Parity
Named by 74 essays across 10 fields — each of them below, with the objects they name alongside it.
Seven bridges, and the invention of throwing things away
Euler solved a puzzle about a Prussian city by deleting the city. What survived the deletion was a new branch of mathematics.
Pascal's triangle, in two colours
Shade the odd numbers in Pascal's triangle and a fractal appears. Nothing was designed to produce it, and the same shape arrives independently from a completely different construction.
The primes on a spiral, and a pattern nobody ordered
Wind the whole numbers outward in a square spiral, mark the primes, and they line up on diagonals. The observation is a hundred years old and there is still no proof it means anything.
Numbers that wrap
A clock does arithmetic. It has finitely many numbers, addition never leaves it, and multiplication behaves entirely differently depending on one property of the size of the dial.
Six people at a party
Among any six people, three are mutual acquaintances or three are mutual strangers. Five is not enough, and the arrangement that saves five is a pentagon. Beyond that the numbers become unknowable.
A walk that always comes home, until it does not
Step left or right at random, forever, and the walk returns to where it started with certainty. On a grid it also returns. In space it does not, and about a third of walks leave and never come back.
Counting one rectangle, twice
Whether seven is a square modulo eleven, and whether eleven is a square modulo seven, are two unrelated-looking questions. Their answers are linked, and the link is a rectangle of dots counted along its rows and then along its columns.
The square that cannot shrink
The usual proof that the square root of two is irrational is about even and odd numbers. There is a proof about squares instead, in which a supposed solution is folded into a smaller one — and the folding is a drawing.
The question nobody can answer
Halve it if it is even, triple it and add one if it is odd. Every number anyone has tried comes down to one. Nobody can prove they all do, and the reason is not that the problem is hard to state.
The map that puts neighbours side by side
Reorder the rows of a truth table so that neighbouring squares differ in one letter, and finding a short formula stops being algebra and becomes the problem of covering a shape with rectangles.
Distance is a picture
A message is a corner of a cube and an error is a step along an edge. Everything a code can do is decided by how far apart the corners it uses are — and that is a fact about a drawing.
Finding the error without reading the message
Three parity checks on a seven-bit word produce three bits. If they are all zero nothing is wrong; otherwise they are the number of the position that broke. The message is never consulted, because the answer does not depend on it.
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.
A walk that changes one thing at a time
Counting from nothing to fifteen in binary changes four digits at once somewhere in the middle. There is another order through the same sixteen words in which every step changes exactly one — and it is a closed walk on a four-dimensional cube.
Three colours force a triangle
Cut a triangle into small ones and colour the corners under one restriction. However the cutting and the colouring are done, some small triangle ends up with all three colours — and the number of them is always odd.
The crossings that will not come out even
Draw a rearrangement as strings from one row of pegs to another and count where they cross. The count depends on how the strings are drawn; whether it is odd or even does not, and that single bit is what makes determinants exist and a sliding puzzle unsolvable.
The only function that behaves like a volume
Ask for a function of the columns of a matrix that scales when a column scales, vanishes when two columns agree, and gives one on the identity. Three conditions, and there is exactly one such function in every dimension.
The terms that cancel almost everything
Multiply out the product of 1 − q, 1 − q², 1 − q³ and so on, and nearly every coefficient is zero. What survives is a single plus or minus one at 1, 2, 5, 7, 12, 15 — and the reason is a way of pairing partitions off so that each pair cancels.
Two pieces, in every dimension
A closed curve cuts the plane in two. A closed curve in space cuts nothing at all, and it takes a closed surface to do the job — which is the shape of the general theorem, and the reason the word "dimension" means anything.
A cycle for every pair
A cyclic sequence in which every window of two consecutive symbols is a different pair of things. For five things it exists and for four it does not, and in both cases there are exactly as many pairs as there are places to put them.
The two supplements, and where the eight comes from
The main law relates two odd primes to each other and says nothing about −1 or about 2. Those two are settled separately, by their own counts, and the answers arrive modulo four and modulo eight — which is a clue about where the whole subject is really taking place.
Almost every number comes down
The Collatz conjecture is open and a great deal about it is not. Whether a number falls below its own start in the first few steps is decided entirely by its remainder on division by a power of two, and the share of numbers for which it happens can be counted exactly.
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.
