Counterexample
Named by 74 essays across 11 fields — each of them below, with the objects they name alongside it.
Twenty-four out of two hundred and fifty-six
Aristotle's syllogisms are four sentence forms in four arrangements, which makes 256 patterns of argument. Fifteen of them are valid. Nine more become valid if you assume the things being talked about exist, and the gap between those numbers is a two-thousand-year-old disagreement.
Every row, or one column
For every person there is someone who loves them, and there is someone who loves everyone, are the same six words in a different order. Draw the relation as a grid and they become two obviously different questions — one about rows, one about columns.
The field with four elements
The integers modulo four are not a field: two times two is zero and two has no reciprocal. There is nevertheless a field with four elements, and building it means giving up on counting as the way to make arithmetic finite.
The thirty-six officers
Six regiments send six officers each, one of every rank. Arrange all thirty-six in a square so that each row and each column holds every rank once and every regiment once. Euler could not, guessed why, and was wrong about the reason.
The majority that goes in a circle
Every voter hands in a ranking, and a ranking is transitive by construction. Compare the candidates two at a time and let the majority decide each pair, and the verdicts need not fit together into a ranking at all.
Five rules and five winners
Twenty-seven ranked ballots, five entirely reasonable ways of counting them, and five different candidates declared the winner. Every count is correct, every rule is defensible, and the answer turns out to be a property of the rule rather than of the ballots.
A lie that pays
A ballot is usually read as a report of a preference. This one reads it as a move, and walks every move one voter has — all six rankings, the winner each produces, and the ones that beat honesty.
Nobody has a reason to run away
A matching is stable when no two people on opposite sides would both rather have each other than what they have — a condition that names nothing to build and everything to rule out. The surprise is that something always satisfies it, however perverse the rankings are made.
No stable rule is safe from a lie
A stable matching always exists, and the side that proposes gets the best one it could hope for. This essay closes the story with the result that spoils it — one participant's whole strategy space searched, four submissions found that beat the truth, and a theorem saying no rule anywhere escapes.
Which side of the line is inside
A closed curve with no self-crossings divides the plane into an inside and an outside. Nobody doubts it, almost nobody can prove it, and on a curve wound tightly enough nobody can see which side a given point is on either.
Area by counting dots
Draw a polygon with every corner on a grid of dots. Count the dots strictly inside, add half the dots on the edge, subtract one — and the answer is the area, exactly, with no measuring anywhere.
The staircase that is not the diagonal
A staircase can be made to follow a quarter circle as closely as anyone likes. Its length is 2 at every stage and the arc's length is 1.5708, and no amount of refinement closes the gap — which is a fact about length rather than about staircases.
Nine points on one circle
Three midpoints, three feet of altitudes and three more midpoints. Nine points defined in three unrelated ways, on an arbitrary triangle, and all nine sit on one circle — checked here on two hundred and forty triangles as well as on the drawn one.
A curve with a corner at every point
Continuity means a curve can be drawn without lifting the pen. Differentiability means it has a tangent. The first was assumed to nearly imply the second until 1872, when Weierstrass exhibited a curve that is continuous everywhere and has a tangent nowhere — and it is a sum of cosines.
Three trisectors and a triangle nobody expected
Cut every angle of a triangle into three. The trisectors nearest each side meet in three points, and those three points are always the corners of an equilateral triangle — for every triangle there is, with no exceptions and no reason anybody finds obvious.
The solid where the answer is not two
A slab with a hole through it has flat faces, straight edges and sixteen corners, and its alternating sum is zero. It is not a trick and not a degenerate case — it is the object that shows the theorem had a hypothesis nobody had written down.
Five colours, and a chain that can be followed
The four-colour theorem cannot be checked by a person. The five-colour theorem can, in a page, and the argument that does it is the one Kempe thought had settled four — with the exact step where it fails visible in the picture.
Where the fixed point escapes
The theorem asks for a set that is closed, bounded and free of holes. Drop any one of the three and a map appears that moves every single point — and in each case the point that should have stayed still can be seen leaving.
An average that never settles
The average of many independent quantities is supposed to steady as their number grows. For one famous distribution it does not steady at all — the average of a thousand draws has exactly the same distribution as a single draw, and no amount of further averaging changes it.
How fast the bell arrives
The limit theorem says a standardised sum approaches the bell curve and says nothing about when. The rate is one over the square root of the number of terms, the constant in front is made of the third moment, and both are visible.
