385 puzzles puzzles, each one credited to the book, paper or primary source it comes from. Free to read and swipe on Savvy.
A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost? Answer fast — then check yourself.
— Shane Frederick, Cognitive Reflection and Decision Making
A patch of lily pads doubles in size every day. It takes 48 days to cover the entire lake. How long does it take to cover half the lake?
— Shane Frederick, Cognitive Reflection and Decision Making
Four cards lie on a table showing E, K, 4 and 7. Each has a letter on one side and a number on the other. Rule: if a card has a vowel on one side, it has an even number on the othe…
— Peter Wason, Reasoning about a rule
A car is behind one of three doors; goats behind the other two. You pick door 1. The host — who knows where the car is — opens door 3, revealing a goat, and offers you the chance t…
— Steve Selvin; Marilyn vos Savant, A Problem in Probability (letter); popularised in Parade's 'Ask Marilyn'
How many people do you need in a room before it's more likely than not that two of them share a birthday?
— Richard von Mises, Über Aufteilungs- und Besetzungswahrscheinlichkeiten (the classic 'birthday problem')
Two doors: one leads to freedom, one to doom. Two guards: one always tells the truth, one always lies — you don't know which is which. You may ask one guard a single question. What…
— Raymond Smullyan, What Is the Name of This Book? (knights-and-knaves puzzles)
You must ferry a wolf, a goat and a cabbage across a river. The boat holds only you and one passenger. Left alone together, the wolf eats the goat, and the goat eats the cabbage. H…
— Alcuin of York, Propositiones ad Acuendos Juvenes ('Problems to Sharpen the Young')
You have two ropes and a lighter. Each rope takes exactly one hour to burn, but they burn unevenly — half the rope might take five minutes or fifty. How do you measure exactly 45 m…
— Peter Winkler, Mathematical Puzzles: A Connoisseur's Collection
The old city of Königsberg had seven bridges linking two islands and both riverbanks. Citizens tried for years to find a walk crossing every bridge exactly once. Can it be done?
— Leonhard Euler, Solutio problematis ad geometriam situs pertinentis
Cut an equilateral triangle into as few pieces as possible so they can be rearranged into a perfect square. Henry Dudeney's solution goes further than the challenge asks — his four…
— Henry Dudeney, The Canterbury Puzzles
The 15 puzzle has fifteen numbered tiles sliding in a 4x4 frame. In 1880, puzzle-maker Sam Loyd offered $1,000 to anyone who could take a puzzle with tiles 14 and 15 swapped and sl…
— Sam Loyd (popularizer); Noyes Chapman (inventor), The 15 Puzzle
You have 12 coins that look identical, but one is counterfeit and weighs slightly differently — you don't know if it's heavier or lighter. Using a balance scale only three times, h…
— Martin Gardner (popularizer), Mathematical Games column
Lewis Carroll imagined the Tortoise refusing to accept a simple logical conclusion from Achilles unless Achilles also wrote down the rule that lets you draw it — and then refusing …
— Lewis Carroll, What the Tortoise Said to Achilles
If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?
— Shane Frederick, Cognitive Reflection and Decision Making
Take 6 army regiments, each sending 6 officers of 6 different ranks. Can all 36 officers be arranged in a 6x6 square so that every row and column contains one officer of each rank …
— Leonhard Euler, Recherches sur une nouvelle espèce de quarrés magiques
A dying man leaves his three sons 30 glass flasks: 10 full of oil, 10 half-full, and 10 empty. Divide both the oil and the flasks so each son receives exactly the same amount of oi…
— Alcuin of York (attrib.), Propositiones ad Acuendos Juvenes
In a rectangular room 30 feet long, 12 feet wide and 12 feet high, a spider sits on one end wall, a foot below the ceiling and centred. A fly sits on the opposite wall, a foot abov…
— Henry Ernest Dudeney, The Canterbury Puzzles
A card shows a circle of Chinese warriors printed around a rotating disc set into a rectangular background of the Earth. Rotate the disc slightly one way and you can count thirteen…
— Sam Loyd, edited by Martin Gardner, Mathematical Puzzles of Sam Loyd
A judge tells a prisoner he'll be hanged at noon on one weekday next week, and that he won't know which day until the executioner knocks. The prisoner reasons it can't be Friday (h…
— Martin Gardner, The Unexpected Hanging and Other Mathematical Diversions
A suitor must choose between a gold casket and a silver casket; one contains the treasure. The gold casket reads: 'The treasure is in this casket.' The silver casket reads: 'Exactl…
— Raymond Smullyan, The Lady or the Tiger? And Other Logic Puzzles
Can a knight, moving only in its usual L-shape, tour an entire 8x8 chessboard, landing on every square exactly once? Indian and Arabic scholars had explored the pattern for centuri…
— Leonhard Euler, Solution d'une question curieuse qui ne paroit soumise à aucune analyse
During the Jewish–Roman War in 67 AD, the historian Flavius Josephus and 40 companions trapped in a cave reportedly agreed that rather than surrender, they'd stand in a circle and …
— Flavius Josephus, The Jewish War
Five shipwrecked sailors gather a pile of coconuts to split in the morning. In the night, each sailor in turn wakes, divides the pile into five equal piles with exactly one coconut…
— Ben Ames Williams, Coconuts (short story)
Three gods, A, B and C, are called (in some order) True, False and Random — True always answers truthfully, False always answers falsely, and Random answers truthfully or falsely a…
— George Boolos, The Hardest Logic Puzzle Ever
Two trains, 60 miles apart, race toward each other on the same track — one at 20 mph, the other at 30 mph. The instant they start, a tireless fly departs from the front of one trai…
— Traditional; popularised via a John von Neumann anecdote, Two Trains Puzzle
SEND + MORE = MONEY. Each letter represents a different digit 0–9 (no leading zeros), and the addition has to work exactly as written. Find the digits.
— Henry Dudeney, The Strand Magazine, Vol. 68
In one family, each daughter has exactly as many brothers as she has sisters, and each son has exactly twice as many sisters as he has brothers. How many sons and daughters are in …
— Boris Kordemsky, The Moscow Puzzles: 359 Mathematical Recreations
Three barbers, Allen, Brown, and Carr, share a shop that always has at least one of them minding it. Allen is so nervous he never leaves without Brown coming too. A logician tells …
— Lewis Carroll (Charles Lutwidge Dodgson), A Logical Paradox
You're holding one of two sealed envelopes and know one contains exactly twice as much money as the other. Before opening it, you reason: my envelope has some amount X, so the othe…
— Maurice Kraitchik, La Mathematique des Jeux (Mathematical Recreations)
A king's dungeon has two doors. Behind one waits a lady; behind the other, a hungry tiger. Door One reads: "IN THIS ROOM THERE IS A LADY, AND IN THE OTHER ROOM THERE IS A TIGER." D…
— Raymond M. Smullyan, The Lady or the Tiger? and Other Logic Puzzles
A ladder has 100 steps. One dove perches on the first step, two doves on the second, three on the third, one more bird on each successive step all the way up, until the hundredth s…
— Alcuin of York (attributed), Propositiones ad Acuendos Juvenes (Problems to Sharpen the Young)
A pile of objects is counted out in groups. Counted three at a time, two are left over. Counted five at a time, three are left over. Counted seven at a time, two are left over. Wha…
— Sun Zi (attributed; exact identity unknown), Sunzi Suanjing (The Mathematical Classic of Sun Zi)
Four travelers must cross a rickety bridge at night, sharing a single torch that must accompany every crossing; at most two people may cross at once, and a pair moves at the slower…
— Saul X. Levmore and Elizabeth Early Cook, Super Strategies for Puzzles and Games
Take a standard 8x8 chessboard and cut off two diagonally opposite corner squares, leaving 62 squares. You have 31 dominoes, each covering exactly two adjacent squares. Can you til…
— Max Black, Critical Thinking: An Introduction to Logic and Scientific Method
Five houses in a row are each painted a different colour, and owned by a person of a different nationality who drinks a different beverage, smokes a different brand, and keeps a di…
— Anonymous, Life International
Albert is told only the month of Cheryl's birthday, Bernard only the day. Albert says: 'I don't know when her birthday is, but I know Bernard doesn't know either.' Bernard replies:…
— Joseph Yeo Boon Wooi / SASMO, Singapore and Asian Schools Math Olympiad
Three houses each need a line running to three separate utility companies — water, gas and electricity. Can you draw all nine connecting lines on a flat sheet of paper so that no t…
— Henry Dudeney, Amusements in Mathematics
Change one letter at a time — never rearranging, adding or removing letters — so that every intermediate step is itself a real word, until CAT becomes DOG. What's the shortest chai…
— Lewis Carroll, Doublets: A Word-Puzzle
You have four cubes, each face painted in one of four colours differently arranged on every cube. Stack all four into a tower so that all four colours appear exactly once on each o…
— Frank Armbruster (1967 version); lineage traced to the 1900 Katzenjammer Puzzle by Frederick A. Schossow, Instant Insanity
An epitaph preserved in the Greek Anthology gives the mathematician Diophantus's life story entirely in fractions: a boyhood, a youth, a bachelorhood, a son born five years into ma…
— Compiled by Metrodorus; original epigram author unknown, Greek Anthology, Book XIV, Epigram 126
You have an 8-litre jug full of water, plus empty 5-litre and 3-litre jugs. Using only pouring between the three — no measuring lines, no guessing — split the water into two equal …
— W. W. Rouse Ball, Mathematical Recreations and Essays
Two black frogs and two white frogs sit on a strip of seven lily pads, black on the left, white on the right, one empty pad in the middle. Moving one frog at a time — sliding onto …
— Henry Ernest Dudeney, Amusements in Mathematics
A dying man leaves 960 shillings and a pregnant wife. His will says: if she bears a son, the son gets 9/12 of the estate and she gets 3/12; if a daughter, the daughter gets 7/12 an…
— Alcuin of York (attrib.), Propositiones ad Acuendos Juvenes
At any party of six people, prove that there must be either three people who all know each other, or three who are all strangers to one another. Why does a party of only five peopl…
— Frank P. Ramsey, On a Problem of Formal Logic
Mr. Smith has two children, and at least one of them is a boy. What's the probability that both are boys?
