100 Fun Facts About High School Math

In the famous birthday paradox, what is the smallest group size with a better-than-even chance of a shared birthday?

23 people are enough for a 50.7% chance, because they form 253 different pairs who might match, far more than the 22 comparisons any one of them makes.

In the Monty Hall problem, which move wins the car two times in three?

Switching to the other door wins two times in three. When Marilyn vos Savant said so in Parade magazine, thousands of readers, many signing as PhDs, wrote in to tell her she was wrong.

Which Greek, whose followers were reportedly forbidden to eat beans, gives his name to the right-triangle rule a² + b² = c²?

Pythagoras's theorem even has a proof by a US president: James Garfield published one in 1876, while he was still a congressman, built around a trapezoid.

NASA's Jet Propulsion Laboratory steers spacecraft between planets using just 15 decimal places of which constant?

π to 15 decimal places is all the laboratory needs for its most exact work, because that pins the edge of a circle billions of miles across to within the length of a little finger.

After twenty heads in a row, what is the probability that a fair coin lands heads on the next flip?

The chance stays 50% because a coin has no memory; believing a tail is 'due' is the gambler's fallacy, also called the Monte Carlo fallacy after a 1913 roulette run of 26 blacks.

Which saves a shopper more: one 40% price cut, or a 20% cut followed by 20% off the reduced price?

The single cut. The second 20% comes off an already-reduced price, so the double cut leaves 64% of the original (0.8 × 0.8) and saves only 36%.

What is the repeating decimal 0.999…, the limit of the sequence 0.9, 0.99, 0.999 and so on, in ordinary arithmetic?

Exactly 1, not a whisker less: tripling 1/3 = 0.333… gives 0.999… on one side and 1 on the other, so the two decimals are simply two names for one number.

What do the numbers 1 to 100 add up to, famously found fast by pairing 1 with 100, 2 with 99?

The total is 5,050 because the fifty pairs 1 + 100, 2 + 99 … 50 + 51 each make 101, and 50 × 101 = 5,050.

Which 9th-century Baghdad scholar's Latinised name gave us 'algorithm', and his book Al-Jabr the word 'algebra'?

Al-Khwarizmi wrote Al-Jabr as a practical handbook for what people need in inheritance, lawsuits and trade, and much of the book works through the rules of Islamic inheritance.

Which mathematician from Pisa brought the sequence 1, 1, 2, 3, 5, 8 to Europe in 1202 via a puzzle about rabbits?

Fibonacci did not invent the sequence: Indian scholars had described it centuries earlier, and his 1202 Liber Abaci mattered more for spreading the Hindu-Arabic numerals across Europe.

What is b² − 4ac called, the expression under the quadratic formula's square root, whose sign counts the real roots?

A negative discriminant means the quadratic's graph never touches the x-axis, so its two roots are complex conjugates rather than real numbers.

In the line equation y = mx + b, what is m, a quantity that is undefined for a vertical line?

The slope's m is a mystery: nobody knows why the letter was chosen, and the suggestion that it stands for the French monter, 'to climb', has no evidence behind it.

Which acronym do American schools teach for the order of operations, the rule that makes 2 + 3 × 4 equal 14?

PEMDAS puts multiplication ahead of addition, so 2 + 3 × 4 is 2 + 12 = 14 and not the 20 that working left to right gives; British pupils learn the same rule as BODMAS and Canadians as BEDMAS.

A colony doubles every hour; by what factor has it grown after ten hours, a number every computer scientist knows?

Ten doublings give 2¹⁰ = 1,024, which is why a kilobyte traditionally meant 1,024 bytes rather than a round 1,000.

Which curve, the graph of every quadratic function, did Galileo show is also the path of a thrown projectile?

A parabola reflects every ray arriving parallel to its axis through one focus, which is why satellite dishes and car headlight reflectors use the shape.

Which French philosopher's 1637 treatise La Géométrie united algebra and geometry?