A limit that forgets to be continuous
Every one of the functions x, x², x³, … is as smooth as anything could be, and every column of the picture settles down. What they settle on has a jump in it — and the quantity that sees the difference is the largest gap anywhere, which is a number about the whole graph rather than about any point of it.
A line with as many points as a square
Interleave the decimal places of two numbers and one number comes out; take every other place back and the two return. The square has no more points than the segment, and dimension turns out to be invisible to counting.
A room that cannot be lit
Mirror the walls of a room and put a lamp inside it. Every point should be lit, since light bounces forever — and there are rooms with a dark spot no ray from the lamp ever reaches.
The triangle nobody can settle
Does every triangular billiard table have a path that closes on itself? Acute triangles do, right triangles do, triangles with rational angles do — and for the rest the question has been open since it was asked.
A curve that has area
The Jordan curve theorem assumes three things and nothing else — continuous, closed, no self-crossing. Everything else the eye supplies is false of some curve that satisfies all three, including the assumption that a curve is thin.
Every loop is a circle in disguise
Separating the plane is the weak half of what the eye believes about a closed curve. The strong half is that the inside is a disc — that the whole plane can be bent until the curve is a round circle — and for a polygon that is a construction rather than an argument.
A ball whose outside is not one
Alexander's sphere separates space into two pieces, exactly as the theorem promises. Its inside is an ordinary ball. Its outside is not, and the obstruction is a tree of clasped horns whose tips never stop.
Two out of three, and never all three
Stay inside every region's quota, never take a seat away when the house grows, never take one from a region that grew faster. Each pair is achievable. All three together are not, and the proof is that no rule anywhere manages it.
The theorem that has no version in space
A lattice polygon's area is decided completely by two counts of dots. The obvious guess is that a lattice solid's volume is decided by the same two counts in three dimensions, and there is a family of tetrahedra with identical counts and every volume that says otherwise.
Where the guarantee stops
Convexity converts every downhill method into a correct one, and its absence removes the guarantee entirely rather than degrading it. What is left is a collection of partial answers, and knowing which of them apply to a given problem is most of what non-convex optimisation is.
The puzzle that is exactly half solvable
A sliding puzzle sold with two tiles swapped is not a hard puzzle; it is an impossible one, and the proof is a quantity that no slide can change. The same argument, run three times at once, says that one arrangement of a scrambled cube in twelve is reachable.
Three points, however many there are
A point inside the hull of a thousand points is inside the hull of three of them. Any four points split into two groups whose hulls meet. And a family of convex sets, every three of which have a common point, has one common to all — three, in each case, being one more than the dimension.
A wall between two bodies
Two convex sets that do not meet can be told apart by a single straight line, and the line is a certificate — one object, checkable in a moment, proving something about every point of both. Remove convexity from either and no line exists, which is what the hypothesis was for.
![Partial sums of x − x²/2 + x³/3 − … on [0, 1]. Partial sums of the power series x − x²/2 + x³/3 − … drawn on the interval from 0 to 1 with the function the series sums to inside the interval, and the values at x = 1 marked.](/f/partial-sums-of-x-x2-2-x3-3-on-0-1-e402dfbd.svg)
A sum read from inside
The series 1 − 1/2 + 1/3 − 1/4 + … adds to log 2, and the reason is not in the series. Its power series equals log(1 + x) inside the interval, and the value at the edge is read off by continuity. Abel's theorem says when that reading is honest — and the series 1 − 1 + 1 − …, which has no sum, is read the same way as a half.
The table inside every quota
Give seats to districts and parties at once, and every cell of the table has a fair share it ought to round from. A table rounding every cell to its floor or its ceiling, with every total exact, always exists. The biproportional method does not always choose one: here it gives a party 2 seats where its fair share is 3.088.
Where the rounding runs out
In two dimensions a table of seats inside every fair share always exists. Add a third family of totals — every district and party split between groups — and it need not. Sixteen halves in a three-by-three-by-three table meet every total, and no whole table does it without a seat where the fair share is nothing, because the halves close a loop of seven.
When several pairs share the roads
For one pair of places, the most that can travel between them equals the cheapest cut that separates them. Give three pairs a hub of three roads to share and the two numbers come apart: the pairs can send 3/2 between them, while separating every pair costs 2. Two pairs still meet their cut, but only by splitting units in half.
One tree for every cut
A network of six places has fifteen pairs, and each pair has its own cheapest cut. All fifteen can be read off a tree with five numbers on it: the cheapest cut between any two places is the smallest number on the tree's path between them. Gomory and Hu proved in 1961 that such a tree always exists, and building it takes five cuts, not fifteen.
