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Essays arrive in groups rather than one at a time, and a group usually opens up a subject not covered before. Between one group and the next nothing changes, so a reader who has seen the most recent group has seen everything.
12 November 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
Algebra
A quarter turn quiets the echoes
Lay a sequence of signs on a shifted copy of itself and add up the squares of the overlaps. Random signs leave the total near n²/2, and the squares modulo a prime, started a quarter of the way along, cut it to a twelfth. Exhaustive search finds far quieter short sequences, and no infinite family is proved to do better than 6.342.
Algebra
Multiply until every sign is positive
A form with a negative coefficient can still be positive wherever its variables are, and proving it usually takes an argument. Pólya found a certificate that needs none: multiply by a power of x + y and wait until every coefficient is positive. The power needed is finite exactly when the form never touches zero, and it grows like the reciprocal of how close it comes.
Analysis
A ball rebuilt everywhere but its centre
Rebuild a solid ball from its frequencies, keeping those below a cut-off, and read the result at the centre, where the ball is perfectly smooth. In the plane the answer settles on 1. In space it swings between 0.36 and 1.64 for ever, because the whole surface of the ball is the same distance from the centre and its jump arrives there in step.
Analysis
The curve that bends as far as it has gone
Say how sharply a curve bends at each distance along it, and the curve is fixed. Bending in proportion to distance gives Euler's spiral, whose arms are infinitely long and end a finite distance away. Bending that repeats decides at once whether the curve ever closes, and taken in steps the spiral is a Gauss sum.
Applied
The debts the Mishna divided
A man dies owing his three wives 100, 200 and 300. The Mishna divides an estate of 100 equally, of 300 in proportion, and of 200 as 50, 75 and 75, and for nearly two thousand years nobody could say why. The table is the nucleolus of a cooperative game, and the only division in which every pair splits its share as two people split a contested garment.
Applied
Twice as many goods as people
Indivisible goods with random values cannot always be shared so that nobody envies anybody. With one good each it works only when everyone's favourite is different; with twice as many goods as people it almost always works. Deciding each case exactly shows the chance climbing steeply in between — and a simple draft getting there only when the goods divide evenly.
Computation
Thirteen convex tangrams
The seven tangram pieces make thousands of figures, and only thirteen of them are convex. The proof turns a puzzle with sliding pieces into a finite list: every piece is made of the same small triangle, any convex figure puts them all on one grid, and that grid allows exactly twenty convex outlines of the right area. A search tries all twenty and finds that thirteen can be made.
Computation
Wrong guesses that arrive together
Seven players in red or blue hats each see every hat but their own, and must guess their own colour or pass. Every guess is a coin toss, yet with the Hamming code the team wins seven times in eight — not by making guesses better, but by making the wrong ones arrive together. The best chance for any number of players is fixed by the fewest losses that cover the cube.
Discrete
How many ways a regular graph pairs off
Give every point of a bipartite graph the same number of edges and Hall's theorem promises a perfect matching. It promises far more: at least ((d − 1)^(d−1)/d^(d−2))ⁿ of them, by Schrijver's theorem, and at most (d!)^(n/d), by Brégman's. Listing all 297,200 cubic graphs on six and six finds only four different counts, and random graphs turn out to live near the bottom.
Discrete
One colour more than the busiest point
Colour the edges of a graph so that edges meeting at a point differ. The busiest point forces as many colours as it has edges, and Vizing proved that one more is always enough. So every graph is in one of two classes, and deciding which is hard — yet almost every graph is in the first, and the ones in the second carry an odd piece that cannot be paired off.
Dynamics
Two blocks and a wall count π
Slide a heavy block towards a light one resting by a wall, and count every collision until they part for good. A heavy block 100 times the mass gives 31; ten thousand times, 314; a million, 3,141. The digits are π because the two blocks are secretly one billiard ball in a wedge, and a ball in a wedge bounces as many times as the wedge's angle fits into half a turn.
Dynamics
A conserved quantity that is not there
In 1964 Michel Hénon and Carl Heiles asked whether a star in a galaxy obeys a third conservation law besides energy and angular momentum. They cut its orbit by a plane and looked. At low energy every orbit drew a clean curve, as if the law held; at higher energy the curves dissolved into dust. Integrating their model again shows the share of chaotic orbits climbing from nothing at energy 1/12 to almost all at 1/6.
Geometry
Four perpendiculars to an ellipse
From a point near the centre of an ellipse, four lines meet the curve at right angles; from a point far away, only two. Apollonius found the boundary between the cases twenty-two centuries ago. It is the curve traced by the ellipse's centres of curvature, a four-cusped star, and it is also the envelope of all the normals, the fold in a family of lines where two perpendiculars merge and vanish.
Geometry
The most four hinged rods can enclose
Four rods of fixed lengths, hinged at their ends, can be pushed into a whole family of quadrilaterals. The one enclosing the most area has its corners on a circle, and Brahmagupta gave its area in 628 as a product of four terms. Order the rods any way round the circle and the area is the same; add a fifth rod and the circle still wins, but the formula is gone.
Logic
Venn diagrams that turn into themselves
A Venn diagram shows every combination of in and out exactly once. Ask for one that looks the same after a turn of 1/n, each curve rotating onto the next, and a counting argument forbids it unless n is prime. Five congruent ellipses manage it, certified by Euler's formula; four rotated copies cannot, and it took until 2004 to show that every prime can.
Logic
Every tautology survives double negation
Intuitionistic logic refuses excluded middle, Peirce's law and every other classical principle that claims more than can be shown. Glivenko proved in 1929 that it refuses nothing outright: put ¬¬ in front of any classical tautology and the result is intuitionistically valid. Every Kripke model of up to four states confirms it for eight famous laws, a census of 11,451 formulas confirms it one by one, and a model with a growing domain shows where it stops.