The carries decide the divisibility
How many times a prime divides a binomial coefficient is not a fact about the coefficient at all. It is a count of the carries that happen when two numbers are added in that prime's base, which is a question about column addition and has nothing to do with choosing anything.
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.
The piece that cannot pair off
Take the sides away and the obstruction to a matching changes character completely. It is no longer a shortage of partners; it is a parity, and the quantity that measures it counts pieces of odd size rather than vertices of any size.
One gadget defeats every refinement
Colour refinement fails on two triangles against a hexagon; its two-dimensional version fixes that and fails on a pair of strongly regular graphs. For every k there are two graphs the k-dimensional version cannot separate, and they are built from one local piece whose only symmetry is a parity.
The curve that no three points in line define
In a finite plane, take as many points as possible with no three on a line. In odd order the largest such sets have one more point than the order — and every one of them, searched exhaustively in the small planes and proved by Segre for all odd orders, is a conic. In even order every tangent meets at one point, which can be added, and the curves stop being forced.
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.
Infinitely many guessers, finitely many wrong
An infinite line of people each wears a black or white hat, sees every hat in front and none of their own, and must guess their own colour. With a finite line, each guesser is right half the time whatever they agree in advance. With an infinite line and the axiom of choice, they can agree a strategy under which all but finitely many are right — and nobody can carry it out.
A walk that splices in its own detours
Euler proved that a walk crossing every bridge once needs every landmass to have an even number of bridges, and then stated, without proof, that this was enough. The missing half took 137 years, and it is not an argument but a procedure: walk until stuck, notice that stuck can only mean home, and splice in a detour from anywhere with edges left. The procedure never fails, and the reason fits in one sentence about arriving and leaving.
The streets a postman walks twice
A postman must walk every street of a district and come back. If every corner has an even number of streets, no street needs walking twice. If not, some must — and the ones repeated always join the odd corners in pairs. Pricing every way of pairing them finds the shortest round; pairing the nearest corners first does not.
Opposite labels that have to meet
Cut a square into triangles, label every corner +1, −1, +2 or −2, and insist only that opposite points of the edge get opposite labels. Somewhere inside, an edge must join a label to its negative. The proof counts quarter-turns round a diamond — an odd number on the boundary, zero in any triangle that avoids opposites — and making the triangles smaller turns the count back into the theorem about opposite points on the Earth.
Every place changes back
A closed walk through every corner of a cube changes one place at each step, and each place, having changed, must change back before the walk returns home. So every place changes an even number of times — which is why no walk on three places can share the work evenly, why perfect sharing is possible only when the number of places is a power of two, and what sorts the 1,344 walks on the 4-cube into exactly four kinds.
The walk through the middle levels
On seven places, the words with three ones and the words with four number thirty-five each. Is there a closed walk through all seventy, changing one place at a time and never leaving those two levels? On five places the answer is 24 walks, on seven and nine a search finds one in moments — and whether one exists for every odd length was open for thirty years, until Torsten Mütze proved in 2016 that it always does.
Half the cube and √n neighbours
Choose more than half the corners of an n-dimensional cube, any way at all, and some chosen corner has at least √n chosen neighbours. That statement about a cube settled a thirty-year question about how sensitive a truth function must be to its inputs, and its proof is a matrix of plus and minus ones whose square is n times the identity. A search over every choice for the 4-cube finds the bound exactly: nine corners, and some corner always has two chosen neighbours.
A contradiction that is only a sum
Put a variable on every edge of a graph and ask each vertex for an odd or an even number of true edges, with the demands adding up to odd. Add all the demands and every edge is counted twice, so the left side is zero and the right side is one: the contradiction is a single sum. Resolution cannot add. It has to reach the same conclusion clause by clause, and on a graph where every group of vertices has many edges leaving it, that takes exponentially long.
Six points in space and a pair that must link
Put six points anywhere in space and join every pair with a straight segment. Split the six into two triangles — there are ten ways — and at least one of the ten pairs of triangles is linked like two rings of a chain. No placement avoids it. The reason is a parity: moving an edge through another changes exactly two of the ten linking numbers, so their sum stays odd whatever is done.
A rent nobody envies
Three housemates, three rooms that are not alike, one rent. Every way of splitting the rent is a point of a triangle; ask, at each point of a fine grid, which room one housemate would take at those prices, taking turns so that each small triangle has one corner for each of them. Sperner's lemma then promises a small triangle where all three would choose different rooms — and as the grid is refined, the envy at that triangle shrinks to nothing.