— Martin Gardner, Mathematical Games: Problems Involving Questions of Probability and Ambiguity
100 numbered prisoners face 100 boxes, each secretly holding one prisoner's number in random order. Each prisoner may open just 50 boxes before leaving everything exactly as they f…
— Anna Gál and Peter Bro Miltersen, The 100 Prisoners Problem
Hamilton's 1857 board game challenges players to trace a path along the edges of a dodecahedron — twelve pentagonal faces, twenty corners — that visits every corner exactly once be…
— William Rowan Hamilton, The Icosian Game
Sam Loyd's 1858 puzzle card shows two mules with two riders, but the picture is cut into three strips that can be rearranged. Slide them into the wrong arrangement and the riders a…
— Sam Loyd, Famous Trick Donkeys
A bag contains a single counter, known to be either black or white with equal probability. You add one white counter, shake the bag, and draw out a counter at random — it's white. …
— Lewis Carroll, Curiosa Mathematica, Part II: Pillow Problems
Write the digits 1 through 9 in ascending order, then insert plus and minus signs between some of them (digits may be pushed together to form bigger numbers, but never reordered or…
— Henry Ernest Dudeney, Amusements in Mathematics
An apprentice repairing a clock accidentally fits two hands of identical length and appearance, so there's no way to tell which is the hour hand and which is the minute hand just b…
— Boris A. Kordemsky, The Moscow Puzzles: 359 Mathematical Recreations
Three identical boxes each hold two coins: one box holds two gold coins, one holds two silver, and one holds one of each. You pick a box at random and draw one coin without looking…
— Joseph Bertrand, Bertrand's box paradox
Ten prisoners stand in a line, each seeing every hat in front of them but not their own or anyone behind them. Hats are red or blue, assigned in any pattern. Starting from the back…
— Martin Gardner (popularizer), Induction puzzles
You meet three islanders: one is a knight who always tells the truth, one a knave who always lies, and one a spy who can say either. Red says 'I am a knight.' Blue says 'Red is tel…
— Raymond Smullyan, What Is the Name of This Book?
Lewis Carroll loved building chains of odd-sounding premises, called sorites, that hide one inevitable conclusion. Take just three of his: no kitten that loves fish is unteachable;…
— Lewis Carroll (Charles Lutwidge Dodgson), Symbolic Logic
In his 1202 book Liber Abaci, Fibonacci posed a thought experiment: start with one newborn pair of rabbits, assume every pair matures in a month and produces one new pair every mon…
— Leonardo of Pisa (Fibonacci), Liber Abaci
Move a stack of discs from one peg to another, one disc at a time, never placing a larger disc on top of a smaller one, using a spare peg to help. Legend has it that monks in a tem…
— Édouard Lucas, Tower of Hanoi
Three missionaries and three cannibals must cross a river in a boat that holds at most two people. If cannibals ever outnumber missionaries on either bank, the missionaries get eat…
— Standard treatment in recreational mathematics and AI, Missionaries and cannibals problem
Several children have been playing and some (at least one) have mud on their forehead. Each child can see everyone else's forehead but not their own, and no one may say who is mudd…
— Standard treatment in epistemic logic, Common knowledge (logic)
Consider a single sentence: 'This sentence is false.' If it's true, then what it says must hold, meaning it really is false. But if it's false, then its own claim to be false is wr…
— Attributed to Eubulides of Miletus, Liar paradox
Using exactly four 4s each time, plus any of +, −, ×, ÷, √, !, decimal points and parentheses, form an expression equal to every whole number from 0 to 10. (4 ÷ 4 + 4 ÷ 4 = 2 gets …
— Standard treatment in recreational mathematics, Four fours
The average human head holds well under 200,000 hairs, and essentially no one has more than 500,000. London has roughly 9 million residents. Without knowing anything about any spec…
— Peter Gustav Lejeune Dirichlet, Pigeonhole principle
A rope is tied snugly around the Earth's equator, roughly 40,000 km long. You splice in one extra metre of rope and pull it into a perfect circle, evenly raised above the ground al…
— Traditional puzzle, first recorded by William Whiston, String girdling Earth
Place eight queens on a standard 8×8 chessboard so that no two threaten each other — no two may share a row, a column, or a diagonal. How many different ways can it be done?
— Max Bezzel; first solved by Franz Nauck, Eight queens puzzle
Rule a floor with parallel lines spaced exactly as far apart as a needle is long. Drop the needle randomly, over and over, and record how often it crosses a line. What does the fra…
— Georges-Louis Leclerc, Comte de Buffon, Essai d'arithmétique morale
Hold one coin still on a table. Roll an identical coin around its rim, without slipping, until it returns to its starting position. How many full rotations does the moving coin mak…
— Martin Gardner, Mathematical Carnival
Three married couples need to cross a river in a boat that holds at most two people at a time. No woman may ever be in the company of a man who isn't her husband — on either bank o…
— Alcuin of York; extended by Niccolò Tartaglia, Propositiones ad Acuendos Juvenes
The Grand Hotel has infinitely many rooms, numbered 1, 2, 3 and onward, and every single one is occupied tonight. A new guest arrives asking for a room. Can the manager accommodate…
— David Hilbert (originating lecture); popularised by George Gamow, One Two Three... Infinity
Fifteen schoolgirls walk to school each day in five groups of three. Can you schedule the groups across seven days so that no two girls ever end up in the same group more than once…
— Thomas Penyngton Kirkman, Query VI, The Lady's and Gentleman's Diary
Two men are driving their oxen down a road. The first says to the second, 'Give me two of your oxen, and I'll have as many as you.' The second replies, 'Now give me back my two oxe…
— Alcuin of York, Propositiones ad Acuendos Juvenes
A retired professor claims to have found a magic shortcut for cube roots: just add up a number's digits, and that sum is its cube root. Test it on 512 — its digits sum to 8, and 8 …
— Henry Dudeney, 536 Puzzles and Curious Problems
A rope runs over a frictionless pulley. A weight hangs from one end; a monkey hangs from the other, exactly balancing it. The monkey starts climbing the rope. What happens to the w…
— Lewis Carroll, The Diaries of Lewis Carroll
Take 15 tokens and split them into any piles you like — say 10, 3 and 2. Now repeat this move forever: take one token from every pile, and gather all the tokens you removed into on…
— Martin Gardner, Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications
In Smullyan's Transylvania, every inhabitant is a human (always truthful) or a vampire (always lying) — and separately, either sane (correct about what's true) or insane (convinced…
— Raymond Smullyan, What Is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles
A diesel ship sets out on a long voyage. When it's 180 miles from shore, a seaplane — flying exactly ten times as fast as the ship travels — is sent out from shore to deliver the m…
— Boris Kordemsky, The Moscow Puzzles: 359 Mathematical Recreations
In an election, candidate A beats candidate B by a final tally of p votes to q votes (with p > q). If the ballots are drawn and counted one at a time in a completely random order, …
— Joseph Louis François Bertrand, Bertrand's ballot theorem
In a certain village there is exactly one barber. He shaves all — and only — the men who do not shave themselves. Every man in the village is shaved either by himself or by the bar…
— Bertrand Russell, The Philosophy of Logical Atomism
Five philosophers sit around a circular table, alternating between thinking and eating a spaghetti dish that requires two forks. There is exactly one fork between each pair of neig…
— Edsger W. Dijkstra, Hierarchical Ordering of Sequential Processes
A hare stands at one end of a 150-foot field; a hound stands at the other and gives chase. Each leap of the hound covers 9 feet; each leap of the hare covers only 7 feet. Leaping a…
— Alcuin of York (attributed), Propositiones ad Acuendos Juvenes
Take any flat map divided into regions — countries, counties, anything — where neighboring regions share a real border, not just a single point. Color every region so no two neighb…
— Kenneth Appel and Wolfgang Haken, Four color theorem
A Greek epigram attributed to Archimedes describes the Sun god's cattle grazing on Sicily in four colors — white, black, yellow, and dappled bulls and cows — and lays out a set of …
— Archimedes (attributed), Archimedes's cattle problem
Two players take turns removing any number of objects, at least one, from a single pile they choose among several laid out at the start. Whoever takes the last object wins. With th…
— Charles L. Bouton, Nim, A Game with a Complete Mathematical Theory
Assume a pub has at least one person who might, at some point, order a drink. Logician Raymond Smullyan claimed there's always at least one patron such that, if that particular per…
— Raymond Smullyan, What Is the Name of This Book?
Take a long strip of paper, fold it into a flat hexagon, and glue the ends. It looks like it has just two faces. But pinch three alternating corners together and fold it open from …
— Martin Gardner, Flexagons
You have an unlimited supply of stamps in two denominations that share no common factor, say 5 cents and 7 cents. You can combine any number of each. Some amounts are impossible to…
— James Joseph Sylvester, Mathematical Questions with their Solutions
Start with any positive whole number. If it's even, divide it by two. If it's odd, multiply it by three and add one. Repeat the process on whatever number you get. Try it with a fe…
— Jeffrey C. Lagarias (survey); conjecture by Lothar Collatz, The 3x+1 Problem and Its Generalizations
An ant starts at one end of a taut rope 1 km long, crawling forward at a steady 1 cm per second. But the whole rope also stretches uniformly, at 1 km per second, carrying the ant f…
— Traditional; popularised by Martin Gardner, Ant on a rubber rope
You're hiring for one position and will interview candidates one at a time in a random order. After each interview you must accept or reject on the spot — no calling anyone back la…
— John P. Gilbert, Frederick Mosteller, Recognizing the Maximum of a Sequence
A lamp starts off. You flip its switch after 1 minute, then after another 1/2 minute, then 1/4 minute, then 1/8, and so on — infinitely many flips, all packed into exactly two minu…
— James F. Thomson, Tasks and Super-Tasks
Imagine two separate gambling games, each individually rigged so you lose money on average over time. Play only Game A, and you lose. Play only Game B, and you lose. But if you swi…
— Juan M. R. Parrondo, Parrondo's paradox
In 1973, UC Berkeley's graduate admissions showed men getting admitted at a noticeably higher overall rate than women, and looked like blatant discrimination. But when researchers …
— P. J. Bickel, E. A. Hammel, J. W. O'Connell, Sex Bias in Graduate Admissions: Data from Berkeley
Two allied generals camp on hills overlooking an enemy city, and can only win by attacking at exactly the same time. Their only communication is a messenger who might be captured c…
— E. A. Akkoyunlu, K. Ekanadham, R. V. Huggins, Some Constraints and Tradeoffs in the Design of Network Communications
Consider this phrase: 'the smallest positive integer not nameable in fewer than twenty syllables.' Count its own syllables — there are fewer than twenty. But the phrase claims to n…
— Bertrand Russell (paradox credited to G. G. Berry), Mathematical logic as based on the theory of types
Take two red blocks, two blue blocks, and two yellow blocks and line all six up in a row. Can you arrange them so that exactly 1 block sits between the two reds, exactly 2 blocks s…
— C. Dudley Langford, Mathematical Gazette
For Christmas, Mrs. Perkins receives a patchwork quilt made of 169 small squares of silk, all sewn into one large 13-by-13 square. She decides to unpick it and re-cut it into the s…
— Henry Ernest Dudeney, Amusements in Mathematics
In 1898 Sam Loyd published a maze built to defeat the classic trick of solving mazes backward from the exit. Starting on the heart at the center of a numbered grid, you move exactl…
— Sam Loyd, Sam Loyd's Cyclopedia of Puzzles
Picture four unusually numbered dice, A, B, C, and D. They're built so that A beats B more often than not, B beats C more often than not, and C beats D more often than not - yet D …
— Martin Gardner, Scientific American
Five apples sit in a basket. Can you divide them among five girls so that each girl receives an apple of her own - and yet one apple is still left behind in the basket?
