A red bowling ball sits in front of 10 bowling pins in a triangle formation on a yellow background.
Math, Revealed
Bowling for Nobels
How a centuries-old math puzzle helped us see inside the human brain.
Each installment of “Math, Revealed” starts with an object, uncovers the math behind it and follows it to places you wouldn’t expect. Sign up here for the weekly Science Times newsletter.
June 30, 2025A red bowling ball sits in front of 10 bowling pins in a triangle formation on a yellow background.
As a kid, I loved going bowling with my dad.
The same 10 bowling pins, seen from a higher angle, are arranged in a triangular formation on a yellow background.
I was particularly fascinated by how the 10 pins were placed at the start of each frame, neatly arranged in an equilateral triangle with rows of one, two, three and four pins.
Ten is called a triangular number for this reason.
Three bowling pins, seen from above, arranged in a small triangle formation on a yellow background.
Smaller triangular numbers are also visible: The first two rows make 1 + 2 = 3.
Three more bowling pins added in a row above two other rows, creating a triangle of six pins, on a yellow background.
The first three rows make 1 + 2 + 3 = 6.
(So 3 is a triangular number, as is 6.)
Animation of more rows added to the six-pins triangle formation, creating an ever-growing triangle, on a yellow background.
Bigger triangular numbers are also possible, though you won’t see them at the bowling alley.
After 10, the next one is 1 + 2 + 3 + 4 + 5 = 15, familiar from the setup at the start of a game of pool.
The triangular numbers go on forever: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, …
A white bust of the philosopher Gottfried Wilhelm Leibniz, featuring a long curly wig and cravat, is centered against a yellow background.
They have been studied for centuries, but perhaps their most glorious day occurs in 1672.
Gottfried Wilhelm Leibniz, 26 years old, is an aspiring polymath eager to learn what the best mathematicians in Europe are working on. He turns to Christiaan Huygens, an established superstar, who gives him a problem that has come up in Huygens’s work on games of chance.
A bust of Gottfried Wilhelm Leibniz facing slightly to the left, on a yellow background.
The question is how to add up the reciprocals of all the triangular numbers:
1 + 1/3 + 1/6 + 1/10 + 1/15 + ⋯
A bust of Gottfried Wilhelm Leibniz moved to the left side of the photo, facing to the right.
At first glance, it seems impossible — the sum is endless. Yet Leibniz spots a remarkable pattern: Each number he wants to add is related to the difference of two consecutive fractions 1/1, 1/2, 1/3, 1/4, …, like this:
1 = 2(1/1 – 1/2)
1/3 = 2(1/2 – 1/3)
1/6 = 2(1/3 – 1/4)
1/10 = 2(1/4 – 1/5)
1/15 = 2(1/5 – 1/6)
and so on.
A bust of Gottfried Wilhelm Leibniz, now wearing a pink party hat, on a yellow background.
Next, Leibniz looks at the stack of equations above — infinitely many! — and adds them from top to bottom. On the left of the equal sign, he gets 1 + 1/3 + 1/6 + ⋯
1 = 2(1/1 – 1/2)
1/3 = 2(1/2 – 1/3)
1/6 = 2(1/3 – 1/4)
1/10 = 2(1/4 – 1/5)
1/15 = 2(1/5 – 1/6)
The bust of Gottfried Wilhelm Leibniz, moved to the right side of the photo, facing left and wearing a green party hat.
On the right, a tremendous amount of simplification occurs, because 1/2, 1/3, 1/4, 1/5 , and all the other simple fractions appear twice — once with a positive sign and once with a negative sign — so they cancel each other out.
1 = 2(1/1 – 1/2)
1/3 = 2(1/2 – 1/3)
1/6 = 2(1/3 – 1/4)
1/10 = 2(1/4 – 1/5)
1/15 = 2(1/5 – 1/6)
The bust of Gottfried Wilhelm Leibniz, moved to the right side of the photo, facing left, wearing a turquoise party hat.
The only exception is the first term, 1/1, which has no counterpart to cancel it. So, after the smoke clears from all the cancellations, the only term that survives on the right is 2 × ( 1/1 ) = 2, which must equal the sum of all the terms on the left.
1 = 2(1/1 – 1/2)
1/3 = 2(1/2 – 1/3)
1/6 = 2(1/3 – 1/4)
1/10 = 2(1/4 – 1/5)
1/15 = 2(1/5 – 1/6)
Two busts of Gottfried Wilhelm Leibniz on a yellow background.
Thus, Leibniz concludes:
1 + 1/3 + 1/6 + 1/10 + … = 2
An animation of colorful confetti falling around a single bust of Gottfried Wilhelm Leibnitz wearing a pink party hat on a yellow background.
Marvelous!
Eventually Leibniz generalizes the cancellation trick to a much wider class of infinite sums. He affectionately refers to this more powerful, systematic framework as “my calculus.” Nowadays we just call it “calculus.”
A medical model of a human head cut away to show the brain and sinuses. A strand of blue string passes from the outside of the head through the brain and comes out the other side. On the left side of the head the strand is thicker than on the right side.
In the centuries since, calculus has helped humanity solve vital problems in science, technology and medicine.
Take CT scans. When a standard X-ray beam is sent through a patient’s head, it gets partially absorbed by bone, brain matter and various soft tissues — each dimming its intensity by a different amount.
The detector on the far side records the total dimming, but not which tissues were responsible or where they were. The result: a blurry shadow with no internal detail.
A medical model of a human head with multiple strings of different colors coming from a single point on the left side and passing through the head. As the strings emerge on the right side of the head, they become thinner.
A CT scan does more.
In the late 1960s, Godfrey Hounsfield, a British engineer, had a bold idea: Fire thousands of beams through the skull from different angles, record the results and use math to reconstruct the interior.
The detector records a total dimming for each beam — and each total is different depending on the beam’s angle and what it encountered.
A medical model of the human head rotates around until the cross section showing the brain faces forward.
Whereas Leibniz’s task was to find the sum of infinitely many known terms, Hounsfield wondered: Could the process be run in reverse? With enough total dimmings, from enough directions, could we work backward to deduce the unknown absorption at each point along the many beams — and use that to see inside the brain?
A close-up into the brain of a medical model of a human head.
Most radiologists thought the idea was crazy. But one doctor was willing to listen. He handed Hounsfield a jar containing a human brain with a tumor in it and challenged him to image it with his prototype scanner.
Hounsfield soon brought back images of the brain that pinpointed the tumor — and areas of bleeding within it.
In 1979, Hounsfield won the Nobel Prize in Physiology or Medicine for his role in the development of computer-assisted tomography. He shared it with Allan Cormack, a physicist who had worked out the mathematical theory for it years earlier, all based on calculus and its offshoots.
A red bowling ball at the center of 10 scattered and falling pins on a yellow background.
But it was triangular numbers that got the ball rolling in the first place.