Two weak sources make one fair bit
No fixed rule can turn every weakly random source into fair bits: for any rule, some source with almost full unpredictability makes it constant. Two independent sources are different. Multiply their bits in pairs, add, and keep the parity — and if the two together carry more unpredictability than the length of one, the result is nearly fair, whatever else the sources do. The reason is that the table of those parities is balanced on every large rectangle.
Thirty-one moves from solved
Parity settles which half of a sliding puzzle's arrangements can be reached and is silent about how far away any of them is. Searching every reachable arrangement of the three-by-three tray answers the second question exactly — two arrangements sit thirty-one moves out — and parity turns up again, this time as a law about distance.
Which graphs let the tokens go anywhere
A sliding puzzle is a graph with a token on every vertex but one. Richard Wilson found in 1974 what every such puzzle can reach, and the answer has a surprise in it — the half the tray is stuck with is not a fact about permutations at all, but about the board being two-coloured — and one exception, a graph of seven vertices that reaches exactly 120 of 720.
No odd number of equal triangles
A square can be cut into two triangles of equal area, or four, or any even number. It cannot be cut into three, or five, or any odd number — whatever shapes the triangles take. Paul Monsky's proof of 1970 has no geometry in its engine at all: it colours the points of the plane by how divisible their coordinates are by two.
Where the Collatz map is a coin
Extend the Collatz map from the whole numbers to the 2-adic integers — binary strings that run on for ever to the left — and it stops being mysterious. It becomes, after a change of coordinates, the simplest chaotic system there is: shifting a string of coin tosses one place. Everything about it is then known, and none of it says anything about the whole numbers, which is the most instructive failure in the whole story.
Two halves a sieve cannot tell apart
Split the whole numbers into those with an even number of prime factors and those with an odd number. To a sieve — which sees only how many numbers each small divisor divides — the two halves look almost exactly alike. Yet sifting out the small primes leaves every prime in one half and nothing at all in the other. That is Atle Selberg's parity example, and it is the reason no sieve can prove there are infinitely many twin primes.
What the signs allow
Descartes' rule reads the signs of a polynomial's coefficients and bounds its positive roots by how often they change — a bound set by the number of terms, not the degree, and proved by nothing more than Rolle's theorem. Each count the rule allows can be had. But not every pair of counts, positive and negative, that the rule allows together can be had together: the first combination that never occurs is at degree four, and the list of impossible ones is still being worked out.
How many gates a truth table needs
Every truth function can be built from gates, and the natural measure of a function is the fewest gates that build it. For three letters the whole answer can be computed: no function needs more than four. For n letters, a count shows that almost every function needs about 2ⁿ/n gates, and a construction shows that none needs more. And for any function anyone can actually write down, the best proof in fifty years says it needs 3.1n.
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.
Seven points and a knot they cannot avoid
Put seven points anywhere in space and join every pair with a straight segment. There are 360 closed paths that visit all seven points once each, and at least one of them is knotted — however the points are placed. The reason is a parity, as it was for six points and a linked pair: a number read off each path's knot adds up, over all 360, to something odd. Six points are not enough; seven always are.
Seven graphs that must link
Six points joined in every way cannot be placed in space without two disjoint triangles hooking together. Trade any triangle for a three-pointed star, or a star back for a triangle, and the property survives; doing it in every possible way from K₆ reaches exactly seven graphs, the Petersen graph among them, and stops. Every placement of every one of them links, by the same parity count. Remove any edge from any of them, and placements that link nothing appear at once. The seven are the whole answer: a graph must link exactly when it contains one of them.
The formula that cannot share
A circuit computes the parity of n letters with a chain of n − 1 exclusive-or gates. A formula in and, or and not, which may use each result only once, needs n² letters at least — a bound Khrapchenko proved by counting the cube's edges between true and false — and exhaustive search over every formula of four letters finds parity sitting exactly on it.
Is the partition count even half the time?
The number of partitions of n is even for 50.0% of the n up to half a million, its runs of one parity are as long as a coin's, and nothing proves that the share is a half — the best theorems only show there are at least about √n of each. Modulo 5 and 7 the zeros carry Ramanujan's congruences and something more: an excess that follows whether 1 − 24n is a square.