— Boris A. Kordemsky, The Moscow Puzzles: 359 Mathematical Recreations
Jerry took a test along with the rest of his class. When the marks came back, it turned out Jerry had scored the 15th highest mark in the class - and also the 15th lowest mark in t…
— Maggie E. Toplak, Richard F. West, and Keith E. Stanovich, Assessing miserly information processing: An expansion of the Cognitive Reflection Test
A milkmaid starts at a stool, must walk to a straight stream to fill her pail, then carry it to a barn door. The stream doesn't lie directly between them — it's off to one side. Sh…
— Henry Dudeney, Amusements in Mathematics
Emily's father has three daughters. The first is named April, the second is named May. What is the third daughter's name?
— Keela Thomson and Daniel Oppenheimer, Investigating an Alternate Form of the Cognitive Reflection Test
A cask holds 7,200 pints of wine and springs three cracks at once. Through the first crack, a third plus a sixth of the wine leaks out. Through the second, exactly a third leaks ou…
— Alcuin of York (attrib.), Propositiones ad Acuendos Juvenes
A wall clock strikes six o'clock, and you time it: exactly 30 seconds pass between the first stroke and the last. At the same steady pace, how many seconds will the same clock take…
— Boris Kordemsky, The Moscow Puzzles
A Greek cross — five equal squares arranged in a plus shape — sits on your table. Using only four straight cuts, can you slice it into pieces that reassemble, edge to edge with no …
— Henry Dudeney (ed. Martin Gardner), 536 Puzzles and Curious Problems
Nine dots are arranged in three rows of three, like a tic-tac-toe grid with no lines drawn. Using one continuous stroke of exactly four straight lines, can you pass through every d…
— Sam Loyd, Sam Loyd's Cyclopedia of Puzzles
A bamboo stalk stands 10 feet tall. It snaps partway up, and the broken top bends over until its tip touches the ground exactly 3 feet from the base of the stalk. The unbroken lowe…
— Unknown (anonymous compilation, classic edition c. 1st century CE), The Nine Chapters on the Mathematical Art (Jiuzhang Suanshu)
A fifth part of a swarm of bees settles on a blossom of kadamba, and a third part on a flower of silinda. Three times the difference between these two numbers flies off to a flower…
— Bhaskara II, Lilavati
You're running a race and you pass the person currently in second place. What position are you in now?
— Keela S. Thomson and Daniel M. Oppenheimer, Investigating an Alternate Form of the Cognitive Reflection Test
A man and his wife, exactly equal in weight, want to cross a river with their two children, each of whom weighs exactly half as much as a parent. They have only one small boat, and…
— Alcuin of York, Propositiones ad Acuendos Juvenes
A rooster costs 5 coins, a hen costs 3 coins, and 3 chicks together cost 1 coin. You must buy exactly 100 birds for exactly 100 coins, buying at least one rooster, one hen, and one…
— Zhang Qiujian, Zhang Qiujian Suanjing (Zhang Qiujian's Mathematical Manual)
In Lewis Carroll's 'A Tangled Tale,' two travelers leave at 3 o'clock and get home at 9, walking a level road, up a hill, and down again. On the level they walk 4 miles an hour, up…
— Lewis Carroll, A Tangled Tale
Cut a large square into smaller squares — every piece a square, no two the same size, and no wasted space. Mathematicians call this 'squaring the square,' and for decades no one co…
— Brooks, Smith, Stone and Tutte; popularized by Martin Gardner, Squared square
In 1769, Leonhard Euler conjectured that you need at least n whole numbers, each raised to the nth power, to add up to another perfect nth power — an extension of Fermat's Last The…
— L. J. Lander and T. R. Parkin, A Counterexample to Euler's Conjecture on Sums of Like Powers
A farmer had 15 sheep, and all but 8 died. How many sheep does the farmer have left?
— Maggie E. Toplak, Richard F. West and Keith E. Stanovich, Assessing Miserly Information Processing: An Expansion of the Cognitive Reflection Test
A sequence of rings hangs looped on interlinked pillars, threaded through a bar. The rules only let you slide one ring at a time under a strict condition based on the rings next to…
— Gerolamo Cardano, De subtilitate
In this card game, a dealer picks a hidden rule for which cards can legally follow which. The other players take turns playing a card; a legal play is kept, an illegal one is expos…
— Robert Abbott, popularized by Martin Gardner, Eleusis (card game)
Two players take turns removing counters from two piles: any number from just one pile, or — the twist — an equal number from both piles at once. Whoever takes the last counter win…
— Willem Abraham Wythoff, A modification of the game of Nim
Take 1 + 1/4 + 1/9 + 1/16 + 1/25 and so on, adding the reciprocal of every perfect square out to infinity. The sum obviously keeps growing more slowly and seems to approach some fi…
— Leonhard Euler, De Summis Serierum Reciprocarum
A gold casket reads: 'The portrait is in this casket.' A silver casket reads: 'The portrait is not in this casket.' A lead casket reads: 'The portrait is not in the gold casket.' Y…
— Raymond Smullyan, What Is the Name of This Book?
Pegs sit on every square of an infinite checkerboard below a horizontal line. Each move takes one peg and jumps it over an adjacent peg in a straight line, removing the peg that wa…
— John Conway, Elwyn Berlekamp, Richard Guy, Winning Ways for Your Mathematical Plays
Call a word 'autological' if it describes itself, like 'short' (which is short) or 'English' (which is English). Call it 'heterological' if it doesn't, like 'long' (which isn't lon…
— Kurt Grelling and Leonard Nelson, Bemerkungen zu den Paradoxien von Russell und Burali-Forti
Cut a right triangle into four pieces along specific lines, then rearrange those exact same four pieces into a slightly different triangle shape. Measure the area both times using …
— Wikipedia, Missing square puzzle
Imagine a checker program that takes any other program and its input, and reliably answers whether that program will eventually stop, or run forever in an infinite loop. Could such…
— Alan Turing, On Computable Numbers, with an Application to the Entscheidungsproblem
In 1933, Danish polymath Piet Hein was sitting in on a lecture about quantum physics when he sketched a set of seven pieces, each made from three or four unit cubes glued together …
— Martin Gardner, Mathematical Games: Piet Hein's Puzzling Soma Cube
Seat men and women alternately around a circular table, with the women's seats already fixed, so that nobody ends up next to their own husband or wife. For four couples, how many v…
— Édouard Lucas, Théorie des Nombres
Most tiling patterns — bathroom floors, brick walls — eventually repeat in a grid. In the 1970s, physicist Roger Penrose found a set of just two tile shapes, nicknamed 'kites' and …
— Roger Penrose, Pentaplexity: A Class of Non-Periodic Tilings of the Plane
What day of the week was 4 July 1776? Mathematician John Conway devised a mental algorithm, the Doomsday rule, that lets a practiced person answer questions like this for any date …
— John Conway & Richard Guy, The Book of Numbers
There are over 43 quintillion possible scrambles of a Rubik's Cube, and yet mathematicians proved every single one can be undone in at most 20 face turns — a number puzzlers nickna…
— Tomas Rokicki, Herbert Kociemba, Morley Davidson & John Dethridge, God's Number is 20
Zeno of Elea argued that motion itself is impossible: to travel any distance, you must first cover half of it, but before that, half of that half, and so on forever — an infinite n…
— Aristotle, reporting Zeno of Elea's argument, Physics, Book VI
Alcuin of York's 9th-century puzzle collection has a man ordering 90 measures of grain moved 30 leagues to another house. He has only one camel, strong enough to carry 30 measures …
— Alcuin of York, Propositiones ad Acuendos Juvenes (Problems to Sharpen the Young)
In Sam Loyd's Cyclopedia of Puzzles, the winner of an archery contest scored exactly 100 points using six arrows, each landing in one of six rings worth 16, 17, 23, 24, 39, or 40 p…
— Sam Loyd, Sam Loyd's Cyclopedia of 5,000 Puzzles, Tricks and Conundrums
In 1751, Leonhard Euler wrote to his friend Christian Goldbach with a geometry question: cutting a convex polygon into triangles using only non-crossing diagonals, how many differe…
— Leonhard Euler, Letter to Christian Goldbach, September 4, 1751
In one of Henry Dudeney's puzzles, a retired professor claims to have found a trick for extracting cube roots: add up a number's digits, cube that total, and you land back on the n…
— Henry Dudeney, Modern Puzzles
A short-order cook stacks pancakes of all different sizes in a messy pile. The only tool available to straighten them into a neat stack — largest on the bottom, smallest on top — i…
— Jacob E. Goodman (writing as 'Harry Dweighter'), The American Mathematical Monthly
A near-perfect predictor has already decided what to put in a hidden box, based on forecasting what you're about to do — and is right almost every time. In front of you sit that hi…
— Martin Gardner, based on a problem devised by physicist William Newcomb, Mathematical Games (Scientific American)
Take any convex polyhedron, a cube, a pyramid, a soccer-ball shape, and count its vertices (V), edges (E) and faces (F). Try computing V minus E plus F for a few different shapes b…
— Leonhard Euler, Elementa Doctrinae Solidorum
Can a square be divided into a finite number of smaller squares, using every size only once, with no gaps or overlaps? For decades mathematicians weren't even sure it was possible.
— R. L. Brooks, C. A. B. Smith, A. H. Stone, and W. T. Tutte, The Dissection of Rectangles into Squares
A 5th-century Chinese puzzle: a rooster costs 5 coins, a hen costs 3 coins, and three baby chicks together cost 1 coin. You must spend exactly 100 coins to buy exactly 100 birds to…
— Zhang Qiujian, Zhang Qiujian Suanjing (The Mathematical Classic of Zhang Qiujian)
In Lewis Carroll's 1895 dialogue, Achilles gets the tortoise to accept two premises: 'A: things equal to the same are equal to each other' and 'B: the two sides of this triangle ar…
— Lewis Carroll, What the Tortoise Said to Achilles
Five sailors gather a pile of coconuts, then sleep planning to divide them fairly at dawn. In the night, each sailor in turn wakes, divides the pile into 5 equal piles with exactly…
— Ben Ames Williams, Coconuts
In Nim, players alternate removing any number of objects from a single pile, and whoever takes the last object wins. With several piles at once, is there a reliable way to know whe…
— Charles L. Bouton, Nim, A Game with a Complete Mathematical Theory
An Euler brick is a rectangular box where every edge and every diagonal drawn across each face is a whole number — the smallest known example has edges 44, 117 and 240. Can you als…
— Standard number-theoretic accounts, Euler brick
A cereal company hides one of 50 different collectible toys at random in every box. Assuming each of the 50 is equally likely and you buy boxes one at a time, roughly how many boxe…
— Standard probability theory accounts, Coupon collector's problem
You're offered two urns to bet on. Urn A has exactly 50 red balls and 50 black balls. Urn B has 100 red and black balls in a completely unknown ratio. You must bet on a colour bein…
— Daniel Ellsberg, Risk, Ambiguity, and the Savage Axioms
Three cryptographers are told their dinner has already been paid for — either by one of them, anonymously, or by an outside party like their employer. They want to determine which …
— David Chaum, The Dining Cryptographers Problem: Unconditional Sender and Recipient Untraceability
Roughly 100 billion humans have been born so far, and you're one of the more recent of them. If you treat your position in that count as a random draw from everyone who will ever l…
— Brandon Carter, The Anthropic Principle and its Implications for Biological Evolution
You bring home 100kg of potatoes, which are 99% water by weight. You leave them out and they dry slightly, down to 98% water. How much do they weigh now?