René Descartes published it as an appendix to his Discourse on the Method, and in it named 'imaginary' numbers: roots one could imagine, he wrote, though no real quantity matched them.

Which Scottish laird published the first logarithms in 1614, so astronomers could multiply big numbers by adding?

John Napier's logarithms, said Laplace, 'doubled the life of the astronomer' by cutting the labour of many months to a few days.

Which number, about 1.618 and written with the Greek letter phi, is the positive solution of x² = x + 1?

The golden ratio is the hardest number to approximate with fractions: its continued fraction is all 1s, so approximations such as 13/8 converge as slowly as possible.

Which constant, about 2.718, is the base of natural logarithms and the limit of (1 + 1/n)ⁿ as n grows without bound?

The constant e was found by Jacob Bernoulli in 1683 while studying compound interest: compounding a 100% yearly rate ever more often tops out at a factor of 2.718.

Which number, the length of a 1 × 1 square's diagonal, was probably the first ever proved to be irrational?

√2 carries a grim legend: Hippasus, the Greek said to have shown it cannot be written as a fraction, was supposedly drowned at sea for it, though the ancient accounts differ on his crime.

How many degrees do a flat triangle's three angles add up to, a total that grows larger on a sphere?

180 is the total only on a flat surface: on a sphere the three angles add up to more, anywhere up to 540°, and the bigger the triangle's area, the bigger the surplus.

What is the sum of the interior angles of an octagon, the shape of the familiar red stop sign?

An octagon's angles total 1,080° because it splits into six triangles from one corner; each corner of a regular octagon measures 135°.

Besides the equilateral triangle and square, which regular polygon is the only one that tiles a floor by itself?

Hexagon floors put three tiles at every corner, and the honeycomb is no accident: Thomas Hales proved in 1999 that no way of dividing a surface into equal cells uses less wall.

Which polygon, the shape of Britain's 50p coin, is the smallest regular one that compass and straightedge cannot draw?

Heptagon coins arrived in 1969 with Britain's 50p, whose sides are curved so that it is the same width whichever way it is measured and rolls through vending machines like a round coin.

Which side of a right triangle is the denominator of both SOH and CAH in the SOHCAHTOA mnemonic?

Hypotenuse, the name of the longest side, comes from the Greek for 'stretching under', because the side stretches across beneath the right angle it faces.

How far apart are the points (0, 0) and (6, 8) on a coordinate grid, by the distance formula?

10, because √(6² + 8²) = √100: the line joining the points is the long side of a 6-8-10 right triangle, the famous 3-4-5 with every side doubled.

An arc subtends 40° at a circle's centre; what angle does it subtend at a point on the remaining circumference?

20°: the angle at the circumference is always half the angle at the centre on the same arc, wherever the point sits, which is why any angle drawn in a semicircle is a right angle.

Which astronomer nested the five Platonic solids inside one another in 1596 to model the Solar System?

Johannes Kepler thought the five solids explained why there were just six planets: nested with a sphere around each, they marked out six shells, one for every planet then known.

Whose Elements, compiled around 300 BC in Alexandria, contains the classic proof that the prime numbers never run out?

Euclid's argument is short: multiply any list of primes together and add 1, and the result must have a prime factor that the list left out.

Which Greek, reputed to have predicted a 585 BC eclipse, is credited with proving every angle in a semicircle is 90°?

Thales is said to have measured the height of Egypt's pyramids from their shadows, at the moment when his own shadow was as long as he was tall.

In a right triangle, which trig function is the opposite side divided by the adjacent side?

Tangent is Latin for 'touching': drawn on a unit circle its line only touches the circle, while its partner the secant, from the Latin for 'cutting', slices through it.

Which rule for any triangle's third side collapses to a² + b² = c² when the included angle is a right angle?

The law of cosines appears in Book II of the Elements around 300 BC as a statement about rectangle areas, long before sines or cosines existed.

