Mathematics

56 cards about Mathematics, with explanations, methods and source links.

Cards about Mathematics

• How many people does it take before two of them share a birthday?

  There are 365 days in a year, so it feels like a room would need hundreds of people before two birthdays collide. But the thing being counted is not people, it is pairs, and every pair is its own chance to match. So how big does the room have to be?

  Source: Britannica

• How long is the coast of Britain?

  Measuring a coastline means first choosing a ruler, which sounds like a technical footnote. A ruler 100 kilometers long gives Britain roughly 2,800 kilometers of coast. Halve the ruler and the answer climbs to roughly 3,400. So how long is the coast?

  Source: Britannica

• Is there a way to stack oranges more densely than the greengrocer does?

  Greengrocers stack oranges in layers where each fruit sits in the hollow between the fruits below it, without thinking about it twice. Kepler’s conjecture, from 1611, said no other arrangement of identical spheres fills more space. How long did proving that take?

  Source: Hales, Annals of Mathematics

• Did Alan Turing really prove the halting problem attributed to him?

  The halting problem asks whether a general algorithm can decide in advance whether a program will ever finish or run forever, and it is almost always credited to Turing’s 1936 paper. A piece of logical and historical work went back to the paper itself. What is actually in it?

  Source: Hamkins & Nenu, arXiv

• How can you estimate pi by dropping needles on a floor?

  Picture a floor with parallel lines and a needle thrown onto it again and again, with no circle anywhere in sight. The share of throws that land across a line depends on the length of the needle and the spacing of the lines. What else is hiding in that ratio?

  Source: Wiley StatsRef

• Who reaches the top, the most talented or the luckiest?

  Talent is spread across people roughly along a bell curve: most in the middle, a few at the edges. Success is spread quite differently, with a few holding almost everything. If success came from talent, the gap between the two curves needs explaining. What explains it?

  Source: Pluchino, Biondo & Rapisarda, Advances in Complex Systems

• Why is a weather forecast so sensitive to a tiny change?

  Edward Lorenz was studying a simple mathematical model of flow in the atmosphere, and a model like that holds no randomness at all: the same starting conditions always lead to the same outcome. Then he entered a slightly rounded number. What happened?

  Source: Lorenz, Journal of the Atmospheric Sciences

• How do you prove an ant measures distance by its own steps?

  Desert ants return in a straight line to a nest that is nearly invisible, so plainly they are measuring distance. The question is with what. The researchers lengthened some ants’ legs with pig bristles and shortened others. What happened to them?

  Source: Wittlinger, Wehner & Wolf, Science

• A bee sees a blank card. Is it, to the bee, less than one?

  Treating "nothing" as a number is a conceptual leap that took people thousands of years, and children learn it late. Bees were trained to fly to whichever picture held fewer shapes, and then shown a blank surface against a single shape. What did they choose?

  Source: Howard et al., Science

• How does the list of funds at work change the risk in your savings?

  Splitting money equally sounds balanced: if five funds are offered, each one gets a fifth. But the list of options an employer chose is not an investment plan, and nobody designed it for you. What does that do to risk?

  Source: Benartzi & Thaler, American Economic Review

• What happened to a share after more people searched for it online?

  Attention is not information about a company’s profits, and it does decide which shares enter a small investor’s list of options at all. To measure it almost in real time, search frequency served as a clock, and then came the question of what followed a jump in searches. What was the sequence?

  Source: Da, Engelberg & Gao, The Journal of Finance

• What happens to an important report when many companies publish on the same day?

  An earnings report is supposed to move a share price when a company reports profit different from what was expected, and that is the founding assumption of an efficient market. But investor attention is limited, and on some days dozens of companies publish at once. What happens then?

  Source: Hirshleifer et al., The Journal of Finance

• Why does a personal gain on one share offering affect the next one?

  An initial public offering is the moment a company starts selling shares on an exchange, and logic says an investor would judge it on the company and the market. Here the state of the market was separated from what each investor personally experienced. Which of them predicted demand?

  Source: Kaustia & Knüpfer, The Journal of Finance

• Does the choice of a share pass between neighbors?

  Neighbors share a city, employers and news, so a similarity in their portfolios proves nothing at all about whether they ever spoke. Those shared factors were controlled for, and actual purchases compared. What was left after the controls?

  Source: Ivković & Weisbenner, Review of Financial Studies

• When does a lesson about money actually change behavior?