— Standard recreational mathematics accounts, Potato paradox
Visiting a sick Srinivasa Ramanujan in hospital, G. H. Hardy mentioned he'd arrived in taxi number 1729, calling it 'rather a dull number'. Ramanujan disagreed instantly. What did …
— Robert Kanigel, The Man Who Knew Infinity
Imagine a monkey pressing keys on a typewriter completely at random for an infinite amount of time. Mathematicians have proven something that sounds absurd: given infinite time, it…
— Émile Borel, Mécanique Statistique et Irréversibilité
Consider the never-ending series 1 − 1 + 1 − 1 + 1 − 1 ... Group the terms one way, (1−1) + (1−1) + ..., and it looks like it sums to 0. Group them another way, 1 − (1−1) − (1−1) −…
— Guido Grandi, Quadratura circuli et hyperbolae per infinitas hyperbolas geometrice exhibita
A, B and C fight a three-way duel, firing in turn. A hits 1 in 3 shots, B hits 2 in 3, and C never misses. Each may aim at anyone. A goes first. Counter-intuitively, mathematicians…
— Martin Shubik, Game Theory and Related Approaches to Social Behavior
You have a two-pan balance scale and may place weights on either side. What is the smallest set of four whole-number weights that lets you balance any integer weight from 1 to 40 a…
— Claude Gaspard Bachet de Méziriac, Problèmes plaisants et délectables qui se font par les nombres
Player A starts with £9, Player B with £1. Each round, they flip a fair coin for £1, stopping only when one player is wiped out. It's a fair coin, so surely the game itself is fair…
— Christiaan Huygens, De Ratiociniis in Ludo Aleae
Take the curve y = 1/x for x ≥ 1 and spin it around the x-axis. The resulting trumpet-shaped solid has a finite volume — you could fill it with a finite amount of paint. But its su…
— Evangelista Torricelli, De solido hyperbolico acuto
In 1742, Christian Goldbach wrote to Leonhard Euler with a simple claim: every even number greater than 2 is the sum of two prime numbers (4=2+2, 8=3+5, 100=3+97). Every even numbe…
— Christian Goldbach, Letter to Leonhard Euler
A warden puts 100 prisoners in isolation. Each day he picks one at random and sends them alone into a room with a single lightbulb, which they may switch on or off. Any prisoner ma…
— Jean-Paul Dehaye, Sean Ford & Henry Segerman, One hundred prisoners and a lightbulb
Draw an isosceles triangle with its two base angles at 80°. From one corner, a line at 30° cuts across to the opposite side; from the other corner, a line at 20° does the same. Usi…
— Edward Mann Langley, A Problem
Given a list of cities and the distances between them, find the shortest possible route that visits every city exactly once and returns to the start. For a handful of cities it's e…
— Clay Mathematics Institute, P vs NP Millennium Prize Problem
A standard pair of dice, numbered 1-6 each, produces the classic bell-shaped spread of totals from 2 to 12. Can you find a different pair of six-sided dice, still only positive num…
— Martin Gardner, Mathematical Games column
In Chapter 7 of Alice's Adventures in Wonderland, the Mad Hatter asks Alice this riddle at the tea party — and never gives an answer. Alice guesses, gives up, and the story simply …
— Lewis Carroll, Alice's Adventures in Wonderland
Picture a road network where drivers each choose whichever route looks fastest to them individually. Mathematician Dietrich Braess showed in 1968 that adding a new shortcut can tem…
— Dietrich Braess, Über ein Paradoxon aus der Verkehrsplanung
Take any three solid shapes, scattered anywhere in space, in any sizes — say, the bread, ham and cheese of an oddly assembled sandwich. Is there always a single flat plane that sli…
— Hugo Steinhaus and Stefan Banach, Ham sandwich theorem
Some problems are easy to verify but seem brutally hard to solve — like finding a short route through a huge map of cities, versus just checking whether a given route is short enou…
— Clay Mathematics Institute, P versus NP problem
In 1859, Bernhard Riemann proposed a precise pattern governing where prime numbers should cluster among all the other numbers, tied to the zeros of a function called the Riemann ze…
— Bernhard Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse
Around 1637, Pierre de Fermat wrote in the margin of a book that no three positive whole numbers a, b and c can satisfy a^n + b^n = c^n for any whole number n greater than 2 — and …
— Pierre de Fermat; proved by Andrew Wiles, Fermat's Last Theorem
Look at a drawing of a triangle made from three beams that each look straight and connect at right angles — yet the whole shape loops back on itself in a way no real solid object c…
— Lionel Penrose, Roger Penrose, Penrose triangle
At a party of six people, any two either know each other or are strangers. Prove that there must always be at least three people who all know each other, or three people who are al…
— Frank P. Ramsey (underlying theorem), Theorem on friends and strangers
Write the positive integers in a square spiral starting from 1 at the centre, then circle only the prime numbers. Do the circled primes scatter randomly across the grid, or is ther…
— Stanislaw Ulam, Ulam spiral
Ancient geometers spent over two thousand years trying to construct, using only an unmarked straightedge and compass, a square with exactly the same area as a given circle. Countle…
— Ferdinand von Lindemann, Squaring the circle
Legend says the plague-stricken citizens of Delos consulted an oracle, who told them to double the size of their cubic altar to Apollo. Simply doubling each edge would make the vol…
— Pierre Wantzel, Doubling the cube
Imagine a mail carrier who must walk every street in a neighbourhood at least once and return to the post office, covering as little repeated ground as possible. If every street co…
— Guan Meigu (Mei-Ko Kwan), Graphic Programming Using Odd or Even Points
Soldiers stand in a circle. Starting from one person, every second (or kth) person is eliminated as the count goes around, again and again, until only one remains. Where should you…
— Flavius Josephus, Josephus problem
Shuffle a deck. Silently pick a number from 1 to 10, count that many cards from the top, and look at the card you land on — that's your 'key card'. Take its value (face cards count…
— Martin Kruskal, Kruskal's count
Take a strip of paper, give it a single half-twist, and tape the ends together to form a loop. Now run scissors all the way around the middle of the strip, cutting it lengthwise as…
— August Ferdinand Möbius, Möbius strip
At a party of any size, some guests shake hands with some others and not with the rest. Prove that no matter how the handshakes happen to fall, there must always be at least two pe…
— Leonhard Euler, Handshaking lemma
Sleeping Beauty agrees to an experiment: she's put to sleep, and a fair coin is flipped. On heads, she's woken once, on Monday. On tails, she's woken twice, once on Monday and once…
— Adam Elga, Self-locating belief and the Sleeping Beauty problem
1 is interesting because it's the first number. 2 is interesting because it's the smallest prime. 3 is the first odd prime, and so on. Now imagine the set of numbers that are NOT i…
— Mathematical folklore, Interesting number paradox
A bridge player is dealt 13 cards and announces, 'I have an ace.' What's the probability she has at least one more ace? Now suppose instead she'd said, 'I have the ace of spades.' …
— Martin Gardner, Scientific American, 'Mathematical Games'
I paid a shilling for some apples, but they were so small the seller threw in two extra for the same money. That worked out to exactly a penny a dozen less than the price he'd firs…
— Henry Ernest Dudeney, Amusements in Mathematics
Stack cannonballs into a square pyramid: 1 ball on top, 4 beneath that, 9 beneath that, and so on down to a square base N balls per side. Is there any pyramid height for which that…
— Edouard Lucas, Cannonball problem
A traveler arrives in one city, where the money in his purse doubles, then he spends 12 denari. He moves on to a second city, where his money doubles again, then he spends another …
— Leonardo of Pisa (Fibonacci), Liber Abaci
A cage holds nothing but pheasants and rabbits. Counting heads from above gives 35. Counting feet from below gives 94. How many pheasants and how many rabbits are in the cage?