Which curves have a length at all
A length is defined as a supremum over inscribed polygons, which behaves because every refinement is longer than the last. It is also sometimes infinite — and the condition separating the two cases is a sum of absolute differences that either settles or does not.
The length belongs to the journey
Three maps from an interval with exactly the same image, and three different lengths. The picture of a curve is the set of points it passes through, and that set does not determine how far anything travelled along it.
A rotation in different coordinates
At an unlocked parameter the map is not merely like a rigid rotation; it is one, after a change of coordinates built out of a single orbit. The theorem needs the map to be smooth enough, and the map that shows why is one whose orbit leaves a hole.
No single input can move it far
Independence was never the hypothesis doing the work. A quantity built from many separately drawn inputs concentrates whenever changing one of them moves it only a little — and that covers quantities which are not sums of anything and have no formula at all.
One circle touching four
The nine-point circle touches the inscribed circle and each of the three escribed ones. Nothing in its construction mentions them, the two families of centres are built from different kinds of number, and the tangency is four exact equalities between distances and radii.
A centre is three weights
Write each classical centre as a weighted average of the corners and a coincidence becomes a determinant. The Euler line is then one number rather than a construction, the whole catalogue becomes mechanical, and the reason one centre is missing from it is visible in the weights.
A ring that no pairing can break
Put everybody in one pool and a stable pairing may not exist. Allow rings as well as pairs and something stable always exists — and the pairs-only answer fails exactly when that stable arrangement contains a ring of odd length. Two sides make every ring even, which is the whole reason the two-sided theorem holds.
No local rule can count the votes
A ring of cells, each holding 0 or 1, has to agree on whichever value is in the majority — every cell seeing only its neighbours. The best-known rule gets it right most of the time and wrong near a tie; no rule of any radius gets it right always. Yet two rules run one after the other do, on every ring, and the first of them is the traffic rule.
One cell short of a transversal
A transversal of a Latin square picks one cell in every row and every column with every symbol different. The cyclic squares of even order have none, and that was settled by a parity argument centuries old. Whether every square of odd order has one is a conjecture from 1967 that nobody has proved; whether every square comes within one cell of having one was settled only in 2023, and only for squares large enough.
A limit can jump at every fraction
A sequence of continuous functions can settle, point by point, on a function that is discontinuous at every rational number. It cannot settle on one that is discontinuous everywhere — the indicator of the rationals needs two limits in a row, and Riemann's integral cannot follow the second. The line between the two is Baire's theorem, and it measures smallness by gaps rather than by length.
Uniform, except on a small set
The functions xⁿ settle on their limit at every point and never uniformly — the trouble is all in a strip next to 1. Throw the strip away and the convergence is uniform on what is left, however thin the strip. Egorov proved that this always happens on an interval, Lusin proved the matching fact about a single function, and a bump sliding off along the whole line shows why both need a set of finite length to start from.
The ripples that make a series run away
Adding up the first N terms of a Fourier series is the same as averaging the function against one fixed wiggly curve. Its area is always one, but the area of its absolute value grows like the logarithm of N, without limit — and that single number is enough to force a continuous function, with no jump and no corner anywhere, whose Fourier series diverges at a point. Averaging the partial sums removes the negative ripples, and with them the whole problem.
Three places cut apart
Separating two places as cheaply as possible is solved exactly by a flow. Separating three from one another is a different problem: no flow measures it, the pairwise answers do not add up to it, and the best shortcut known in 1994 — cut each place off on its own and throw the dearest cut away — is guaranteed only to within a third of the truth.
Two graphs the eigenvalues cannot tell apart
A graph's matrix has eigenvalues, and they count a surprising amount of the drawing: its edges, its triangles, every closed walk of every length. They do not count everything. A star with four arms and a square beside a lone point have the same eigenvalues exactly, although one of them is in two pieces — and on six points ten of the 156 graphs have a twin of this kind.
![The slope of x² sin(1/x): discontinuous at nought, and never jumping. The derivative of x^2 sin(1/x) on [−0.12, 0.12], with the value at nought marked and the level 0.5 crossed 77 times.](/f/the-slope-of-x2-sin-1-x-discontinuous-at-nought-and-never-60857308.svg)
A slope can swing but never jump
A function can have a slope at every point without that slope changing continuously: x² sin(1/x) has slope nought at the origin and a slope that swings between −1 and 1 however close to the origin it is taken. What a slope cannot do is jump. Darboux proved in 1875 that a derivative takes every value between any two of its values, so a step is never a derivative — and the only way a slope can be discontinuous is by oscillating.
