60 Fun Facts About Hard Math

Who published the first successful proof of Fermat's Last Theorem in 1995?

Fermat had written that he had a proof too large to fit in the margin of his copy of Arithmetica.

How many Millennium Prize Problems did the Clay Mathematics Institute name in 2000?

They include the Riemann hypothesis, P versus NP, Navier-Stokes and the Poincaré conjecture, the only one solved so far.

The prize presented every four years at the International Congress of Mathematicians goes only to those under what age?

It is presented every four years at the International Congress of Mathematicians; Maryam Mirzakhani became the first woman to win it in 2014.

Which honour did Grigori Perelman refuse in 2006, saying he did not want to be 'on display like an animal in a zoo'?

In 2010 he also declined the first Clay Millennium Prize, worth a million dollars.

The record prime discovered in October 2024 has how many decimal digits?

It is the Mersenne prime 2^136,279,841 - 1, found by a GIMPS volunteer running cloud machines; at discovery it was the largest prime known.

Most record-breaking primes are Mersenne primes. A Mersenne prime is one less than what?

They can be checked with a special primality test that is far faster than the general one.

Graham's number arose as an upper bound in which field of mathematics?

Ronald Graham used it explaining his work to the popular science writer Martin Gardner.

Paul Erdős, who published at least 1,525 papers, gave his name to a number measuring what?

He co-wrote with more than 500 people, so a great many mathematicians have an Erdős number of 1 or 2.

In 1976 the four colour theorem became the first major theorem proved with the help of what?

Kenneth Appel and Wolfgang Haken checked a vast number of "reducible configurations"; many mathematicians initially refused to accept a proof no human could verify.

Euler's 1736 proof about the seven bridges of Königsberg founded which branch of mathematics?

The city is now Kaliningrad, and the puzzle also foreshadowed topology.

Why is 1729 called the Hardy-Ramanujan number?

Hardy had called the taxi number dull; Ramanujan replied that it was very interesting indeed.

How many problems did David Hilbert set for the 20th century in his famous 1900 list?

The first of them, the continuum hypothesis, turned out to be independent of the standard axioms of set theory.

In which year did Kurt Gödel publish his incompleteness theorems?

They show that any sufficiently powerful consistent system contains true statements it cannot prove.

In the Monty Hall problem, what is the probability of winning the car if you switch doors?

Marilyn vos Savant's 1990 Parade column giving this answer drew thousands of angry letters, many from PhDs.

How many people must be in a room before the chance that two share a birthday exceeds 50%?

The trick is that you are comparing every possible pair, not one person against everyone else.

Euler solved the Basel problem in 1734, summing the reciprocals of the squares. What is the sum?

The problem had been posed by Pietro Mengoli in 1650 and had defeated the leading mathematicians of the age; Euler was 28.

6 is the first perfect number, equal to the sum of its proper divisors. What is the second?

Its divisors 1, 2, 4, 7 and 14 add up to 28; the next two are 496 and 8128, and no odd perfect number has ever been found.

220 and 284, each the sum of the other's proper divisors, are the smallest pair of what numbers?

The divisors of 220 sum to 284 and the divisors of 284 sum to 220.

The unsolved Collatz conjecture, introduced in 1937, generates sequences nicknamed what?

Halve even numbers, triple-plus-one odd numbers, and the values rise and fall like hail in a cloud before, apparently always, hitting 1.

How many Platonic solids exist?

Tetrahedron, cube, octahedron, dodecahedron and icosahedron; Plato matched them to the classical elements in the Timaeus.

The Kepler conjecture, proved by Thomas Hales in 1998, concerns the densest way to do what?

The greengrocer's pyramid stacking is optimal, filling about 74.05% of space; the proof needed enormous computer checking.

The Abel Prize, often called the Nobel of mathematics, is awarded each year by whom?

King Oscar II wanted to fund it in 1902, but the break-up of the Sweden-Norway union scuppered the plan for a century.

Évariste Galois, founder of group theory, died at 20 after what?

The reasons remain obscure; he wrote "I need all my courage to die at twenty" the night before.

Ramanujan, the self-taught genius Hardy brought to Cambridge, died in 1920 at what age?

Hardy said some of his theorems "defeated me completely; I had never seen anything in the least like them before".

Euclid's Elements is a collection of definitions, postulates and proofs in how many books?

It draws on Hippocrates of Chios, Eudoxus and Theaetetus and covers geometry, number theory and incommensurability.

According to Cicero, Archimedes' tomb was topped with which two shapes from his favourite proof?

Cicero found the neglected tomb overgrown near the Agrigentine gate while serving as quaestor in Sicily.

'Algorithm' and 'algebra' both trace back to which 9th-century Baghdad scholar?

Algebra comes from the title of his book Al-Jabr; algorithm comes from the Latin form of his name.

Which 7th-century Indian mathematician was the first to formalise zero as a number, in 628?

The Brāhmasphuṭasiddhānta also gave rules for arithmetic with negative numbers.

Which number system's defining formula did William Rowan Hamilton carve into a Dublin bridge in 1843?

The carving on Brougham (Broom) Bridge has faded, but mathematicians still make an annual Hamilton Walk to it.

