50 Fun Facts About College Math

William Sealy Gosset devised the Student's t-test for small samples while brewing for which Irish beer company?

Guinness preferred its staff to publish under pen names, so Gosset signed his 1908 paper as Student, and the test has carried the alias ever since.

The word calculus comes from a Latin term for which small object, once used as a counter on Roman abacuses?

Pebble is a meaning that survives in medicine, where a calculus is a stone formed in the body, such as a kidney stone.

In topology, a coffee mug counts as the same shape as which food, since one can be molded into the other?

A doughnut and a mug each have exactly one hole, and topology treats two shapes as equal when one can be stretched into the other without cutting or gluing.

Graph theory began with a 1736 proof that no walk could cross each of which Prussian city's seven bridges once?

Königsberg, now Kaliningrad in Russia, defeated every walker because all of its land masses were touched by an odd number of bridges, as Leonhard Euler showed.

Stanisław Ulam's random-sampling method, born at Los Alamos, took its code name from which gambling resort?

Monte Carlo was where Ulam's uncle borrowed money to gamble; the idea itself came to Ulam while he played solitaire and wondered how often a deal would come out.

Gottfried Leibniz spent years in a bitter priority dispute over who invented calculus with which English scientist?

Isaac Newton was president of the Royal Society when its committee ruled in his favor, and the committee's report was written by Newton himself.

Which French lawyer wrote in a book's margin that he had a marvelous proof the margin was too narrow to hold?

Pierre de Fermat never shared the proof, and his claim stood unproven for 358 years until Andrew Wiles finally cracked it in the 1990s.

A 1976 computer-assisted proof showed any flat map needs at most how many colors so no neighboring regions match?

Four colors always suffice, but proving it meant a computer checking well over a thousand map configurations one by one, which took over a thousand hours.

Which game theory puzzle, devised at the RAND Corporation in 1950, has two rational players each betray the other?

The prisoner's dilemma got its name later, when Albert Tucker recast the RAND payoffs as prison sentences to make the puzzle easier to explain.

Which prize, often described as the Nobel of mathematics, goes only to mathematicians under the age of 40?

The Fields Medal honors the Canadian mathematician John Charles Fields and was first awarded in 1936, to Lars Ahlfors and Jesse Douglas.

In a first calculus course, the derivative of the sine function sin x turns out to be which other function?

Cos x gives the slope of the sine curve at every angle, which is why a sine wave is steepest where it crosses zero and flat at its peaks.

The fundamental theorem of calculus shows that differentiation and which other operation essentially reverse each other?

Integration measures the area under a graph while differentiation measures its slope, and proving that each undoes the other is what fused them into one subject.

In calculus, which result finds the derivative of one function nested inside another, such as (x² + 1)⁵?

The chain rule multiplies the outer function's derivative by the inner one's; it seems to have been first used by Gottfried Leibniz, whose notation still expresses it.

Which method for limits of the form 0/0 is named for a French marquis who learned it from tutor Johann Bernoulli?

L'Hôpital's rule reached print in the marquis's 1696 textbook, the first on calculus; he paid Bernoulli an annual allowance for the right to use his discoveries.

Which figure, made by spinning the curve y = 1/x about an axis, has finite volume but infinite surface area?

Gabriel's horn could be filled with a finite quantity of paint that would still not be enough to coat its surface, a puzzle known as the painter's paradox.

Which calculus result about limits also goes by the nickname of the two policemen and a drunk?

The squeeze theorem pins down a function's limit by trapping it between two others that head to the same value, like officers steering a drunk between them.

On a smooth curve, what name is given to the spot where the second derivative changes sign and the bend reverses?

An inflection point needs the sign to flip: where the second derivative touches zero without changing sign, the spot is called an undulation point instead.

Which infinite expansion, introduced in 1715, rebuilds a function from its derivatives at a single point?

A Taylor series taken around zero is also called a Maclaurin series, after Colin Maclaurin, who made heavy use of that special case in the 18th century.

The simplest Riemann sums estimate the area under a curve by adding up many thin strips of which shape?

Rectangles work because the strips can be made ever thinner: as their number grows, the total closes in on the exact area, which is the idea behind the integral.

Which function with base e is famous in calculus for exactly equaling its own derivative at every point?

The exponential function is the unique one that equals its own derivative and takes the value 1 at zero, so its graph's slope always matches its height.

A square matrix can be inverted only when which single number computed from its entries is not zero?

The determinant is also the factor by which a matrix scales every volume, so a value of zero means space has been flattened and the step cannot be undone.

The eigen in eigenvalue, a core idea of linear algebra, is borrowed from which language's word for own?

German eigen is a cognate of the English word own, and it was first attached to these special values in 1904.

Introduced by Arthur Cayley in 1858, which matrix operation turns rows into columns by flipping over the diagonal?

The transpose leaves a symmetric matrix exactly as it was, because a symmetric matrix already mirrors itself across its main diagonal.

Which method for solving linear systems appears in the ancient Chinese text Nine Chapters on the Mathematical Art?

Gaussian elimination is named for Carl Friedrich Gauss, who devised a notation for it in 1810, long after Chinese scholars had written the method down.

Which idea from combinatorics guarantees that among 367 people at least two must share a birthday?

The pigeonhole principle works because only 366 birthdays exist; it also proves that at least two people in London have the same number of hairs on their heads.

Georg Cantor labeled the sizes of infinite sets, beginning with the counting numbers, using which Hebrew letter?