A sum that forbids half the pairings
Put two 1s, two 2s, …, two ns in a row so that between the two ks there are exactly k other numbers. For n = 3 there is one way, 312132; for n = 4 one way; for n = 5 and 6 none at all, and the reason is a single sum — adding up the positions of every entry in two ways forces n(3n − 1)/2 to be even. A search confirms the count forbids nothing that exists and finds 26 arrangements at n = 7 and 108,144 at n = 12, but the sum is all anyone knows about why.
The boards on which every light goes out
In Lights Out, pressing a light toggles it and its neighbours, and the goal is to turn every light off. Over the field with two elements the puzzle is a system of linear equations, and whether every pattern can be cleared depends only on whether one matrix is invertible. On a 3 × 3 board it is; on 5 × 5 a quarter of patterns can be cleared; on 39 × 39 one in four billion. Which sizes fail is decided by a common factor of two polynomials.
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.
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.
The boards a knight can tour
A knight can visit every square of a chessboard once and land a move from where it began. On a 5 × 5 board it cannot, on a 4 × 100 board it cannot, and on a 3 × 8 board it cannot, though on 3 × 10 it can in sixteen ways. Allen Schwenk found in 1991 the complete list of rectangles without a closed tour, and the reasons on it are of two kinds — a colouring that counts squares, and a second colouring that a tour would have to make agree with the first.
Two random shuffles reach every shuffle
Pick two ways of rearranging n letters at random and allow them to be repeated in any order. Almost always the two together reach every rearrangement there is, or every even one — and when they fail, the usual reason is a coincidence about a single letter that both happen to leave alone. Eugen Netto guessed it in 1882 and John Dixon proved it in 1969.
Every light on, from every light off
In Lights Out, pressing a light toggles it and its neighbours. On a five-by-five board only one pattern in four can be reached from all off — and yet all on is always one of them. The same is true on every board and on every network of lights whatever, provided the influence runs both ways. The proof is a single observation about symmetric matrices over a field with two elements, and it fails, as a three-arrow example shows, the moment influence runs one way.
Signs that make a determinant count
Hall's theorem says whether a board can be covered by dominoes. How many ways it can be covered is a different question, and for a general graph a hopeless one — the count is a permanent, and nobody expects an efficient way to compute permanents. On a flat board it is a determinant. Put a minus sign on the vertical pairs in every second column and the thirty-six coverings of a four-by-four board stop cancelling and add up; the chessboard has 12,988,816. A hole in the board breaks the rule in a way a cut repairs, and on K₃,₃, which cannot be drawn flat, no choice among its 512 signings works at all.
Every random game has an odd number of equilibria
Find every equilibrium of eleven thousand random games — pure and mixed, by trying every pair of strategy sets the players might mix over — and the totals are 1, 3, 5, 7, … and never an even number. The count is a sum of signs: equilibria of index +1 outnumber those of −1 by exactly one in every game. The average grows about 28% with each strategy added, and searching for the game with the most turns up the coordination game's 2ⁿ − 1, a pattern that holds only up to five strategies.
When a press skips its own light
In Lights Out every board can be turned from all off to all on, because each press toggles its own light. Change the rule so a press toggles only the neighbours and the guarantee is gone: 310 of the 3,600 boards up to sixty by sixty cannot be lit. A single comparison of powers of two predicts every one of them, and on general graphs the answer depends on whether the number of vertices is odd or even, for a reason that is one line of algebra.
What a formula keeps when letters are fixed
Fix half the letters of a formula at random and most of its leaves become irrelevant: on the sixteen-leaf formula for the parity of four letters, two fixings leave four. How fast formulas shrink under random fixings is the whole engine behind the best lower bounds anyone has proved for formula size — Subbotovskaya's 3/2 in 1961, Håstad's 2 in 1998 — and the reason that engine stops at n³.
Half the chance for every silent set
In the lights game where a press toggles only the neighbours, a random network can be lit about 56 times in a hundred when it has an even number of vertices and 44 when odd. The numbers are 0.560562 and 0.439438, they add to one, and they come from a chain: each new vertex raises or lowers the dimension of the silent sets by one, and each dimension halves the chance.
A curve of pixels needs two kinds of touching
On a screen a closed curve is a ring of pixels, and whether it has an inside depends on which pixels count as touching. Let diagonal pixels touch, and the inside leaks out through the corners; forbid it, and a curve can fence in a region while being twelve separate dots. Of the 1,048,575 sets of pixels in a small window, every simple closed curve obeys the Jordan curve theorem exactly when the curve and the rest of the plane are given different rules.