— Sun Zi (attributed; exact identity unknown), Sunzi Suanjing (The Mathematical Classic of Sun Zi)
Archimedes designed a square dissected into 14 oddly-shaped flat pieces, a sort of ancient tangram. He then asked a purely combinatorial question that had nothing to do with pictur…
— Archimedes, Ostomachion (fragmentary text, via the Archimedes Palimpsest)
On an island, 100 people have blue eyes and know the local rule: anyone who works out their own eye color must leave that midnight. No one ever leaves, because everyone can see eve…
— Terence Tao, The blue-eyed islanders puzzle
X and Y are whole numbers with 1 < X < Y and X + Y ≤ 100. Mr. S is told only their sum; Mr. P is told only their product. Mr. P says: 'I don't know X and Y — but I know that you do…
— Hans Freudenthal, Sum and Product Puzzle
Two players each choose a sequence of three coin-flip outcomes, like Heads-Heads-Tails. A coin is flipped repeatedly until one player's sequence appears, and that player wins. Coun…
— Walter Penney, Problem 95: Penney-Ante
In the 1650s, French gambler Chevalier de Méré bet on rolling at least one 6 in four rolls of a die (which he consistently won) and on rolling double-6 in twenty-four rolls of two …
— Blaise Pascal, Correspondence with Pierre de Fermat
Zeno imagined three equal rows of bodies: one stationary, and two others moving past it and each other at equal speed in opposite directions. In the time it takes a moving row to p…
— Aristotle (describing Zeno of Elea), Physics, Book VI
Newton imagined a cannon on an impossibly tall mountain, firing horizontally. A slow ball falls to the ground nearby; a faster one lands farther away; fired fast enough, the ball's…
— Isaac Newton, A Treatise of the System of the World
Conway's Game of Life runs on just two rules about which cells live or die based on their neighbours, applied to a grid, generation after generation. Most starting patterns fizzle …
— John Horton Conway (rules); Richard K. Guy (glider), Conway's Game of Life
Three prisoners, A, B and C, are told one of them has been picked at random for pardon. Prisoner A asks the warden — who knows the answer but can't reveal it — to name one of the o…
— Martin Gardner, Mathematical Games column
A trader has 3,000 bananas at one edge of a 1,000km desert and a camel that can carry at most 1,000 bananas at a time — but eats one banana per kilometre travelled, loaded or not. …
— N. J. Fine, Jeep problem
A farmer buys a pig for $60, sells it for $70, buys it back for $80, then sells it again for $90. Without adding it up on paper, what's your gut sense: did the farmer come out ahea…
— Shane Frederick (original test); extended by later researchers, Cognitive reflection test
Draw a circle with an equilateral triangle inscribed inside it. Now pick a chord of the circle completely at random. What's the probability the chord is longer than a side of the t…
— Joseph Bertrand, Bertrand paradox (probability)
If buses arrive at a stop every 10 minutes on average but at irregular intervals, common sense says a random arrival should wait 5 minutes on average — half the gap. Why is the rea…
— Wikipedia contributors, formalized in queueing theory, Inspection paradox
A snail is invited by a swallow to lunch at a spot exactly one league away. There's just one problem: the snail can crawl no faster than one inch a day. Alcuin of York posed this t…
— Alcuin of York, Propositiones ad Acuendos Juvenes
A ruler offers to reward the inventor of chess with wheat: one grain on the board's first square, two on the second, four on the third, doubling all the way to the 64th square. It …
— Traditional (earliest recorded by Ibn Khallikan, 1256), Wheat and Chessboard Problem
Henry Dudeney set this chessboard challenge in 1900: place as many pawns as possible on an 8x8 board so that no three of them ever fall on a single straight line — at any angle, no…
— Henry Dudeney, No-three-in-line problem
Pierre de Fermat challenged Evangelista Torricelli with a deceptively simple question: inside any triangle, find the single point that minimizes the total distance to all three cor…
— Pierre de Fermat (posed); Evangelista Torricelli (solved), Fermat point
A pentomino is any shape made from five equal squares joined edge to edge — there are exactly twelve distinct ones, once you rule out rotations and reflections as duplicates. The c…
— Solomon W. Golomb (Martin Gardner, popularizer), Pentomino
In 1933, Esther Klein posed a puzzle to her friends in Budapest: take any five points on a plane, with no three of them in a straight line. She claimed you can always pick four of …
— Esther Klein (posed); Paul Erdos and George Szekeres (proved), Happy ending problem
Ancient Greek geometers could bisect any angle perfectly using just a compass and an unmarked straightedge, and assumed trisecting one into three equal parts would be just as easy.…
— Heinrich Dörrie, 100 Great Problems of Elementary Mathematics: Their History and Solution
Two people want to share a cake, but neither trusts the other to cut it fairly, and there's no outside referee to measure exact halves. Neither has to value every part of the cake …
— Hugo Steinhaus (with Stefan Banach and Bronisław Knaster), Fair Cake-Cutting
A bicycle rides across a patch of wet sand, leaving two overlapping tire tracks — one from the front wheel, one from the rear. The two curves cross each other several times, and ne…
— Boris A. Kordemsky, The Moscow Puzzles
Ten men and ten women each privately rank everyone of the opposite sex from most to least preferred. You have to pair all twenty of them off into couples. Is it always possible to …
— David Gale and Lloyd Shapley, College Admissions and the Stability of Marriage
Ask everyone in your friend group how many friends they have. On average, in almost any real social network, each person's friends have more friends than they do — and this holds t…
— Scott L. Feld, Why Your Friends Have More Friends Than You Do
Gather a huge pile of real-world numbers — river lengths, stock prices, populations, electricity bills, anything — and look only at the very first digit of each one. You'd expect t…
— Frank Benford, The Law of Anomalous Numbers
Arrange the numbers 1 through 9 in a 3x3 grid so that every row, every column, and both diagonals add up to the exact same total. Chinese legend says the pattern first appeared on …
— W. S. Andrews, Magic Squares and Cubes
In one of Lewis Carroll's puzzle stories, a lady is asked her age. She replies that eight years ago, her age was three times what her son's age would be six years from now, and her…
— Lewis Carroll, A Tangled Tale
3 and 5, 11 and 13, 17 and 19, 29 and 31: pairs of prime numbers separated by exactly two keep turning up as you count higher, even as primes generally become rarer. The twin prime…
— Yitang Zhang, Bounded gaps between primes
In 1904, Henri Poincaré asked whether any three-dimensional space with no holes in it (in a precise topological sense) can always be stretched and deformed into a perfect sphere, t…
— Grigori Perelman, Poincaré Conjecture
In his 1514 engraving Melencolia I, Albrecht Dürer included a 4x4 magic square where every row, column, and diagonal adds up to 34. Look at the bottom middle two cells and you'll f…
— Albrecht Dürer, Melencolia I
Problem 79 of the Rhind Mathematical Papyrus, copied around 1650 BC, describes a chain of sevens: 7 houses, each with 7 cats, each cat catching 7 mice, each mouse eating 7 grains o…
— Ahmes (scribe), Rhind Mathematical Papyrus
You'll meet 100 candidates one at a time, in random order, and see each one's value immediately. You must accept or reject on the spot, no going back. What strategy gives you the b…
— Martin Gardner, Mathematical Games (Scientific American column)
You have three unmarked jugs holding 5, 11 and 13 ounces, and a container with 24 ounces of liquid. Using only these jugs to pour liquid between them, can you divide the 24 ounces …
— Claude Gaspard Bachet de Méziriac, Problèmes plaisants et délectables qui se font par les nombres
In classic English peg solitaire, you start with every hole filled except the centre, and jump pegs over each other to remove them, aiming to finish with a single peg. You can fini…
— Wikipedia contributors, Peg solitaire
Five houses in a row, each a different colour, owned by people of five nationalities, who each drink a different beverage, smoke a different brand and keep a different pet. Fifteen…
— Life International magazine, Zebra Puzzle
You're dealt four cards: 3, 3, 8 and 8. Using each number exactly once, combined with any mix of addition, subtraction, multiplication and division, can you reach a total of exactl…
— Robert Sun / Suntex International, 24 (puzzle)
Urn A has 50 red balls and 50 black balls. Urn B has 100 balls total, red and black in an unknown ratio. You win money if you draw a colour you bet on in advance. Most people, offe…
— Daniel Ellsberg, Risk, Ambiguity, and the Savage Axioms
An ancient graffito found at Pompeii uses exactly five Latin words, each five letters long: SATOR, AREPO, TENET, OPERA, ROTAS. Arranged correctly into a 5×5 grid, the square reads …
— Unknown (earliest known example found at Pompeii), Sator Square
The 'Nine Chapters on the Mathematical Art' poses this: three sheaves of a good crop, two of mediocre and one of bad together yield 39 measures of grain; two good, three mediocre a…
— Unknown (compiled over centuries, editions from the Han dynasty), The Nine Chapters on the Mathematical Art
A jeep's tank holds enough fuel to travel exactly 1 unit of distance, and it can carry no more. Fuel is freely available at base, and the jeep can make multiple trips, dropping fue…
— Nathan J. Fine, The Jeep Problem
Several army divisions, each led by a general, surround an enemy city and can only communicate by sending messengers. They must unanimously agree to attack together or retreat toge…
— Leslie Lamport, Robert Shostak and Marshall Pease, The Byzantine Generals Problem
Place an ant on an infinite grid of white squares. On a white square, it turns right, flips the square to black, and steps forward; on a black square, it turns left, flips the squa…
— Chris Langton, Langton's ant
On a street, houses are numbered consecutively starting from 1. On one particular house, the sum of all house numbers before it exactly equals the sum of all house numbers after it…
— Robert Kanigel (recounting P. C. Mahalanobis's account), The Man Who Knew Infinity
The 15 Puzzle — fifteen numbered tiles sliding in a 4x4 grid with one gap — became a national craze in 1880. Puzzle showman Sam Loyd claimed for the rest of his life that he'd inve…
— Noyes Chapman (inventor); falsely claimed by Sam Loyd, 15 Puzzle
Three prisoners, A, B and C, are told one of them — chosen at random — will be pardoned. Prisoner A, knowing the warden can't reveal his own fate, asks instead to be told the name …
— Martin Gardner, Mathematical Games
Take a long strip of paper, fold it into a ring of triangles, and glue the ends: you've built a hexaflexagon. Pinch three alternating corners together and it opens out flat again —…
— Arthur H. Stone and colleagues; popularized by Martin Gardner, Flexagon
Almost everyone's first answer is 10 cents. It feels obviously right, arrives instantly, and is wrong. Can you find the actual price of the ball, and see why your gut answer fails …
— Shane Frederick, Cognitive Reflection and Decision Making
Five houses in a row are each a different colour, and each is home to a person of a different nationality, who drinks a different drink, smokes a different brand, and keeps a diffe…
— Life International magazine (original author unknown), Zebra Puzzle
A prisoner is told he'll be hanged at noon on one weekday next week, but that the hanging will come as a total surprise — he won't know the day until the executioner knocks. He rea…
— Origin disputed (Sweden, 1940s); popularized by Martin Gardner, Unexpected Hanging Paradox
Raymond Smullyan's island has two kinds of people: the sane, who believe only true things, and the insane, who believe only false things, and everyone always says exactly what they…
— Raymond Smullyan, What Is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles
Sam Loyd's Pony Puzzle gives you a picture of a small pony sliced into six interlocking pieces. Rearranged one way, the pieces show the original pony. Rearranged another, the same …
— Sam Loyd, The Pony Puzzle
Lewis Carroll's final and most argued-over 'pillow problem': a bag contains two counters, each independently either black or white, with no other information given. Carroll offered…
— Lewis Carroll (Charles Dodgson), Pillow Problems
Three guests split a $30 hotel bill, $10 each. The manager realizes it should have been $25 and sends the bellhop back with $5. The bellhop pockets $2 and gives each guest $1 back.…
— R. M. Abraham, Diversions and Pastimes
Alcuin's puzzle: two merchants pool 100 coins and buy pigs at five for two coins, 250 pigs total. Selling them all together at the same rate they paid, they can't make a single coi…
— Alcuin of York, Propositiones ad Acuendos Juvenes
In Alcuin's puzzle, two walkers see a flock of storks and try to guess how many there are. They reason: if there were three times as many storks, plus half of a third of that tripl…
— Alcuin of York, Propositiones ad Acuendos Juvenes
A heap of a million grains of sand is still a heap after you take away one grain. Do that again and again, one grain at a time, and by the same reasoning it should always still be …
— Diogenes Laertius, Lives of the Eminent Philosophers
In Bhaskara's classic version, the square of one-eighth of a troop of monkeys is playing in a forest, while the remaining twelve monkeys sit chattering on a nearby hilltop. How man…
— Bhaskara II, Lilavati
A mule complains to a donkey: 'If you gave me one of your sacks, I'd be carrying exactly twice as many as you. But if I gave you one of mine instead, we'd be carrying the same numb…
— Metrodorus (compiler), Greek Anthology (Anthologia Palatina), Book XIV
In Sprouts, two players take turns joining dots with a curved line (never crossing another line, and no dot may ever get more than three line-ends), adding a fresh dot on every lin…
— Martin Gardner, Scientific American (Mathematical Games column)
Divide the number ten into two parts. Multiply the two parts together, and separately multiply one of the parts by itself. It turns out the self-multiplied result is exactly four t…
— Muhammad ibn Musa al-Khwarizmi, The Compendious Book on Calculation by Completion and Balancing
Given a triangle, draw three circles inside it so each circle touches the other two and each also touches two of the triangle's sides, the natural way to pack three mutually-tangen…
— Gian Francesco Malfatti, Memoria sopra un problema stereotomico
A hallway one unit wide turns a sharp right angle. What is the largest-area two-dimensional shape — the 'sofa' — that can be slid flat around the corner without ever lifting it? Th…
— Leo Moser, Moving furniture through a hallway (Problem 66-11)
An eccentric art gallery is shaped like a simple polygon with n straight walls, and its guards can each see in every direction but never through a wall. What is the smallest number…
— Václav Chvátal, Chvátal's art gallery theorem
Two balls sit anywhere on a circular billiard table. Using only the cushion — no other balls, no pockets — where on the rail must you aim so that a ball struck from the first point…
— Ibn al-Haytham (Alhazen), Book of Optics (Kitab al-Manazir)
A necklace has beads of k different colors, strung in any order, with an even number of beads of each color. Two thieves want to split it so each gets exactly half the beads of eve…
— Noga Alon, Splitting Necklaces
Start with three spheres, all mutually touching each other, sitting inside (or alongside) one more enclosing sphere. Now pack a chain of smaller spheres into the gap, each one touc…
— Frederick Soddy, Soddy's hexlet (independently recorded on an 1822 sangaku tablet)
In the board game Hex, two players alternate placing stones, each trying to link their two opposite sides of a rhombus-shaped board with an unbroken chain. The board can never end …
— John Nash, Strategy-stealing argument
If 6 cats can kill 6 rats in 6 minutes, how many cats does it take to kill 100 rats in 50 minutes? Most people blurt out '100' — Lewis Carroll built this word problem specifically …
— Lewis Carroll, A Tangled Tale
Four glasses sit at the corners of a square turntable, each either right-side up or upside down — but you're blindfolded. Each turn you may touch any two glasses and flip either, b…
— Martin Gardner, Mathematical Games (Scientific American column)
Two freight trains, each a sixth of a mile long, travel toward each other at 60 mph apiece. From the instant their locomotives draw level to the instant their cabooses do, how many…
— Boris A. Kordemsky, The Moscow Puzzles: 359 Mathematical Recreations
In a forest of talking birds, calling one bird's name to another makes it call back the name of some bird in reply. A 'mockingbird' always responds to any bird's name by calling ba…
— Raymond Smullyan, To Mock a Mockingbird and Other Logic Puzzles
Three voters each rank candidates A, B, and C in a different order. Pair them up head-to-head: a majority prefers A over B, and a majority prefers B over C. Surely, then, a majorit…
— Marquis de Condorcet, Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix
A desk calendar shows any day of the month, 01 through 31, using just the front faces of two cubes side by side. Each cube has only 6 faces, so between them they have 12 faces for …
— Martin Gardner, Mathematical Games column
An explorer walks one mile due south, turns and walks one mile due east, turns again and walks one mile due north — and finds himself exactly back at his starting point. He sees a …
— Martin Gardner, Mathematical Games column
In Lewis Carroll's 'A Tangled Tale', a ship's captain explains a puzzle of dates to his passengers: a vessel that circumnavigates the globe travelling west keeps setting its clocks…
— Lewis Carroll, A Tangled Tale
Arrange six pennies in a triangle: three along the bottom, two above them, one on top, each coin touching its neighbours. Now move the coins one at a time into a straight line of s…
— Henry E. Dudeney, 536 Puzzles and Curious Problems
Three boxes hold, respectively, two black marbles, two white marbles, and one of each — but a prankster has swapped every label, so all three boxes are now guaranteed to be wrong. …
— Martin Gardner, Mathematical Games column
Five pirates, ranked by seniority, must split 100 gold coins. The most senior proposes a split; all pirates including the proposer vote, and if at least half approve, it happens — …
— Traditional game theory puzzle; popularized by Ian Stewart, Pirate Game
A rotating sprinkler normally spins because water shoots out of its curved arms. Submerge it and make it suck water in instead, reversing the flow — does it spin the same direction…
— Ernst Mach (originator); popularized by Richard Feynman, Feynman sprinkler
A physicist with an office near the bottom of a building noticed the first elevator to stop at his floor was almost always heading down. A colleague near the top noticed the opposi…
— Marvin Stern, George Gamow, Elevator paradox
Four cards lie on a table, each with a number on one side and a color on the other. You can see: 3, 8, blue, red. To test the rule 'if a card shows an even number, its other side i…
— Peter Cathcart Wason, Wason selection task
You're given three circles of any size, placed anywhere on a page. Using only a compass and straightedge, construct a fourth circle tangent to all three — touching each one without…
— Apollonius of Perga, Problem of Apollonius
Lewis Carroll offers three statements to accept as true: all babies are illogical; nobody who can manage a crocodile is despised; and all illogical people are despised. Taking all …
— Lewis Carroll (Charles Lutwidge Dodgson), Symbolic Logic
Picture a donkey that is equally hungry and equally thirsty, standing exactly equidistant from a pile of hay and a bucket of water. It has no more reason to choose one over the oth…
— Attributed to Jean Buridan, Buridan's ass
Two suspects are arrested and questioned in separate rooms, unable to communicate. Each is offered the same deal: betray your partner and, if they stay silent, you go free while th…
— Merrill Flood and Melvin Dresher (game); Albert W. Tucker (framing), Prisoner's dilemma
Suppose you claim to have listed every real number between 0 and 1, pairing the first with 1, the second with 2, and so on forever, so no such number is left off your list. Georg C…
— Georg Cantor, Cantor's diagonal argument
A friend picks a secret whole number between 1 and 1,000,000, and you may ask as many yes-or-no questions as you like to identify it. There's one catch: somewhere across the entire…
— Alfréd Rényi; Stanisław Ulam, Ulam's game
On a grid of dots, two players take turns drawing one line between two adjacent dots. Whoever draws the fourth side of a small box claims it and immediately takes another turn. Onc…
— Édouard Lucas, Dots and Boxes
Draw a 'ground' line, then sprout any number of connected line segments above it, like a stick-figure hedge. Two players alternate erasing one segment at a time; whenever a segment…
— John Horton Conway, On Numbers and Games
An auctioneer offers a dollar bill to the highest bidder, with one twist: both the highest and second-highest bidder must pay their final bid, though only the winner gets the dolla…
— Martin Shubik, The Dollar Auction Game: A Paradox in Noncooperative Behavior and Escalation
A circular cave has one entrance splitting into two tunnels that meet at a locked door, which opens only with a secret word. Peggy claims she knows the word; Victor wants proof, bu…
— Jean-Jacques Quisquater et al., How to Explain Zero-Knowledge Protocols to Your Children
Consider the sentence: 'If this sentence is true, then Germany borders China.' Assume for a moment that the sentence is true. Then, by what it says, its own truth lets you conclude…
— Haskell Curry, The Inconsistency of Certain Formal Logics
An eccentric billionaire offers you a deal: intend, tonight at midnight, to drink a vial of toxin tomorrow that will make you painfully ill for a day but cause no lasting harm. If …
— Gregory S. Kavka, The Toxin Puzzle
A wealthy 8th-century head of household orders 100 bushels of corn handed out to exactly 100 people under his roof, men, women and children among them. Every man is to receive 3 bu…
— Alcuin of York, Propositiones ad Acuendos Juvenes
In one of Alcuin of York's 8th-century arithmetic riddles for students, two men marry each other's sisters. Years later, each man has a son. Alcuin's question: exactly what family …
— Alcuin of York, Propositiones ad Acuendos Juvenes
Henry Dudeney's landowner has a square plantation of 49 trees planted in neat rows and columns. A storm blows down 4 of them. He decides to fell all but ten of the trees left stand…
— Henry Dudeney, Amusements in Mathematics
A famous 1924 theorem says a solid ball can be cut into a finite number of pieces and reassembled, no stretching, no added material, into two solid balls, each exactly the same siz…
— Stefan Banach and Alfred Tarski, Sur la decomposition des ensembles de points en parties respectivement congruentes
As the story goes, a schoolmaster wanted to keep his class quiet, so he set them a tedious task: add up every whole number from 1 to 100, longhand. He expected it to take the rest …
— Carl Friedrich Gauss (anecdote recorded by Wolfgang Sartorius von Waltershausen), Gauss zum Gedaechtniss
Three pegs hold a stack of discs of different sizes. Move the whole stack from one peg to another, one disc at a time, never placing a larger disc on top of a smaller one. Édouard …
— Édouard Lucas, Récréations mathématiques
Two players alternately call out a positive whole number that cannot be written as a sum of numbers already named (a number may be reused any number of times in that sum). For exam…
— Elwyn Berlekamp, John Conway, Richard Guy, Winning Ways for Your Mathematical Plays
On an infinite chessboard, an 'Angel' of power k can fly up to k squares in any direction each turn, while a 'Devil' can block one square per turn, trying to eventually trap the An…
— John Conway (proofs by Bowditch, Máthé, Kloster), The Angel Problem
A Bongard problem shows two small sets of simple diagrams side by side: every image on the left obeys some hidden rule, and every image on the right breaks it. The task is to work …