Each user pays for its own last link
Several users must be connected to a source, and the cheapest network that does it is a tree. Dividing its cost so that no group of users would rather build its own looks like a hard search, and it has a one-line answer: each user pays for the link that joins it to the tree on its way to the source. No group is ever overcharged — while the average over orders of arrival, the rule that settles so much else, can charge a pair more than its own connection costs.
Five where four were promised
In one unknown, a polynomial with three terms has at most two positive roots, whatever its degree. The natural guess for two equations in two unknowns, each with three terms, was two times two: four. It is wrong. Bertrand Haas found two such equations in 2002 whose curves cross five times in the positive quadrant, all five crowded within a tenth of one corner, and five turned out to be the most there can ever be.
Perfect but for one factor
In 1638 Descartes wrote to Mersenne with an odd number whose divisors add up to exactly twice the number — provided one of its factors, 22021, is counted as a prime. It is not; it is 19² × 61. Nearly four centuries later that number is still the only odd one of its kind known, and a search of every odd number below ten million finds no other. It is the closest anyone has come to an odd perfect number, and what it shows is how an odd perfect number would have to be built.
Every pair side by side, once
Seat an odd number of guests at round tables for as many nights as it takes, the same table sizes every night, so that every two guests sit side by side on exactly one night. For a single table a zigzag turned a notch each night does it for any number of guests. For other table plans the answer is almost always yes — and for six guests at two tables of three, nine at tables of four and five, and eleven at three, three and five, an exhaustive search proves it is no.
Three circles that touch and are not the largest
In 1803 Gian Francesco Malfatti asked how to cut three round columns from a triangular block of marble with as little waste as possible, and answered with three circles each touching the other two and two sides. The circles exist in every triangle and are found by solving three equations. They are never the answer to his question: the plain greedy rule — the inscribed circle first, then the largest circle that still fits, twice — always does better, by 1.4 per cent on the equilateral triangle and by almost double on a thin one.
A vote worth nothing until it was outnumbered
From 1958 to 1973 Luxembourg held one vote of seventeen in the Council of the European Communities and could never once change an outcome. When the Council grew, Luxembourg's share of the votes fell and its share of the power rose from nothing. Power is not a quantity a member holds; it is a property of the whole assembly, and changing the assembly moves it in directions nobody would guess.
A third kind of member
Two sides always have a stable matching, and the proof is a procedure. Add a third kind of member — capitals who rank numbers, numbers who rank letters, letters who rank capitals — and every market small enough to check still has a stable grouping. Ten million markets of three have at least ten each. Nobody can prove it continues, because the structure the two-sided proof stands on is gone.
A contradiction that stays where it is
In classical logic one contradiction proves everything: from p and not-p, any q at all. Add a third truth value — both true and false — and count it as holding, and the explosion stops. The classical laws all survive as laws; what goes is exactly the reasoning that carries a contradiction somewhere it was not. Run the other way, the same three tables make a logic of gaps instead of gluts.
Abundant, and still not a sum of its parts
Seventy's proper divisors add to seventy-four, more than seventy itself — and yet no selection of them adds to exactly seventy. The reason is that the excess, four, cannot be made from the small divisors one, two and five. Numbers like seventy are called weird; there are thirty-six below twenty thousand, every one of them even, and nobody knows whether an odd one exists.
![Polynomials climbing to the square root, each above the last. The iterates of p ↦ p + (x − p²)/2 at steps 1, 2, 3, 4, 6, 9 on [0, 1] beneath the curve √x, with largest gaps 1.000, 0.500, 0.259, 0.176, 0.108, 0.068.](/f/polynomials-climbing-to-the-square-root-each-above-the-last-ce601e49.svg)
Settling in order settles everywhere at once
A sequence of functions can settle at every point and never settle uniformly — the largest gap stays put while each point escapes it. Dini found the circumstances in which that cannot happen: continuous functions, a continuous limit, every member above the next, and a closed interval. Under those four conditions convergence at each point is convergence everywhere at once, and Weierstrass's iteration for the square root becomes a sequence of polynomials converging uniformly to |x|.
Close in height and nowhere close in slope
Uniform convergence carries continuity to the limit and carries integrals to the limit. It carries slopes nowhere. The functions sin(n²x)/n flatten to nothing while their slopes grow without bound; smooth curves converge uniformly to a corner; a Fourier series converges and its differentiated series diverges. What does carry a slope is uniform convergence of the slopes themselves — and the reason is that integration averages wiggles away while differentiation multiplies them.