Which unit of angle, equal to about 57.3°, divides a full turn into 2π parts?

Radian: one is the angle whose arc is just as long as the circle's radius, and since a circumference is about 6.28 radii long, a full turn holds about 6.28 of them.

Which Greek astronomer, famed for discovering the precession of the equinoxes, is called the founder of trigonometry?

Hipparchus compiled the first known table of chords, an ancestor of the sine table, so that the triangles in his astronomy need not each be solved from scratch.

Which Greek of Syracuse asked for a sphere inside a cylinder to be carved on his tomb, marking his favourite discovery?

Archimedes was killed by a Roman soldier when Syracuse fell in 212 BC; Cicero went looking for the grave 137 years later and found it lost under brambles, recognising it by the carving.

Which Alexandrian librarian, who measured the Earth's circumference, gives his name to a prime-finding 'sieve'?

Eratosthenes sized the Earth with shadows: he compared the midsummer noon shadow at Alexandria with the one at Syene, far to the south, and scaled up the distance between the two cities.

Which Swiss mathematician, author of some 866 publications, was the first to write f(x) for a function applied to x?

Leonhard Euler produced almost half of his work after going blind, relying on a prodigious memory and on the sons and assistants who wrote it all down.

Which German, called the 'Prince of Mathematicians', gives his name to the bell-shaped curve of the normal distribution?

Carl Friedrich Gauss made his name in 1801, when the newly found Ceres vanished behind the Sun and astronomers recovered it almost precisely where he had calculated it would be.

Which self-taught Indian mathematician famously told G. H. Hardy that the taxi number 1729 was far from dull?

Srinivasa Ramanujan, ill in bed when Hardy called, answered at once that 1729 is the smallest number that can be written as a sum of two cubes in two ways: 1³ + 12³ and 9³ + 10³.

Which average do reports of incomes prefer to the mean, since a handful of billionaires can drag the mean upward?

The median is the middle value, so in the list 1, 2, 3, 4, 1,000 it stays at 3 while the mean leaps to 202.

Which measure of spread, the square root of the variance, was named in print by Karl Pearson in 1894?

Standard deviation was Pearson's handier name for the 'root mean square error'; in a bell curve about 68% of values lie within one standard deviation of the mean and 95% within two.

Which Englishman gives his name to the theorem for updating a probability when new evidence arrives?

Thomas Bayes, a Presbyterian minister, never published the theorem: after he died in 1761 his friend Richard Price found the essay among his papers and sent it to the Royal Society.

In how many different orders can five people line up for a photo, the number written as 5! in maths?

120: any of the five can stand first, any of the other four second and so on, which gives 5 × 4 × 3 × 2 × 1; a sixth person in the photo would push the count to 720.

Which French prodigy, who began building a calculator in his teens, helped found probability theory in 1654 letters on gambling?

Blaise Pascal did not discover the triangle that carries his name: it was described in China and Persia centuries earlier, and his own treatise on it was printed only after his death.

Which French magistrate's margin note claimed a marvellous proof, too long to fit there, of a theorem unproved until the 1990s?

Pierre de Fermat never wrote the proof down, and it is doubted that he had a sound one: the proof Andrew Wiles finished in the 1990s rests on mathematics from long after Fermat's time.

As of 2025, whose conjecture that every even number above 2 is the sum of two primes remained unproved?

Goldbach's conjecture has been checked by computer for every even number up to 4 × 10¹⁸, and in 2013 Harald Helfgott released a proof of a weaker version about odd numbers, yet the even case stays open.

Which number is the smallest 'perfect' number, one equal to the sum of its proper divisors, with 28 being the next?

6 = 1 + 2 + 3; every perfect number ever found is even, and nobody knows whether an odd perfect number exists.

Which Alexandrian, who described a steam-driven spinning sphere, gives his name to the formula for a triangle's area from its sides?