  Knowing what interest is does not guarantee that anyone saves more or handles debt better, and it is still taught as a general lesson. Researchers pooled dozens of studies on what happened to people after financial training, and on when it was given. So what separated the training that worked?

  Source: Kaiser & Menkhoff, The World Bank Economic Review

• Does an exchange-traded fund make the market more accurate or more volatile?

  An exchange-traded fund is a package of securities you can buy and sell on an exchange like a single share. The common claim is that it makes the market more efficient, and trading in it is not always connected to the companies inside it. What does that do to their prices?

  Source: Ben-David et al., Annual Review of Financial Economics

• How can easier trading in a share raise the danger of a fall?

  A liquid share is one that is easy to buy and sell without moving the price much, which sounds like a clear advantage for investors. To test whether it is, a change in trading rules that made certain shares more liquid was used, and what happened to them afterwards compared. What went up instead?

  Source: Chang et al., Journal of Financial and Quantitative Analysis

• Why do fund managers choose well what to buy, and badly what to sell?

  Picking a share to buy and deciding to sell a share you already hold look like two versions of the same task, and the same person performs both. A study of institutional portfolios measured each of them separately against a simple alternative. What came out?

  Source: Akepanidtaworn et al., The Journal of Finance

• Can the ability to bet against a share restrain financial reporting?

  Short selling lets you profit from a fall in a share price, so whoever uses it goes looking for negative information about the company. The American securities regulator temporarily lifted the restriction on such selling for about a third of a broad index, which created a comparison group. What happened to those companies’ reporting?

  Source: Fang et al., The Journal of Finance

• How does a thin film solve a geometry problem without calculating?

  Dip a wire frame in soap solution and you get a film that tries to contract. The boundary is fixed, and every interior point can move, so there is a genuine minimization problem here. How does the film solve it?

  Source: Pérez, Notices of the American Mathematical Society

• Why were geometry problems hung in temples in Japan?

  Sangaku are colorful wooden tablets painted with circles, triangles and tangent shapes, alongside the statement of a mathematical problem. Their authors hung them in Shinto shrines and in Buddhist temples. Why there in particular?

  Source: Hosking, Journal for History of Mathematics

• Does more trading really produce a higher return?

  Every trade feels like a chance to correct, to respond or to use information, and sitting still is hard. To test whether it pays, the returns of private investors were compared by how often they traded. Who came out ahead?

  Source: Barber & Odean, The Journal of Finance

• How can most shares lose to Treasury bills while the market still rises?

  We tend to think of the market as an average of many shares rising slowly. But the returns of individual shares are not evenly distributed: many disappear or lag a short Treasury bill, and survivorship bias hides them. So where does all the profit come from?

  Source: Bessembinder, Journal of Financial Economics

• Why do we picture compound interest as weaker than it is?

  With compound interest every gain joins the sum that produces the next gain, so the growth is not a straight line. To find out whether people really take that in, a study paid a representative American sample to estimate growth of that kind across a span of years. By how much did they miss?

  Source: Levy and Tasoff, Journal of the European Economic Association

• Why would anyone pick a fund they expect to earn less from?

  A socially responsible fund chooses its investments on environmental or social grounds alongside the financial ones. The comfortable assumption is that the people holding it simply expect to do just as well. Researchers linked real account records to survey answers and to choices in experiments where money was genuinely at stake. What did they find?

  Source: Riedl and Smeets, The Journal of Finance

• Why do investors prefer a company they know, even when it concentrates the risk?

  Risk and uncertainty are not the same thing. Under risk you can put numbers on the odds, and under uncertainty even the odds are unclear. In a representative survey of American households, researchers measured a distaste for unknown odds with questions built on the Ellsberg urn, then set each score against what that household actually held. What did it predict?

  Source: Dimmock et al., Journal of Financial Economics

• Why can an arithmetic problem hurt before you have even started it?

  Math anxiety is the tension and dread that arrive in front of numbers, and the obvious assumption is that the difficulty lies in the problem itself. A brain imaging study separated two moments: waiting for a math task, and actually doing it. Where did the threat show up?

  Source: Lyons and Beilock, PLOS ONE

• Is a nautilus shell really built on the golden ratio?

  The nautilus is the standard illustration of the golden ratio in nature, and part of the claim is true. The shell does grow as a logarithmic spiral, each chamber larger than the last by a fixed proportion. That makes the claim testable, because a fixed proportion is a number somebody can measure. So what is the number?

