Puzzles From the History of Math
Math isn't new. For thousands of years, people have been challenging each other with riddles. Here are some classics.
Long before there were textbooks, classrooms, or exams, there were puzzles.
Ancient Babylonians carved problems onto clay tablets. Egyptian scribes wrote them on papyrus. Greek philosophers debated them in agoras, and Chinese scholars included them in manuals for civil servants. Mathematics didn't start as a rigid set of rules—it started as a way to solve problems and, crucially, to show off intellectual prowess.
Today, we're taking a time machine back to visit some of these historical brain-teasers. The amazing thing? They're just as tricky today as they were 2,000 years ago. The technology has changed, but the logic hasn't.
Ancient Egyptian Fractions
The Egyptians built the pyramids, but they calculated them using a strange system: they loved unit fractions.
A unit fraction has a 1 on top: 1/2, 1/3, 1/4. The Egyptians insisted on writing almost every fraction as a sum of different unit fractions. They wouldn't write 3/4. They'd write 1/2 + 1/4.
This sounds clumsy, but it was actually practical for dividing resources. If you have 3 loaves of bread and 4 people, slicing every loaf into quarters creates 12 crumbs. But giving everyone half a loaf (using 2 loaves), then quartering the last loaf, is much more efficient.
The Rhind Papyrus (c. 1650 BC) contains 84 problems, many about dividing bread and beer. Problem 79 is a nursery rhyme ancestor: "7 houses; in each house 7 cats; each cat kills 7 mice; each mouse ate 7 grains of barley..." It's a geometric progression puzzle that predates the "As I was going to St. Ives" riddle by three millennia.
Greek Paradoxes and Logic
The Greeks loved geometry, but they also loved to argue. Zeno of Elea (c. 450 BC) created paradoxes designed to prove that motion was impossible.
His most famous is Achilles and the Tortoise. Achilles gives the tortoise a head start. By the time Achilles reaches the tortoise's starting point, the tortoise has moved a little further. By the time Achilles covers that distance, the tortoise has moved again. Zeno argued that since there are infinite catch-up steps, Achilles can never pass the tortoise.
Of course, in reality, he does. But it took 2,000 years for calculus to mathematically explain how an infinite sum of time intervals could add up to a finite result. Zeno wasn't just trolling; he was forcing people to think deeply about infinity.
Medieval Merchants and Alcuin's River
In the Middle Ages, math was often practical—or at least, pretended to be.
Alcuin of York (c. 735–804 AD) wrote a collection called Propositions for Sharpening the Youth. It contains the first known appearance of the River Crossing Puzzle: A man has a wolf, a goat, and a cabbage. He can take only one across the river at a time. The wolf will eat the goat if left alone; the goat will eat the cabbage. How does he cross?
This puzzle (and its variations involving jealous husbands, missionaries, and cannibals) became a staple of logic training. It's not about arithmetic; it's about constraints and state-space search—concepts that are central to modern computer science.
Later, Fibonacci (c. 1202) introduced the Hindu-Arabic numeral system (0-9) to Europe in his book Liber Abaci. Buried inside was a problem about rabbits reproducing, which gave rise to the famous Fibonacci sequence (1, 1, 2, 3, 5, 8...). He wasn't trying to describe nature—he was just setting a puzzle about population growth.
Try These
Ready to test yourself against the ancients? Here are three historical puzzles, rephrased for modern readers.
Puzzle 1: The Diophantus Riddle (Ancient Greece, c. 250 AD)
Diophantus was the "father of algebra," and his tombstone supposedly bore this riddle describing his life:
"God granted him to be a boy for the sixth part of his life, and adding a twelfth part to this, he clothed his cheeks with down. He lit him the light of wedlock after a seventh part, and five years after his marriage he granted him a son. Alas! late-born wretched child; after attaining the measure of half his father's life, chill Fate took him. After consoling his grief with the science of numbers for four years more, he ended his life."
How old was Diophantus when he died?
Hint: Let x be his total age. The sum of the fractions of his life plus the specific years (5 and 4) must equal x.
Puzzle 2: The 100 Fowls Problem (China, 5th Century)
A rooster costs 5 coins, a hen costs 3 coins, and 3 chicks cost 1 coin.
You have 100 coins and you want to buy exactly 100 birds.
How many of each should you buy? (There's more than one answer, but try to find at least one.)
Hint: You need two equations (one for cost, one for count) with three variables. This is a "Diophantine equation," meaning we only care about whole-number solutions.
Puzzle 3: The Spider and the Fly (Dudeney, 1903)
A spider sits on one end wall of a rectangular room (30ft long, 12ft wide, 12ft high). It is 1ft from the ceiling and centered horizontally.
A fly sits on the opposite end wall, 1ft from the floor and centered horizontally.
What is the shortest distance the spider can crawl along the walls/ceiling/floor to catch the fly?
(Warning: It's not just crawling down, along the floor, and up.)
Hint: Imagine unfolding the room like a cardboard box so the walls lay flat. Then draw a straight line.
Final Thought
We tend to think of math as a ladder we climb in school—fractions, then algebra, then calculus. But history shows it's more like a sprawling forest. Different cultures explored different paths, finding puzzles that pleased them.
When you solve these riddles, you're not just doing homework. You're connecting with a curious mind from 500, 1,000, or 2,000 years ago who looked at the world and asked: "I wonder..."
The answers change (we don't use Egyptian fractions much anymore), but the joy of the "aha!" moment is timeless.
Do you have a favourite classic riddle? Ever tried to solve a problem using Roman numerals? (Spoiler: don't.) Share your historical math thoughts in the comments!