Heron's formula needs only the three side lengths, with no height or angle; his books also describe a string of ingenious machines, among them a coin-operated water dispenser.

Niccolò Tartaglia shared his secret method for which kind of equation with Cardano, who promised silence and then published it?

Cubic: Tartaglia, nicknamed 'the stammerer' after a sabre cut to his jaw as a boy, gave Cardano the solution in verse in 1539, and Cardano printed it six years later in his Ars Magna.

The Indian scholar Brahmagupta's 628 treatise set out the first known rules for doing arithmetic with which number?

Zero: his rules for adding, subtracting and multiplying with it still hold, but he also ruled that zero divided by zero is zero, which modern mathematics leaves undefined.

Which symbol did Welshman Robert Recorde introduce in 1557 to avoid writing a tedious phrase over and over?

The equals sign began as a pair of long parallel lines, which Recorde chose because, as he put it, no two things can be more equal.

Which name did nine-year-old Milton Sirotta coin in 1920 for the number 10 to the power 100?

Googol was the boy's suggestion to his uncle, the American mathematician Edward Kasner. A misspelling of the word later gave the search engine Google its name.

Which Oxford mathematics lecturer, born Charles Dodgson, wrote Alice's Adventures in Wonderland under a pen name?

Lewis Carroll taught mathematics at Christ Church, Oxford, for 26 years, and built the pen name by translating his first two names into Latin and back into English.

Which NASA mathematician did John Glenn ask to personally recheck the computer's sums before his Friendship 7 flight?

Katherine Johnson verified the electronic computer's orbit figures, and Glenn refused to fly until she had. Her story reached cinemas in the 2016 film Hidden Figures.

Which learned body made Florence Nightingale its first female member, after she charted army deaths in Crimea?

The Royal Statistical Society was admitting a pioneer of data graphics: her rose-shaped polar area diagram showed far more soldiers dying from illness than from battle wounds.

A bat and a ball cost $1.10 together and the bat costs $1 more; what does the ball cost?

5 cents is right, because the bat then costs $1.05 and the pair $1.10. The instinctive answer is a dime, which is why psychologists use the puzzle to test whether people check their first hunch.

Which one-sided surface, a paper band joined with a half-twist, stays in one piece when cut down the middle?

The Möbius strip, described by two German mathematicians in 1858, turns into one longer loop with extra twists when scissors run along its centre line, instead of falling into two rings.

Which Greek hero can never overtake a tortoise with a head start in a famous paradox by Zeno of Elea?

Achilles must first reach the spot the tortoise has just left, by which time it has crept ahead, and so on without end. Modern answers to the puzzle lean on calculus.

On which game board does a legend place one grain of wheat, then two, then four, doubling on every square?

A chessboard has 64 squares, so the doubling ends with more than 18 quintillion grains in all. The final square alone would hold over nine quintillion.

Which ancient people's base-60 arithmetic, inherited from the Sumerians, gave us the hour of 60 minutes?

Babylonians wrote their base-60 numbers on clay tablets, a handy choice of base because 60 divides evenly by 2, 3, 4, 5 and 6.

Which Englishman called his calculus the method of fluxions, developing it as plague shut Cambridge in 1665?

Isaac Newton wrote his treatise on fluxions in 1671, but it reached print only in 1736, after his death. A fluxion was simply his word for a rate of change.

Which German thinker devised the dy/dx and elongated-S integral notation that calculus classes use?

Gottfried Leibniz drew the integral symbol as a stretched S for the Latin summa, meaning sum, and it first appeared in print in 1686.

Which drawing of overlapping circles, a classroom staple since the 'new math' of the 1960s, shows what sets share?

The Venn diagram dates from around 1880, but its creator, the English logician John Venn, never used that name for his own drawings.

Which daughter of the poet Lord Byron wrote what is often called the first published computer program?

Ada Lovelace published it in 1843 as a method for Charles Babbage's proposed Analytical Engine to calculate a sequence called the Bernoulli numbers.