— Mikhail Bongard, Pattern Recognition
In 1980, political scientist Robert Axelrod invited game theorists to submit computer programs to compete in a round-robin tournament of the iterated Prisoner's Dilemma, where the …
— Robert Axelrod, The Evolution of Cooperation
Draw one large circle and a smaller circle fully inside it, off-centre. Now try fitting a chain of even smaller circles into the ring-shaped gap between them, each one touching its…
— Jakob Steiner, Steiner Chain
Imagine infinitely many prisoners, one for every whole number, each wearing a red or blue hat they cannot see. Each prisoner can see every other prisoner's hat but not their own, a…
— Christopher S. Hardin, Alan D. Taylor, An Introduction to Infinite Hat Problems
Place two white knights on the top corners of a 3x3 chessboard and two black knights on the bottom corners, leaving the centre and edge squares empty. Using only legal knight moves…
— John J. Watkins, on Paolo Guarini di Forlì's 1512 puzzle, Across the Board: The Mathematics of Chessboard Problems
A snake's hole sits at the base of a pillar 9 cubits tall, with a peacock perched on top. The peacock spots the snake at a distance from the hole equal to three times the pillar's …
— Bhaskara II, Lilavati
N gentlemen check their hats at a party. At the end of the night, the attendant, having lost track of who owns what, hands every hat back completely at random, one per guest. As th…
— Pierre Rémond de Montmort, Essay d'analyse sur les jeux de hazard
The classic liar paradox ('this sentence is false') relies on a single sentence pointing directly at itself. Now imagine an infinite list of sentences, where sentence 1 says 'every…
— Stephen Yablo, Paradox Without Self-Reference
Draw three circles of any size so that each one touches the other two, then draw a fourth circle nestled in the gap between them, also touching all three. Given only the sizes of t…
— Frederick Soddy, restating René Descartes' 1643 theorem, The Kiss Precise
According to legend, the Phoenician princess Dido fled to North Africa and asked a local king for land to settle. He mockingly agreed to give her only as much ground as a single ox…
— Virgil, Aeneid
In Lewis Carroll's 'A Tangled Tale,' an aunt and her niece each board a train circling the same loop line in opposite directions — one direction takes 3 hours to complete the circu…
— Lewis Carroll, A Tangled Tale
In 1693, Prince Rupert of the Rhine wagered that a hole could be cut through a cube large enough to let a second cube of the exact same size pass all the way through it, without sp…
— Prince Rupert of the Rhine (problem); John Wallis and Pieter Nieuwland (solutions), Prince Rupert's Cube
A merchant has six sealed barrels holding 15, 16, 18, 19, 20 and 31 gallons. Five contain wine, one contains beer, but none are labelled. He sells one buyer some wine, and a second…
— Henry Ernest Dudeney, Amusements in Mathematics
Two glasses hold equal amounts of wine and water. Take a spoonful of wine and stir it into the water glass. Then take a spoonful of that new mixture and stir it back into the wine …
— Martin Gardner (popularizer), Hexaflexagons and Other Mathematical Diversions
In one of Alcuin of York's medieval puzzle problems, a circular city has a circumference of 8,000 feet. Every house built inside it must measure exactly 30 feet by 20 feet. How man…
— Alcuin of York, Propositiones ad Acuendos Juvenes
You're offered two broken clocks. One has stopped completely and never moves at all. The other still runs, but loses exactly one minute every day. If what matters is how often a cl…
— Lewis Carroll (Charles Lutwidge Dodgson), The Rectory Umbrella
At each step, ten balls numbered consecutively are added to a vase and one ball is removed, always the lowest-numbered ball still inside. If this repeats infinitely many times in a…
— John E. Littlewood; Sheldon Ross, Ross–Littlewood paradox
Two men agree: whoever has the cheaper necktie has to give it to the other as a prize. Each reasons the same way — "If I lose, I lose the value of my tie; if I win, I gain more tha…
— Traditional; related to Maurice Kraitchik's two-envelope problem, Necktie paradox
Take any triangle, however lopsided, and construct an equilateral triangle pointing outward on each of its three sides. Mark the center of each of those three new triangles. What s…
— Traditionally attributed to Napoleon Bonaparte, Napoleon's theorem
Suppose you have a coin that lands heads with some unknown probability, not necessarily 50/50. Using only that biased coin, how can you produce a perfectly fair 50/50 outcome, guar…
— John von Neumann, Fair coin
Imagine a ball completely covered in hair, and you try to comb every single hair so it lies flat against the surface with none sticking straight up. Is there always at least one sp…
— Henri Poincaré (early work); Luitzen Brouwer (general proof, 1912), Hairy ball theorem
Six is divisible by 1, 2, and 3 — and 1+2+3 happens to equal 6 exactly. The next such number is 28 (1+2+4+7+14=28). These are called perfect numbers. Is there a way to generate eve…
— Euclid; Leonhard Euler (Euclid–Euler theorem), Perfect number
Two people wager equal stakes on a fair coin-flip game, first to 6 points wins the whole pot. They're forced to quit early, with the score at 5 to 3. Splitting the pot 5:3 (by poin…
— Blaise Pascal and Pierre de Fermat, Correspondence of 1654 (on the Problem of Points, first posed in Pacioli's Summa de Arithmetica, 1494)
The Grand Duke of Tuscany noticed something odd at the gaming tables: rolling three dice and adding them up, a total of 10 turns up more often than a total of 9 — even though there…
— Galileo Galilei, Sopra le Scoperte dei Dadi (Concerning an Investigation on Dice)
A woman dies leaving a husband, a son, and three daughters, and a will that bequeaths a stranger exactly 1/8 plus 1/7 of her whole estate. By the inheritance law of the time, the h…
— Muhammad ibn Musa al-Khwarizmi, Al-Jabr (The Compendious Book on Calculation by Completion and Balancing)
In the Forest of Forgetfulness, the Lion lies every Monday, Tuesday, and Wednesday and tells the truth the other four days. The Unicorn lies every Thursday, Friday, and Saturday an…
— Raymond Smullyan, What Is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles
Two colonels must each secretly split a fixed number of troops across several battlefields, deciding all at once with no knowledge of the other's plan. Whoever puts more troops on …
— Emile Borel, La theorie du jeu et les equations integrales a noyau symetrique
Everyone in a group privately writes down a whole number from 0 to 100. Whoever's number is closest to two-thirds of the average of everyone's guesses wins. There's no hidden infor…
— Alain Ledoux, Jeux et Strategie magazine tiebreaker game
Ancient Greeks had no word for a number larger than a 'myriad myriad' (100 million) -- anything bigger was simply 'uncountable.' So when someone claimed the sand needed to fill the…
— Archimedes, The Sand Reckoner
For 2,000 years after Euclid, mathematicians assumed only a handful of regular polygons could ever be drawn using just a compass and an unmarked straightedge. In 1796, a 19-year-ol…
— Carl Friedrich Gauss, Disquisitiones Arithmeticae
Austrian wine merchants priced a barrel by pushing a measuring rod diagonally through the bunghole to the far bottom rim, and charging purely by that one length -- never mind the b…
— Johannes Kepler, Nova Stereometria Doliorum Vinariorum
Isaac Newton posed a puzzle in his algebra textbook: a pasture's grass grows at a constant rate every day, even as cows eat it down. Given how many cows it takes to clear one field…
— Isaac Newton, Arithmetica Universalis
Stack identical spheres as densely as possible, the way a grocer piles oranges into a pyramid. Is there any arrangement, however irregular, that packs them more tightly than that i…
— Johannes Kepler, Strena Seu de Nive Sexangula
Pierre de Fermat claimed dozens of theorems in the margins of his books without ever writing down a proof, leaving mathematicians centuries of homework. But for one claim -- that n…
— Pierre de Fermat, Observations on Diophantus
In 1535, Antonio Fior challenged Niccolò Tartaglia to a contest: each gave the other thirty problems to solve. All of Fior's problems required cubic equations of the form x³ + px =…
— Gerolamo Cardano, Ars Magna
Rather than testing each number individually for divisibility, how could you find every prime number up to, say, 100 in one systematic sweep, without doing any division at all?
— Eratosthenes of Cyrene, Sieve of Eratosthenes
Long before anyone had a general algebraic formula for cubic equations, how could you find a real, measurable solution to one at all?
— Omar Khayyam, Treatise on Demonstration of Problems of Algebra
Can every fraction of the form 4/n, where n is a whole number of 2 or more, always be rewritten as the sum of exactly three unit fractions (fractions with 1 on top)? For example, 4…
— Paul Erdős and Ernst G. Straus, Erdős-Straus conjecture
In 1900, David Hilbert asked for a single algorithm that could look at any polynomial equation with integer coefficients and correctly say, in a finite number of steps, whether it …
— David Hilbert; solved by Yuri Matiyasevich, Hilbert's tenth problem
There are infinitely many whole numbers, and a strictly bigger infinity of real numbers. Georg Cantor asked in the 1870s: is there any set whose size sits strictly between those tw…
— Georg Cantor; independence shown by Kurt Gödel and Paul Cohen, Continuum hypothesis
Add 1/1² + 1/2² + 1/3² + 1/4² and so on, forever. The sum clearly settles on some finite number — but what number? Mathematicians across Europe failed to find it for nearly a centu…
— Leonhard Euler, De Summis Serierum Reciprocarum
A man wants to host 'a very small dinner party' and invites his father's brother-in-law, his brother's father-in-law, his father-in-law's brother, and his brother-in-law's father. …
— Lewis Carroll, A Tangled Tale
Three round buns, all the same size and thickness, need to be divided into shares so that four boys receive exactly equal amounts. Cuts can be made anywhere, but the puzzle asks fo…
— Henry Dudeney, Amusements in Mathematics
Imagine an infinite, unchanging universe evenly filled with stars in every direction. Looking far enough along any line of sight, you should eventually hit a star's surface, meanin…
— Heinrich Wilhelm Olbers, Ueber die Durchsichtigkeit des Weltraumes
Special relativity says a moving clock runs slow. But from the traveling twin's point of view, it's the Earth twin who was moving — so shouldn't the Earth twin be the one who ages …
— Paul Langevin, L'évolution de l'espace et du temps
The divisors of 220, excluding itself, are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110. Add them up. Now find the divisors of 284, excluding itself, and add those. What do you notic…
— Attributed to Pythagoras by Iamblichus, Amicable numbers
Pick any four-digit number that isn't made of all the same digit, say 3524. Arrange its digits into the largest possible number and the smallest possible number, then subtract the …
— D. R. Kaprekar, Kaprekar's routine
Ronald Graham needed an upper bound for a question in Ramsey theory about colouring the edges of a hypercube, and the number he came up with is so vast that the observable universe…
— Ronald Graham and Bruce Rothschild, Ramsey's Theorem for n-Parameter Sets
A mathematical hydra is a tree of branching 'heads.' Cut one off, and the hydra sprouts several new copies of whatever was left below the cut, seemingly making your job harder no m…
— Laurence Kirby and Jeff Paris, Accessible Independence Results for Peano Arithmetic
Start with one cow. Every cow, beginning in her fourth year, gives birth to exactly one new calf each year from then on. No cows die. How many cows are there after 20 years, and wh…
— Narayana Pandita, Ganita Kaumudi
Pick any point on a pizza, not necessarily the center, and cut through it at four different angles, evenly spaced 45 degrees apart, producing 8 wedge-shaped slices. Give every othe…
— Larry Carter and Stan Wagon, Proof Without Words: Fair Allocation of a Pizza