Aleph is the first letter of the Hebrew alphabet, and Cantor chose it because he did not want to invent a brand-new symbol for his infinities.

Which mathematician imagined a fully booked hotel with infinitely many rooms that can always fit one more guest?

David Hilbert introduced the hotel in a lecture in the 1920s, and George Gamow's 1947 book One Two Three... Infinity turned it into a popular puzzle.

Which logician showed in 1931 that any consistent system of arithmetic holds true statements it cannot prove?

Kurt Gödel later became a good friend of Albert Einstein, whom he met on his first trip to the United States in 1933.

The classic argument that √2 is irrational assumes it is a fraction and derives an impossibility; what method is this?

Proof by contradiction is also known by the Latin name reductio ad absurdum, which translates as reduction to absurdity.

Proof by mathematical induction is often pictured as a line of which objects, each one knocking over the next?

Dominoes show the two steps: prove the first one falls, then prove each one topples the next. The other popular picture is climbing a ladder rung by rung.

Ladislaus Bortkiewicz made the Poisson distribution famous by counting Prussian soldiers accidentally killed by what?

Horse kicks were rare and random enough to fit the formula neatly, and Bortkiewicz published the tally in his 1898 book The Law of Small Numbers.

Which result explains why averages of many random samples form a bell curve, even when the underlying data do not?

The central limit theorem got its name in 1920 from George Pólya, who called it central because of its importance to the whole of probability.

In statistics, what name is given to the default claim of no effect that a test sets out to reject?

The null hypothesis had a famous early outing when Ronald Fisher tested a lady who claimed she could taste whether the milk or the tea had been poured first.

Which pattern, used by auditors to flag faked data, says about 30% of real-world figures begin with the digit 1?

Benford's law was first spotted in 1881 by astronomer Simon Newcomb, who noticed that the early pages of logarithm tables were far more worn than the rest.

Which unproven claim, first made in a 1742 letter, says every even number above 2 is the sum of two primes?

Goldbach's conjecture was sent in a letter to Leonhard Euler, and computers have since confirmed it for every even number up to 4 × 10¹⁸ without finding a proof.

Which operation, written with an exclamation mark, multiplies a whole number by every positive whole number below it?

Factorial notation was introduced by the French mathematician Christian Kramp in 1808, and by convention the factorial of zero is equal to 1.

Which kind of calculation, set out in the 1801 book Disquisitiones Arithmeticae, wraps numbers around like a clock?

Modular arithmetic is why 4 hours after 9 o'clock is 1 o'clock: on a 12-hour dial, 13 leaves a remainder of 1.

Which famous optimization puzzle asks for the shortest round trip through every city on a list, visiting each once?

The traveling salesman problem gets brutally hard as cities are added, yet in 2006 an exact best tour through 85,900 points on a microchip was computed.

Which unsolved question, worth a $1 million Millennium Prize, asks whether easily checked answers are also easily found?

P versus NP has an everyday flavor: a filled-in Sudoku grid is quick to check, yet nobody knows whether such puzzles can always be solved quickly too.

Which Berkeley student arrived late to a 1939 lecture and solved two open statistics problems he mistook for homework?

George Dantzig's mix-up grew into an urban legend and an opening scene of the film Good Will Hunting; he later developed the simplex algorithm.

In the film Good Will Hunting, math genius Will secretly solves blackboard problems while holding which job at MIT?

Janitor Will Hunting anonymously solves a problem a professor posted for graduate students; the script won Matt Damon and Ben Affleck an Academy Award.

The 2001 film A Beautiful Mind, starring Russell Crowe, tells the life story of which mathematician and Nobel laureate?

John Nash shared the 1994 Nobel Memorial Prize in Economic Sciences for game theory work he had done as a Princeton graduate student.

Euler's identity, once voted the most beautiful theorem in mathematics, says e raised to the power iπ equals what?

−1 is the value, so adding 1 gives exactly zero, tying e, i, π, 1 and 0 into one line; readers of The Mathematical Intelligencer picked it in a 1990 poll.

Which closed surface, first described in 1882, has only one side and so no distinct inside or outside?

The Klein bottle, named for Felix Klein, cannot be embedded in ordinary three-dimensional space without the surface passing through itself.

The Lotka–Volterra differential equations describe the rise and fall of which pair of interacting populations?

Predators and prey cycle out of step in the model, which grew from the puzzle of why predatory fish made up more of the Adriatic catch during World War I.

William Rowan Hamilton cut the defining formula of which number system into the stone of a Dublin bridge in 1843?

Quaternions came to Hamilton on a walk and he carved the formula with his pocket knife; today they handle three-dimensional rotations in computer graphics.

Which playfully named result in topology implies that somewhere on Earth there must always be a cyclone?

The hairy ball theorem says you cannot comb a hairy ball flat without creating a cowlick, and wind over the globe behaves like that hair.

Which famous sum, 1 + 1/2 + 1/3 and so on, grows without bound although its terms shrink to zero?

The harmonic series was proved to diverge around 1350 by Nicole Oresme, and its name comes from the harmonics of a vibrating string in music.

Commutative groups are named for which Norwegian, who proved the general quintic unsolvable and died at 26?

Niels Abel made his discoveries while living in poverty; the adjective abelian is so common that it is usually spelled with a lowercase a.

Which annual exam for North American undergraduates is so hard the median score is often zero or one out of 120?

Putnam Competition winners, known as Putnam Fellows, include two later Nobel laureates in physics, Richard Feynman and Kenneth Wilson.