Which Dutch artist, inspired by mathematics, filled his prints with interlocking birds and fish and impossible stairs?

M. C. Escher believed he had no mathematical ability, yet he swapped ideas with mathematicians such as Roger Penrose and carried out his own research into tessellation.

Which American mathematician and Nobel laureate did Russell Crowe portray in the 2001 film A Beautiful Mind?

John Nash shared the 1994 economics Nobel for game theory he had worked out as a Princeton graduate student, before schizophrenia interrupted his career.

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

The Fields Medal is presented at the International Congress of Mathematicians, and its first two winners, a Finn and an American, were named in 1936.

Which idea did John Wallis give its symbol in 1655, a looping curve that livestock branders call the 'lazy eight'?

Infinity got its sideways figure-eight from Wallis, an English mathematician. The same shape, also called a lemniscate, already decorated combs in the Viking Age.

Which abbreviation, from the Latin for 'that which was to be demonstrated', traditionally closes a mathematical proof?

QED stands for quod erat demonstrandum, itself a Latin version of a Greek phrase. Many modern books print a small solid black square at the end of a proof instead.

What is the fewest colours that can fill any flat map with no neighbours matching, a fact proved by computer in 1976?

Four colours always suffice, a pattern Francis Guthrie noticed in 1852 while colouring a map of the counties of England. Its proof was the first of a major theorem to rely on a computer.

Which of these numbers is one of the 25 primes below 100, with no divisors but 1 and the number?

97 is the last prime before 100, while the tempting 91 is 7 × 13, 93 is 3 × 31 and 95 is 5 × 19.

Which word describes two angles, such as 35 and 55 degrees, that together make a square corner?

Complementary angles complete a 90° corner, which is why the two acute angles of any right-angled triangle always form such a pair.

What do the exterior angles of any convex polygon add up to, whether it has three sides or three hundred?

360° is the total because tracing once around the shape means making exactly one full turn, however many corners share the turning.

What is each interior angle of a regular pentagon, whose five interior angles add up to 540°?

108° comes from sharing the total equally among five corners, and every extra side adds another triangle's worth of angle to a polygon's total.

Which name is given to a straight line joining two points on a circle, the diameter being the longest one?

Chord comes from the Latin chorda, meaning catgut or string, and the short line from its midpoint out to the arc is the sagitta, Latin for arrow.

Which quadrilateral has all sides equal but need not have square corners, like the diamond on a playing card?

Rhombus is the formal name for the diamond or lozenge shape, and a square is simply one whose corners are all 90°.

What fraction of a cylinder's volume does a cone of equal base and height hold, as a pyramid does of its prism?

One third is the share for any solid that tapers from a flat base to a point, whatever the outline of that base.

Doubling every edge of a cube multiplies its surface area by 4; by what factor does its volume grow?

8 is 2 cubed, while area only quadruples, a gap that helps explain why large mammals such as elephants find it harder to keep cool than mice.

Which two numbers solve x² + 5x + 6 = 0, the roots that factoring by inspection finds?

−2 and −3 are the roots, because the left side factors as (x + 3)(x + 2) and a product vanishes only when one of its brackets does.

Which technique rewrites x² + 6x + 5 as (x + 3)² − 4 and is used to derive the quadratic formula?

Completing the square also shows a quadratic graph's turning point at a glance: here the lowest point sits at x = −3, y = −4.

Which mnemonic, first printed in a 1929 American textbook, guides students multiplying (x + 2) by (x + 3)?

FOIL stands for First, Outer, Inner, Last, the order in which the pairs of terms are multiplied before the results are added together.

In the polynomial 7x² − 3xy, which name is given to the 7 and the −3 that multiply the variables?

Coefficient is the word for any such multiplying factor, and it can be a letter too: the a, b and c of ax² + bx + c play the same role.