Log-concave with nothing to make it so
The chromatic polynomial of a graph counts its colourings, and its coefficients rise to one peak and fall, each squared at least the product of its neighbours. For a real-rooted polynomial that would be automatic. But by nine vertices fewer than one graph in twenty has only real chromatic roots, and the coefficients stay log-concave anyway — on every graph tested, and, by June Huh's theorem of 2012, on every graph there is.
A partition piled into a corner
A partition is a row of numbers that never increases. Let it never increase down the columns too, and it becomes a pile of cubes pushed into the corner of a room. MacMahon counted the piles with one change to Euler's product — each part size k in k colours — and in a box with a product over its cells. The second direction moves the growth from e^(√n) to e^(n^(2/3)), puts ζ(3) where π was, and makes a count that grows with a box's area. One direction further, his guess fails at six.
No graph is poorer in squares than chance
Take any graph with a fixed share of its possible edges and count how often four points chosen in order close up into a ring. However the edges are arranged — in cliques, in two-sided blocks, at random — the count never falls below what a random graph of the same density gives. Sidorenko conjectured that the same holds for every pattern without an odd cycle. For a triangle it is false; for a path or a ring of four it takes two lines; for K₅,₅ with a ten-cycle removed nobody knows.
Syllogisms that say most
The syllogism knew four words — all, no, some, some not — and two thousand years of logic decided which of its 256 forms were valid. Add 'most', a word everyone uses in arguments, and the 256 become 864. A diagram with marks for 'empty' and 'occupied' can no longer decide them, because 'most' compares sizes; but writing the number of things in each region turns every form into a linear programme, and the programme decides all of them at once. Thirty-one are valid, including one Augustus De Morgan found in 1847: if most M are S and most M are P, then some S are P. And 'most' cannot be passed along a chain: ninety-nine per cent of S can be M, ninety-nine per cent of M can be P, and no S be P at all.
How many pieces to shrink a shape
Cut a disc in two and one half still holds a full diameter; cut it in three and every piece is narrower. In 1933 Karol Borsuk asked whether every bounded shape in d dimensions can be made narrower with d + 1 pieces, and the antipodal theorem he proved in the same paper says no number smaller will do for a ball. The guess held in the plane and in space, and failed in 1993: sixty years later, a set of sign patterns turned into points showed that some shapes need exponentially many pieces — the first in 1,325 dimensions, and now known in 64.
Never negative and no sum of squares
Any quadratic that never goes below nought can be completed into squares, and the squares are the proof. In 1888 Hilbert showed the same habit must fail at higher degree without exhibiting a single failure; the first one was written down 79 years later. Motzkin's x⁴y² + x²y⁴ − 3x²y² + 1 touches nought four times and never crosses it, and one cell of a four-by-four table shows that no squares can make it. Multiply it by 1 + x² + y² and five squares appear.
A chain of majorities leads anywhere
Put voters on a line and the middle one's position beats everything. Put five in a plane and nothing is safe: seven majority votes can walk the group from its own centre to a place every one of them likes less. One small circle, drawn through the lines that split the voters, says exactly how far a single vote can go, and it shrinks as the group grows.
Alone on the track
Runners set off together at different constant speeds. Is each of them, at some moment, far from all the others? A pigeonhole on time gives half the conjectured distance in two lines, and sixty years of work have not closed the other half. The speeds 1, 2, …, k are exactly at the limit, and so, surprisingly, are 1, 3, 4, 7 and 1, 4, 5, 6, 7, 11, 13.
Replace the parabola with a corner
Nobody can prove Hénon's map is chaotic at its classical values, since a stable cycle might hide in a window too thin to find. Replace x² in it by |x| and the whole difficulty disappears: one cone of directions is stretched by every step, in three lines of arithmetic, and across 301 parameters the Lyapunov exponent is a straight rising line with no windows at all.
The triangle a triple point buys
How many triangles can eight straight lines cut out? A count of line segments says at most 16, a parity argument takes off one, and the tables say 15. Yet every one of the 1,232,944 arrangements in which only two lines ever cross stops at 14. The fifteenth triangle exists only if three lines are made to meet at exactly one point, and moving a line by a hundredth loses it.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchContinuityConvergenceInvariantParityPreference profileUniform convergenceExistence proofLimitApproximationDualityImpossibility