Legend says the Pythagorean Hippasus was drowned at sea for revealing the existence of what?

The square root of 2 was probably the first number known to be irrational.

About how many reachable positions does a standard 3x3x3 Rubik's Cube have?

That is 43,252,003,274,489,856,000, counting only positions reachable by turning the faces.

Who coined the word "googol" for 10 to the power 100?

Milton Sirotta came up with it in 1920, possibly inspired by the comic-strip character Barney Google.

The prime number theorem was proved independently in 1896 by Jacques Hadamard and whom?

Both proofs leaned on Riemann's zeta function.

In 1963 Paul Cohen showed the continuum hypothesis is what, relative to the standard ZFC axioms?

It was the first of Hilbert's 23 problems; Gödel had shown in 1940 that it could not be disproved, Cohen that it could not be proved.

Nicolas Bourbaki, author of a famous series of rigorous textbooks, is actually what?

The group, mostly École normale supérieure alumni, formed in 1934-35 to write a new analysis textbook.

Which mathematician, praised by Einstein, has a theorem linking symmetry to conservation laws?

Hilbert and Klein brought her to Göttingen in 1915.

At 19, Gauss proved a regular polygon with how many sides can be built with compass and straightedge?

The heptadecagon result of 1796 convinced him to become a mathematician rather than a philologist.

Sophie Germain corresponded with Lagrange, Legendre and Gauss under what male pseudonym?

She won the Paris Academy's grand prize for elasticity theory and laid groundwork on Fermat's Last Theorem.

Our 60-minute hours and 360-degree circles descend from the base-60 system of which people?

It passed from Sumer to Babylon in the 3rd millennium BC and survives in time, angles and coordinates.

What is the minimum number of pieces in the Banach-Tarski paradox, which turns one ball into two?

The pieces are non-measurable sets, which is why it works on paper and never with a knife.

What is the kissing number in three dimensions, the most identical spheres that can touch a central one?

Newton believed it was 12 and Gregory 13; it took until 1953 to settle.

The one-sided surface made by joining a strip's ends with a half-twist was discovered in 1858 by August Möbius and whom?

The shape had already appeared in Roman mosaics from the third century.

How many prime numbers are there below 100?

The list runs from 2 to 97; the prime number theorem says there are roughly n/ln n primes below n.

John Nash, subject of A Beautiful Mind, shared the 1994 Nobel in Economics for work in which field?

He shared it with John Harsanyi and Reinhard Selten, and in 2015 won the Abel Prize with Louis Nirenberg for partial differential equations.

Benoit Mandelbrot made the first high-quality images of his namesake fractal in 1980 while working for which company?

He was at the Thomas J. Watson Research Center.

Claude Shannon's 1950 lower bound for the number of possible chess games is roughly 10 to what power?

He assumed about 1,000 options per pair of moves over a 40-move game, to show brute force could never solve chess.

The Riemann hypothesis says the non-trivial zeros of the zeta function all have what real part?

It is one of the Millennium Prize Problems and still unproved.

Goldbach's conjecture, sent in a 1742 letter to Euler, claims that every even number greater than 2 is what?

It has been checked far beyond 10^18 but never proved.

In 2013 Yitang Zhang made a breakthrough towards settling which famous open question?

James Maynard, Terence Tao and others quickly shrank his bounded gap, but infinitely many twin primes remains unproved.

The Poincaré conjecture, stated in 1904 and proved by Perelman, is a statement about which object?

It says a finite space that looks locally like ordinary 3D space with no holes must be a 3-sphere.

In which year did Felix Klein first describe the closed, one-sided surface now called the Klein bottle?

Unlike the Möbius strip it has no boundary, and it cannot be built in three dimensions without passing through itself.

To how many decimal places is e, the base of natural logarithms, equal to 2.71828?

It is sometimes called Euler's number or Napier's constant.

What is the sum of the interior angles of a hexagon?

Each angle of a regular hexagon is 120°, a third of a full turn, which is why they tile so neatly.

Pi Day on 14 March was founded in 1988 by physicist Larry Shaw at which institution?

Staff and visitors marched around a circular space and then ate fruit pies.

Which 19th-century number theorist first used the name 'Fibonacci sequence'?

Lucas, best known for the Lucas numbers, coined the name; the sequence itself appears in Fibonacci's 1202 Liber Abaci and in Indian work as early as Pingala.

The Fibonacci numbers were first described around 200 BC by which Indian scholar studying Sanskrit poetry?

Pingala enumerated patterns of long and short syllables; Virahanka and Hemachandra later gave clearer expositions.

Which Welsh mathematician first used the Greek letter π for the circle ratio, in 1706?

Jones introduced π in 1706; Euler's later adoption of the symbol popularised it.

William Shanks's 1873 calculation of 707 digits of π was wrong from which digit onward?

His 1853 error at digit 528 invalidated everything after it, including the 100 extra digits he added in 1873.

Until the early 20th century, Germans called π the 'Ludolphian number' after which mathematician?

Van Ceulen reached 20 digits in 1596 and later 35, a record that earned π its German nickname.

How many publications by Leonhard Euler are collected in the Opera Omnia?

Euler's 866 publications make him arguably the most prolific contributor in the history of mathematics and science.