No way to hand out true and false
Birkhoff and von Neumann read the propositions of quantum mechanics as subspaces, with "and" the intersection and "or" the span. Three lines in a plane already break the distributive law. Kochen and Specker proved something stronger: the propositions cannot be called true and false at all, if every complete measurement is to have exactly one true outcome. Cabello's eighteen directions in four dimensions show why in one line of arithmetic — nine bases, each direction in two of them, and nine is odd — and a search of all 262,144 assignments confirms it.
![Five squares on one octagon. Octagon [[1.0756,0.5516],[0.6082,1.2964],[-0.466,1.2294],[-0.8758,0.3153],[-0.797,-0.3348],[-0.3706,-0.7683],[0.2718,-0.6558],[1.1693,-0.4453]]; squares with sides 1.4975, 1.4588, 1.4407, 1.3929, 1.3921.](/f/five-squares-on-one-octagon-d1904a63.svg)
An odd number of squares on every polygon
Does every closed curve that does not cross itself pass through the four corners of some square? Toeplitz asked in 1911, and for polygons and smooth curves the answer is yes, because the squares can be counted and the count is odd. For rectangles of a fixed shape the count is even, which is why they needed a different proof — and for an arbitrary continuous curve the square is still not proved to exist.
The turns a tangent makes
Drive once round a closed track and count how many full turns the steering has made. A circle is one turn, a figure-eight is nought, and every loop adds one. Whitney and Graustein proved that this one number decides when one track can be bent into another without ever stopping — and it can be read off the crossings of a drawing without following the curve at all.
Thirty-one directions nobody can label
In four dimensions eighteen directions are enough to show that quantum measurements cannot have answers waiting in advance, and a parity count does it in a line. In three dimensions, where Kochen and Specker first proved it, no parity count works and the smallest known set has thirty-one directions. Peres's thirty-three can be checked by hand; Conway and Kochen's thirty-one fall out of a search through every direction with coordinates between −2 and 2; and whether thirty or fewer could ever do is still being settled by exhaustive computation, which has so far ruled out everything up to twenty-four.
Lines that leave half on each side
Scatter ten points on a page and draw every line through two of them that leaves four on each side. There are at least five such halving lines and at most thirteen, and every point lies on an odd number of them. Both facts come from turning a line through half a revolution, the argument that halves two shapes with one cut. How many halving lines n points can have is the k-set problem, and its answer lies somewhere between a little more than n and n to the four-thirds.
How short a paper Möbius band can be
A strip of paper ten times as long as it is wide makes a Möbius band easily. A square does not: paper bends but does not stretch, and a half-twist has to be paid for in length. Fold a strip of length √3 three times at sixty degrees and its ends meet turned over — a Möbius band, flat, in the shape of a triangle. Five folds manage it with 1.6246, and more folds approach π/2, but only by letting the paper pass through itself. Richard Schwartz proved in 2023 that √3 is the true limit for a band that does not.
A second empty square
With one empty square a sliding tray can reach exactly half its arrangements, and the half it misses is guarded by a sign. Leave a second square empty and the guard disappears — every arrangement is reachable, two tiles can be exchanged in four moves, and the far end of the three-by-three tray comes in from thirty-one moves to twenty-four. Searching every tray with every number of holes shows what each extra square of room is worth.
A sort that never looks at the answers
Most sorting decides its next comparison from the last one. A sorting network decides them all in advance: a fixed sequence of swaps-if-out-of-order, wired up before any number arrives. Whether such a thing sorts everything can be settled by trying 2ⁿ strings of noughts and ones instead of n! orderings. Nineteen comparators sort eight numbers, and proving that nothing shorter sorts eleven took decades and a satisfiability solver. Over neighbours only, one single test, the reversed row, decides the whole question.
A game decided before it starts
Draw some crosses on paper. Players take turns joining two free arms with a curve and adding a new cross on it. However they play, a game on n crosses lasts exactly 5n − 2 moves, so the winner is known before the first move: Euler's formula has already decided it. Its close relative Sprouts differs by one rule, leaving a real game whose winner for each size is known only by huge search, and which seems to depend on the size modulo six for no known reason.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchCounting argumentInvariantPermutationCounterexampleExistence proofGraphHypercubePrimesCounting two waysFinite fieldLinear system