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60 free Hard Math trivia questions with answers — science & nature quiz, new questions added Aug 2026.
54 free hard math trivia questions with answers. These hard math trivia questions are for people who found the standard rounds too easy. They cover the great theorems and unsolved problems (Fermat's Last Theorem, the Riemann hypothesis, the Poincaré conjecture, Goldbach, Collatz, P versus NP), the prizes and the people who refused them, the constants and paradoxes (Monty Hall, the birthday problem, Banach-Tarski, Zeno), the history from Babylonian base 60 and Euclid's Elements to Al-Khwarizmi, Gauss, Galois, Ramanujan and Emmy Noether, and a handful of pure calculation questions you can check on paper. The easiest questions here would be the hardest on a general quiz (how many Platonic solids there are, who proved Fermat's Last Theorem). The medium ones ask what the Fields Medal age limit is, why 1729 is famous and how many people you need in a room for a shared birthday to be likely. The hard ones want the year Hilbert set his 23 problems, how many pieces Banach-Tarski needs, who coined "googol" and what Hamilton carved into a Dublin bridge. Every answer has been checked against a reference encyclopaedia, and each question shows its explanation and citation once you have answered.
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Q 01Who published the first successful proof of Fermat's Last Theorem in 1995?
Andrew Wiles
Fermat had written that he had a proof too large to fit in the margin of his copy of Arithmetica.
Q 02How many Millennium Prize Problems did the Clay Mathematics Institute name in 2000?
Seven
They include the Riemann hypothesis, P versus NP, Navier-Stokes and the Poincaré conjecture, the only one solved so far.
Q 03The prize presented every four years at the International Congress of Mathematicians goes only to those under what age?
40
It is presented every four years at the International Congress of Mathematicians; Maryam Mirzakhani became the first woman to win it in 2014.
Q 04Which honour did Grigori Perelman refuse in 2006, saying he did not want to be 'on display like an animal in a zoo'?
The Fields Medal
In 2010 he also declined the first Clay Millennium Prize, worth a million dollars.
Q 05The record prime discovered in October 2024 has how many decimal digits?
41,024,320
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.
Q 06Most record-breaking primes are Mersenne primes. A Mersenne prime is one less than what?
A power of two
They can be checked with a special primality test that is far faster than the general one.
Q 07Graham's number arose as an upper bound in which field of mathematics?
Ramsey theory
Ronald Graham used it explaining his work to the popular science writer Martin Gardner.
Q 08Paul Erdős, who published at least 1,525 papers, gave his name to a number measuring what?
Collaborative distance by co-authorship
He co-wrote with more than 500 people, so a great many mathematicians have an Erdős number of 1 or 2.
Q 09In 1976 the four colour theorem became the first major theorem proved with the help of what?
A computer
Kenneth Appel and Wolfgang Haken checked a vast number of "reducible configurations"; many mathematicians initially refused to accept a proof no human could verify.
Q 10Euler's 1736 proof about the seven bridges of Königsberg founded which branch of mathematics?
Graph theory
The city is now Kaliningrad, and the puzzle also foreshadowed topology.
Q 11Why is 1729 called the Hardy-Ramanujan number?
It is the smallest sum of two cubes in two different ways
Hardy had called the taxi number dull; Ramanujan replied that it was very interesting indeed.
Q 12How many problems did David Hilbert set for the 20th century in his famous 1900 list?
23
The first of them, the continuum hypothesis, turned out to be independent of the standard axioms of set theory.
Q 13In which year did Kurt Gödel publish his incompleteness theorems?
1931
They show that any sufficiently powerful consistent system contains true statements it cannot prove.
Q 21The Kepler conjecture, proved by Thomas Hales in 1998, concerns the densest way to do what?
Pack spheres in space
The greengrocer's pyramid stacking is optimal, filling about 74.05% of space; the proof needed enormous computer checking.
Q 22The Abel Prize, often called the Nobel of mathematics, is awarded each year by whom?
The King of Norway
King Oscar II wanted to fund it in 1902, but the break-up of the Sweden-Norway union scuppered the plan for a century.
Q 23Évariste Galois, founder of group theory, died at 20 after what?
A duel
Q 14In the Monty Hall problem, what is the probability of winning the car if you switch doors?
2/3
Marilyn vos Savant's 1990 Parade column giving this answer drew thousands of angry letters, many from PhDs.
Q 15How many people must be in a room before the chance that two share a birthday exceeds 50%?
23
The trick is that you are comparing every possible pair, not one person against everyone else.
Q 16Euler solved the Basel problem in 1734, summing the reciprocals of the squares. What is the sum?
π²/6
The problem had been posed by Pietro Mengoli in 1650 and had defeated the leading mathematicians of the age; Euler was 28.
Q 176 is the first perfect number, equal to the sum of its proper divisors. What is the second?
28
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.
Q 18220 and 284, each the sum of the other's proper divisors, are the smallest pair of what numbers?
Amicable
The divisors of 220 sum to 284 and the divisors of 284 sum to 220.
Q 19The unsolved Collatz conjecture, introduced in 1937, generates sequences nicknamed what?
Hailstone numbers
Halve even numbers, triple-plus-one odd numbers, and the values rise and fall like hail in a cloud before, apparently always, hitting 1.
Q 20How many Platonic solids exist?
Five
Tetrahedron, cube, octahedron, dodecahedron and icosahedron; Plato matched them to the classical elements in the Timaeus.
The reasons remain obscure; he wrote "I need all my courage to die at twenty" the night before.
Q 24Ramanujan, the self-taught genius Hardy brought to Cambridge, died in 1920 at what age?
32
Hardy said some of his theorems "defeated me completely; I had never seen anything in the least like them before".
Q 25Euclid's Elements is a collection of definitions, postulates and proofs in how many books?
13
It draws on Hippocrates of Chios, Eudoxus and Theaetetus and covers geometry, number theory and incommensurability.
Q 26According to Cicero, Archimedes' tomb was topped with which two shapes from his favourite proof?
A sphere and a cylinder
Cicero found the neglected tomb overgrown near the Agrigentine gate while serving as quaestor in Sicily.
Q 27'Algorithm' and 'algebra' both trace back to which 9th-century Baghdad scholar?
Al-Khwarizmi
Algebra comes from the title of his book Al-Jabr; algorithm comes from the Latin form of his name.
Q 28Which 7th-century Indian mathematician was the first to formalise zero as a number, in 628?
Brahmagupta
The Brāhmasphuṭasiddhānta also gave rules for arithmetic with negative numbers.
Q 29Which number system's defining formula did William Rowan Hamilton carve into a Dublin bridge in 1843?
Quaternions
The carving on Brougham (Broom) Bridge has faded, but mathematicians still make an annual Hamilton Walk to it.
Q 30Legend says the Pythagorean Hippasus was drowned at sea for revealing the existence of what?
Irrational numbers
The square root of 2 was probably the first number known to be irrational.