A barbershop has a single barber, one barber chair, and a small waiting room with a few chairs. If no customers are around, the barber sleeps in his chair. When a customer arrives,…
— Edsger W. Dijkstra, Cooperating Sequential Processes
Three circular rings are arranged so that all three together form a single connected, inseparable knot. But look at any two of the three rings on their own, ignoring the third: are…
— Peter Guthrie Tait, On Knots
Take any finite set of points on a page, as long as they don't all sit on one single straight line. A mathematician asked: must there always exist at least one line that passes thr…
— James Joseph Sylvester (posed); Tibor Gallai (proved), Mathematical Question 11851
Picture a small network of exactly 10 points, each connected to exactly 3 others, drawn so that it looks like a five-pointed star inside a pentagon, with spokes connecting the two …
— Julius Petersen, Sur le théorème de Tait
A man owns 300 pigs and orders them all slaughtered over exactly three days, with an odd number killed on each of the three days. He wants the same done with a smaller herd of just…
— Alcuin of York, Propositiones ad Acuendos Juvenes
Arrange the numbers 1 through 19 into a honeycomb of nineteen small hexagons — rows of 3, 4, 5, 4 and 3 — so that all fifteen straight lines, in every one of the three directions, …
— Martin Gardner, Mathematical Games (Scientific American column)
John can drink an entire barrel of water by himself in 6 days. Mary can drink the same size barrel by herself in 12 days. If they start drinking from one barrel together, at the sa…
— Maggie E. Toplak, Richard F. West, and Keith E. Stanovich, Assessing miserly information processing: An expansion of the Cognitive Reflection Test
The sugar has gone missing, and it was either the Duchess or the Cook — no one else. Everyone agrees on one house rule: whoever stole the sugar is certain to lie about it afterward…
— Raymond Smullyan, Alice in Puzzle-Land: A Carrollian Tale for Children Under Eighty
A 13th-century Chinese mathematician placed the numbers 1 through 33 across four concentric rings around a single central number, 9. He arranged them so that every diameter — a str…
— Yang Hui, Xugu Zhaiqi Suanfa (Sequel to Excerpts of Mathematical Wonders)
A rectangular dance hall has four walls. Place exactly ten chairs along them so that every one of the four walls has the same number of chairs standing against it. No cutting chair…
— Boris A. Kordemsky, The Moscow Puzzles: 359 Mathematical Recreations
An airline loses two travelers' identical antiques and doesn't know their true value. Each privately writes a claim between $2 and $100. Matching claims are both paid in full; mism…
— Kaushik Basu, The Traveler's Dilemma: Paradoxes of Rationality in Game Theory
Doctors reclassify a cancer patient from an earlier, healthier stage to a more advanced one, because a scan finally reveals metastases that were always there but previously invisib…
— Alvan R. Feinstein, The Will Rogers Phenomenon: Stage Migration and New Diagnostic Techniques as a Source of Misleading Statistics for Survival in Cancer
In June 1999, a puzzle called Eternity went on sale: 209 identical-area but irregularly shaped pieces had to be fitted together, edge to edge with no gaps or overlaps, into one lar…
— Christopher Monckton (creator), Eternity puzzle
Start with 1. To build the next line, read the previous line aloud and write down what you say: '1' becomes 'one 1', so line two is 11. That becomes 'two 1s', so line three is 21. …
— John Horton Conway, The Weird and Wonderful Chemistry of Audioactive Decay
A surveyor needs the height of a distant sea island and its distance from shore, but can't cross the water to measure either one directly. All that's available on land: two identic…
— Liu Hui, Haidao Suanjing (Sea Island Mathematical Manual)
In 1911, a young clerk in Madras sent a journal this challenge: work out the value of the square root of 1, plus 2 times the square root of 1, plus 3 times the square root of 1, pl…
— Srinivasa Ramanujan, Question 289
Draw six dots and connect every pair with a line. Two players take turns coloring in one uncolored line at a time, each using their own color. The first player to complete a triang…
— Martin Gardner, Scientific American: Mathematical Games
Using the same seven tangram pieces arranged two different ways, you can build two monk figures that look almost identical — except one of them is clearly missing a foot. Both figu…
— Henry Dudeney, Amusements in Mathematics
Take a strip of eight postage stamps in a row, each joined to the next by a perforation. Using only the existing creases, fold the strip into a single stack the size of one stamp. …
— Henry Dudeney, Modern Puzzles and How to Solve Them
Some shapes have a strange property: you can slice them into several smaller pieces, and every single piece is a smaller copy of the original shape. An L-shaped tromino, for instan…
— Martin Gardner, Scientific American: Mathematical Games
Two ladders of different lengths are propped up across a narrow alley, each leaning from the base of one wall to somewhere up the opposite wall, crossing each other partway up. You…
— Martin Gardner, Mathematical Circus
According to a mathematical retelling of Flavius Josephus's own account, he and 40 fellow soldiers, trapped in a cave by Roman troops in 67 AD, drew lots to kill each other rather …
— Flavius Josephus, The Jewish War
Mr. Smith has two children, and at least one of them is a boy. What's the probability both children are boys? Most people confidently answer 1/2. Martin Gardner posed this exact pu…
— Martin Gardner, Mathematical Games column, Scientific American
Conway's Game of Life plays out on an infinite grid where each cell is alive or dead, and every generation is decided by simple rules based on how many living neighbours each cell …
— Martin Gardner, introducing John Horton Conway's game, Mathematical Games column, Scientific American
Several divisions of an army, each led by a general, surround an enemy city. They must all attack together or all retreat together to avoid disaster, communicating only by messenge…
— Leslie Lamport, Robert Shostak, and Marshall Pease, The Byzantine Generals Problem
A disease affects 1 in 1,000 people. A test for it is 99% accurate, correctly flagging 99% of sick people and correctly clearing 99% of healthy people. You take the test and get a …
— Gerd Gigerenzer, Calculated Risks: How to Know When Numbers Deceive You
Mathematicians once hoped to find one complete, consistent set of axioms strong enough to prove every true statement about arithmetic, with no contradictions. Is that even possible…
— Ernest Nagel and James R. Newman, Godel's Proof
A man dies owing 100, 200 and 300 coins to three creditors, but his estate holds only 100 coins. Ancient rabbinic law splits it equally: 33⅓ each. Raise the estate to 300 coins, an…
— Robert J. Aumann and Michel Maschler, Game Theoretic Analysis of a Bankruptcy Problem from the Talmud
Arrange a deck in a strict repeating four-suit pattern, then have a spectator cut it once and riffle-shuffle it together with itself, just a single time. Deal the shuffled deck off…
— Martin Gardner, Mathematical Games: The Gilbreath Principle
In the 1942 film Mrs. Miniver, a character describes a rose bred to sit in color exactly between a red circle and a pink one: draw a circle, inscribe the largest equilateral triang…
— Martin Gardner, Mathematical Games: The Mrs. Miniver Problem
In Greek myth, the Sphinx guarded the road into Thebes and put the same riddle to every traveller, devouring anyone who failed: what creature walks on four legs in the morning, two…
— Pseudo-Apollodorus, Bibliotheca
A spectator picks five cards at random from a shuffled deck. An assistant looks at them, then hands a magician four of the five, one at a time, face up, with no words exchanged. Th…
— Martin Gardner, Mathematics, Magic and Mystery
You need to link three towns with roads so all three are connected, using the least total length of road. Simply drawing a road between every pair works, but it isn't optimal. A si…
— Jakob Steiner, Steiner tree problem
Two identical bolts sit head to head with their threads meshed together like gears. Hold each bolt firmly so it can't spin in place, then twirl them around each other the way you'd…
— Martin Gardner, My Best Mathematical and Logic Puzzles
Set up a row of bowling pins. Two players take turns: on each turn, knock down either a single pin, or two pins that happen to be standing right next to each other. Whoever knocks …
— Henry Ernest Dudeney, The Canterbury Puzzles
A triangular field has two equal sides of 30 perches each and a base of 18 perches. A ninth-century method for finding its area: add the two equal sides together and take half of t…
— Alcuin of York (attrib.), Propositiones ad Acuendos Juvenes
Lewis Carroll posed this one while lying awake at night: scatter three points completely at random on an infinite flat plane. What is the chance that the triangle they form is obtu…
— Lewis Carroll (Charles Lutwidge Dodgson), Pillow Problems
A cyclist rides a mile with a tailwind in exactly 3 minutes, then turns around and pedals the same mile straight into that same wind, taking exactly 4 minutes. Most people's first …
— Sam Loyd, Sam Loyd's Cyclopedia of Puzzles
Five schoolgirls step on a scale two at a time, in every possible pairing, giving ten different combined weights. None of them is ever weighed alone. From just those ten pair-total…
— Sam Loyd, Sam Loyd's Cyclopedia of Puzzles
In 1693, diarist Samuel Pepys wrote to Isaac Newton asking which is more likely: rolling at least one six with 6 dice, at least two sixes with 12 dice, or at least three sixes with…
— Isaac Newton, Samuel Pepys, Newton-Pepys problem
An ant starts at one end of a 1-km rubber band, crawling at 1 cm per second toward the far end. At the end of every second, the entire band instantly and uniformly stretches by ano…
— Classic mathematical puzzle, Ant on a rubber band
There are three boxes: one holding two gold coins, one holding two silver coins, and one holding one of each. You pick a box at random and draw one coin from it, which turns out to…
— Joseph Bertrand, Calcul des probabilites
Imagine a hotel with infinitely many rooms, numbered 1, 2, 3 and so on, and every single room is occupied. A new guest arrives asking for a room, and the manager agrees without hes…
— David Hilbert, Hilbert's paradox of the Grand Hotel
Consider the sentence: 'This sentence is false.' If it's true, then what it says is accurate, which means it's false. But if it's false, then what it says is inaccurate, which mean…
— Eubulides of Miletus, Liar paradox
Every whole number (1, 2, 3...) can be paired with exactly one perfect square (1, 4, 9...) by squaring it, and every perfect square pairs back to exactly one whole number by taking…
— Galileo Galilei, Two New Sciences
Imagine a fair lottery with 1,000 tickets and exactly one winner. For any single ticket, it's rational to believe it will lose, since it has only a 0.1% chance of winning. But appl…
— Henry E. Kyburg Jr., Probability and the Logic of Rational Belief
Two players alternate deciding whether to take a growing shared pot of money or pass it to the other player, who then faces the same choice with a bigger pot, for a fixed number of…
— Richard McKelvey and Thomas Palfrey, An experimental study of the centipede game
A traveler wants to cross a bridge guarded by an armed man, who warns: "Swear an oath. If it's true, you may pass; if it's false, I'll throw you into the water." The traveler swear…
— Jean Buridan, Sophismata
Four bugs sit at the corners of a square. At the same instant, each starts crawling at the same constant speed directly toward the bug clockwise from it. As every bug moves, the on…
— Pierre Bouguer, Pursuit curve
On August 18, 1913, at a roulette table in Monte Carlo, the ball landed on black 26 consecutive times. As the streak grew, gamblers poured money onto red, certain the wheel was som…
— Historical account, Gambler's fallacy
Take a needle of length 1 and try to rotate it a full 180 degrees, back to its starting line but pointing the opposite way, while sweeping through the smallest possible area of the…
— Soichi Kakeya, Abram Besicovitch, Kakeya set