Which quantity, written with a vertical bar on each side, gives a number's distance from 0, so that |−3| = 3?

Absolute value has been written between two upright bars since the German mathematician Karl Weierstrass introduced the notation in 1841.

Which quick check shows whether a graph is a function, one that a circle fails by giving two y-values?

The vertical line test works because a function may give only one output for each input, so no upright line may cross its graph more than once.

Which line does the curve y = 1/x approach forever, a term from the Greek for 'not falling together'?

An asymptote is never reached in this case: the gap between the curve and the line shrinks toward nothing as x grows, yet it never quite closes.

Which kind of sequence is 2, 6, 18, 54, where each term is the one before multiplied by a fixed amount?

Geometric sequences grow or shrink by a constant multiplier, here 3, and the economist Malthus built his famous essay on population growth on one.

Which total is the most likely when two dice are rolled, the same number that a die's opposite faces add up to?

7 can be thrown in six ways out of 36, more than any other total, which is why it is the pivotal roll at a craps table.

Which counting term, an arrangement in which order matters, describes what a so-called combination lock really needs?

Permutation is the right word because 1-2-3 will not open a lock set to 3-2-1, and 19th-century patents did in fact speak of permutation locks.

Which measure of the data set 2, 2, 3, 7, 11 equals 2, an average that also works for non-numerical data?

The mode is simply the value that turns up most often, so unlike its rivals it can also sum up favourite colours or blood types.

What is the exact value of sin 30°, found by cutting an equilateral triangle in half?

1/2 is the answer because the halved triangle's short side is half its longest side. No other acute angle measured in whole degrees gives such a tidy fraction.

A car covers a route at 60 km/h and returns along it at 20 km/h; what is the car's average speed overall?

30 km/h is correct, not the tempting 40, because the car spends three times as long on the slow leg. Equal distances call for the harmonic mean of the two speeds.

Which number do investors divide by an interest rate to estimate how many years money takes to double?

72 divided by a 9% rate gives 8 years, close to the exact figure of 8.04, and the shortcut already appears in a book printed in Venice in 1494.

Which name is given to 'if Q then P', the flipped form of 'if P then Q' that need not be true?

Converse statements can fail even when the original holds: every human is mortal, yet not every mortal thing is human.

Which calculus operation finds an instantaneous rate of change, turning the squaring function x² into 2x?

Differentiation applied to a moving object's position gives its velocity, and applied a second time it gives the acceleration.

Which teacher in Roman Alexandria, killed by a mob in 415, is the first woman mathematician with a well-recorded life?

Hypatia was the daughter of the mathematician Theon, and many scholars believe she may have edited the surviving text of Ptolemy's great astronomy book, the Almagest.

Which mathematician first proved that infinite sets differ in size, with more real numbers than whole numbers?

Georg Cantor met fierce resistance from contemporaries such as Leopold Kronecker, yet the set theory he created became a foundation of modern mathematics.

Which name does the division symbol ÷ carry, a word once used for marks flagging doubtful lines in manuscripts of Homer?

Obelus comes from the Greek for a roasting spit, and the mark stood for skewering dubious text. Older textbooks used the word for ÷ itself.

Which trig function's name comes from a medieval mistranslation, when an Arabic word was misread as 'bosom' or 'fold'?

Sine began as a Sanskrit word for a bowstring. A 12th-century translator read its vowel-less Arabic spelling as the word for a fold or bosom, which is sinus in Latin.

Which shoemaker's son from Lincoln devised, in 1847, the algebra of true and false behind every digital circuit?

George Boole went on to become the first professor of mathematics at Queen's College, Cork, and engineers later used his algebra to design the logic circuits of computers.

Which word did Benoît Mandelbrot coin in 1975 for rough shapes that repeat their patterns at ever smaller scales?

Fractal thinking grew out of a question Mandelbrot posed in the 1960s: how long is the coast of Britain, when every closer look reveals more wiggles?