  Source: Science Focus, HKUST

• Why is there always one point in the coffee that did not move?

  Stir a cup of coffee and let it settle, without spilling any and without tearing the liquid apart. Intuitively, after a thorough stir everything has moved somewhere else. A theorem in topology says otherwise, and the cup is only the illustration. What does it actually claim?

  Source: Wolfram MathWorld

• How many colors does it take to color any map?

  Color the regions of a map so that no two sharing a border get the same color. A student asked his teacher about it in 1852, and the answer looked obvious enough that proving it should have taken an afternoon. It took 124 years. What finally settled it?

  Source: MacTutor History of Mathematics

• What does math anxiety actually take away from you?

  That math anxiety goes with weaker math performance is well established, and the interesting question is what sits between them. Attentional Control Theory says anxiety pulls the attention system toward whatever feels threatening, which should show up as a cost to working memory. Does the evidence support that?

  Source: Finell et al., Frontiers in Psychology

• Can one shape tile a floor that never repeats?

  A shape that tiles the plane usually tiles it periodically, with a pattern you could slide onto itself. Mathematicians asked for decades whether a single shape could cover the plane while admitting no periodic tiling at all. The problem had a nickname, the einstein, and no answer.

  Source: Smith et al., Combinatorial Theory

• Can every true statement in mathematics be proved?

  For a century mathematicians worked to put all of mathematics on an axiomatic basis, where every truth follows from a handful of stated assumptions. A doctoral student in Vienna had already proved the completeness of the first order functional calculus in 1929. Two years later he published something else.

  Source: MacTutor History of Mathematics

• How do you prove the primes never run out?

  Primes thin out as you count upward, so it is natural to wonder whether they eventually stop. You cannot settle that by listing them, since any list you finish leaves the question open. By the time Euclid’s Elements appeared, around 300 BC, several important results about primes had been proved.

  Source: MacTutor History of Mathematics

• What makes 1729 an interesting number?

  Hardy arrived at Ramanujan’s sickbed having noticed the number of his taxi from London, and announced his disappointment with it before saying hello. It was, he declared, rather a dull number, and he hoped that was not a bad omen. Ramanujan disagreed on the spot.

  Source: Wolfram MathWorld

• What happened after Wiles announced his proof?

  On 23 June 1993, in the third of three lectures at the Isaac Newton Institute in Cambridge, Andrew Wiles wrote Fermat’s Last Theorem on the blackboard, said he would stop there, and sat down. The announcement was the famous part. The next eighteen months were not.

  Source: MacTutor History of Mathematics

• What can you prove by counting boxes?

  Put more objects into boxes than there are boxes, and at least one box holds more than one object. It is the kind of statement that feels too obvious to be worth naming, and it has a name, a date and a long record of results that depend on it.

  Source: Wolfram MathWorld

• Why do mathematicians warn each other off one problem?

  Pick any positive whole number. If it is odd, multiply by three and add one; if it is even, halve it. Repeat. Collatz conjectured in the 1930s that every starting value eventually reaches the number 1, and then loops. Quintillions of examples have been tested without an exception, and nobody can prove it.

  Source: Quanta Magazine

• Can the first digit of a number give away a fraud?

  In many real collections of numbers the leading digit is not evenly spread: ones are common and nines are rare, in the pattern called the Newcomb–Benford law. Auditors use that pattern to flag figures somebody may have invented. The awkward part is that no general theory says which genuine data should obey it.

  Source: Cerioli et al., PNAS

• What do you get if you cut a one-sided surface down the middle?

  Take a paper band, give one end a half twist, reattach the ends, and the result has a single side and a single edge. It is the standard demonstration of a nonorientable surface. What happens when you start cutting it, and how many twists you add first, is where the predictions stop being obvious.

  Source: Wolfram MathWorld

• Which walk through a city turned out to be impossible?

  Königsberg had seven bridges connecting two islands in the river at the center of the city. The puzzle residents passed around was whether a route existed that crossed every bridge exactly once, with no turning back once you had started a crossing. Euler took it up in 1736.

  Source: Cornell University Networks Course

• Why can nobody actually make a Klein bottle?

  The Klein bottle is a closed nonorientable surface with no inside or outside, first described by Felix Klein. You can build one on paper by gluing both pairs of opposite edges of a rectangle, giving one pair a half-twist. The glass ornaments sold under the name are not quite the thing itself.

