now known as ‘Pascal’s triangle’).” The sūtra itself is three words, the reading comes from a commentator more than a thousand years later, and Shah notes that scholars doubt it. So this page marks it as disputed.

Italy. MathWorld: “Niccolò Tartaglia ... published the first six rows of the triangle in 1556.”

France. Pascal wrote his Traité du triangle arithmétique “probably in August” 1654. Cambridge’s record adds: “Not published until 1665,” three years after he died in 1662. He did not invent the triangle, but Cambridge calls his treatise “the justification for calling the arithmetical triangle ‘Pascal’s triangle’.”