  Source: Wolfram MathWorld

• Can every department admit women at a higher rate and the university still not?

  In the early 1970s Berkeley was sued for gender discrimination in graduate admissions. Of 8,442 male applicants for the fall of 1973, 44 percent were admitted, against 35 percent of 4,351 women. Assuming similar qualifications, the aggregate numbers look like exactly what the suit alleged.

  Source: Hershbein, Brookings Institution

• Can three dice each beat the next one around the circle?

  Beating somebody at dice feels like it should be transitive: if A usually beats B and B usually beats C, surely A usually beats C. Martin Gardner presented the famous paradox showing otherwise in 1970, with Efron’s dice. What was left open is how far the trick can be pushed.

  Source: Komisarski, The American Mathematical Monthly

• What if the famous study disproving the hot hand had the maths wrong?

  The hot hand fallacy is one of the best-known results in behavioral science: players and fans believe in streaks, and the canonical study found the belief unsupported in the data. The measure it used compares shots that follow a streak against the rest. That comparison turns out to have a property nobody noticed.

  Source: Miller and Sanjurjo, Econometrica

• Can one ball be cut up and reassembled into two of the same size?

  Cutting something up and putting it back together cannot change how much of it there is. That intuition is so basic it feels like arithmetic. Banach and Tarski published a result in 1924 saying that, for a solid ball in three dimensions, it is false.

  Source: Wolfram MathWorld

• Is there always somewhere on Earth with no wind?

  Comb a tennis ball flat and a tuft stands up somewhere. That sounds like a fact about hair rather than a theorem. In topology it is the statement that no everywhere nonzero tangent vector field exists on the 2-sphere, and it has a consequence for the planet.

  Source: Wolfram MathWorld

• After the host opens a door, does switching help?

  Three doors, two goats and a car. You pick one, and the host, who knows what is behind each, opens another to reveal a goat and offers you the remaining unopened door. Two doors are left, so the instinct is that it cannot matter which you take.

  Source: Wolfram MathWorld

• Is the hexagon really the best way to divide up a surface?

  Bees build in hexagons, and the shape has been held up as nature’s economy since antiquity: Pappus refers to the problem in his fifth book. Admiring a pattern is not the same as proving it optimal. The claim needed is that nothing else does better.

  Source: Wolfram MathWorld

• Can two shapes cover a floor without the pattern ever repeating?

  Tile a floor with squares or hexagons and the design repeats: slide the whole pattern far enough and it lands on itself. For a long time that looked like what tiling meant. Penrose produced a pair of shapes for which it is impossible.

  Source: Wolfram MathWorld

• What is the quickest path for a bead sliding between two points?

  A straight line is the shortest route between two points, and the instinct is that it must also be the fastest for something sliding down under gravity. The brachistochrone problem asks for the curve that takes the least time. The two answers are not the same curve.

  Source: Wolfram MathWorld

• Which digit starts a number most often?

  Nine digits can begin a number, so each should lead about 11.1 percent of the time. In 1881 Newcomb noticed that the first pages of logarithm tables wore out faster than the later ones, which is a strange thing for an evenly used book to do. Benford studied a wide variety of data sets in 1938.

  Source: Wolfram MathWorld

• How many colors does any map need?

  Color a map so that no two countries sharing a border get the same color, and four shades always seem to be enough. Guthrie conjectured it in 1852 and the problem passed through de Morgan into the wider community. Six colors were provable, and five, but four resisted for over a century.

  Source: Wolfram MathWorld

• How does an arrow move if it is motionless at every instant?

  Zeno left four paradoxes dealing with counterintuitive aspects of continuous space and time. The arrow is the sharpest of them: at any given instant a flying arrow has one position, and at that instant it is indistinguishable from a motionless arrow in the same place. So where does the motion live?

  Source: Wolfram MathWorld

• Why does the bell curve turn up in so many different places?

  Heights, measurement errors and exam marks come from completely unrelated processes, and all of them pile up into the same bell shape. That cannot be coincidence, and it is not a fact about any of those subjects. It is a theorem about what happens when quantities are added together.

  Source: Wolfram MathWorld

• How many guests do you need before three of them must all know each other?

  Invite some people to a party. Any two of them either know each other or do not, and you get no say in which. The party problem asks for the smallest number of guests that forces a group of three who all know each other, or a group of three who are all strangers. That number has a name, and the interesting part is how few of them anybody has managed to compute.

  Source: Wolfram MathWorld

More topics