Number
Five powers of two, seven more than a square
2³ − 7 = 1², 2⁴ − 7 = 3², 2⁵ − 7 = 5², 2⁷ − 7 = 11², 2¹⁵ − 7 = 181². Ramanujan asked in 1913 whether there are any more. Testing every exponent up to 10,000 finds none, and Nagell proved in 1948 that none exist, by moving to a number system where 2 splits into two primes and factorisation is still unique. The same five solutions decide when a two-error-correcting code can be perfect.
Number
Two pounds in 73,682 ways
Count the ways to pay an amount exactly with a fixed set of coins and the answer is almost a polynomial in the amount — a polynomial plus waves that repeat with the periods of the coins. With parts 1, 2 and 3 the whole formula fits on a line; with British coins the leading term is still 3% off at four hundred pounds. And the fewest coins is a different question, which the cashier's rule answers correctly only for some coinages.
Probability
Where to look after looking and not finding
A lost object is somewhere on a map of probabilities, and every search that fails redraws the map. Bayes' rule says how: the cell searched loses probability in proportion to how good the search was, and every other cell gains. Looking where the chance of finding it on the next look is highest is provably best, and a search that overrates its own eyesight looks in the wrong place — which is how the wreck of Air France 447 was missed for two years.
Probability
One swap saves every prisoner
A hundred prisoners following the cycles of a shuffle all survive about 31% of the time. Let a helper look into the room first and swap the contents of two boxes, and they survive every time. With fewer boxes each, more swaps are needed — exactly one per cut a long cycle needs — and the number of swaps a random room demands has an exact distribution.
Topology
A tangle that is a fraction
Two ropes, four people, two moves: twist the right-hand pair, or turn everyone a quarter of the way round. Conway noticed that every tangle made this way carries a fraction, that the moves act on it as x + 1 and −1/x, and that the fraction is a complete invariant: two such tangles are the same exactly when their fractions are. Close the tangle and the fraction names a knot.
Topology
A hole that lasts across scales
A cloud of points sampled from a circle has no hole at all, and joined at the right scale it has exactly one — but nobody knows the right scale in advance. Persistent homology joins the points at every scale at once and records how long each hole survives. The circle's hole lasts; accidental holes vanish almost at once; and moving every point by δ moves every survival time by at most 2δ.
Before that
Everything published earlier, newest first. Titles only — the cards are on the full listing.
5 November 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Six lines all at one angle — algebra
- A disc that turns for ever — algebra
- Root two, raised to itself for ever — analysis
- A table that never reaches π — analysis
- A chain of majorities leads anywhere — applied
- The market with the most stable matchings — applied
- Two shares know nothing — computation
- Parallel planes take a vote — computation
- Alone on the track — discrete
- Out before the table says so — discrete
- Replace the parabola with a corner — dynamics
- An orbit stuck to an island — dynamics
- Eight points that refuse the cube — geometry
- The triangle a triple point buys — geometry
- The order of the questions — logic
- The shortest description nobody can certify — logic
- Sums with too few primes — number
- The comet Goldbach left behind — number
- One number for each arm — probability
- A restaurant and a shuffled deck — probability
- A number that counts what it cannot see — topology
- Peaks, pits and passes — topology
2 November 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Never negative and no sum of squares — algebra
- A prime leaves no room for a comb — algebra
- Which powers of x are enough — analysis
- The shortest wall that stops every line — analysis
- Matching workers who will not wait — applied
- A vote worth a square root — applied
- How few distances fix a word — computation
- A sort that never looks at the answers — computation
- The last place in the circle — discrete
- Every two edges must meet — discrete
- A line that fires all at once — dynamics
- Differences round a ring — dynamics
- The largest sofa round a corner — geometry
- How often points can sit on lines — geometry
- Functions that only go up — logic
- Six choices make a seventh — logic
- The largest amount two coins cannot pay — number
- Numbers that never make a prime — number
- A hundred prisoners and one shuffle — probability
- What a random chord is — probability
- A board that cannot be drawn — topology
- A game decided before it starts — topology
29 October 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- A second empty square — algebra
- Sixty rotations as far apart as rotations can be — algebra
- The sum after Leibniz's, made geometric — analysis
- Running a computation backwards with little memory — analysis
- Sellers in a square town — applied
- Two colonels and a linear programme — applied
- Dudeney's four pieces are the fewest — computation
- Two lost bits — computation
- The longest climb in a cube — discrete
- Three arrows in and three out — discrete
- The silver mean at the flat point — dynamics
- Two humps and one entropy — dynamics
- The price of curvature in Morley's triangle — geometry
- The longest fence of diameter one — geometry
- The narrowest refutation — logic
- What an incomplete set of connectives can say — logic
- Divisors that take a prime whole — number
- Nineteen fourth powers, and then sixteen — number
- The best middle when the shape is unknown — probability
- The matches left in the other pocket — probability
- Where random discs change sign — topology
- How many pieces to shrink a shape — topology
23 October 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, number, probability, topology and logic
- Two fields that every prime confuses — algebra
- A line of cubics that are all cyclic — algebra
- A rule that counts lets the sum escape — analysis
- No set fills half of every interval — analysis
- How often the rules find the majority favourite — applied
- How many stable matchings a market has — applied
- A code that survives a lost bit — computation
- A sum that never leaves its circle — computation
- Two numbers approximated at once — discrete
- The tour arrives with the last second neighbour — discrete
- Newton's method without a derivative — dynamics
- The entropy in a single orbit — dynamics
- The search that reuses a point — geometry
- A quarter of the bases at most — number
- Most numbers are missing from the table — number
- Arcs that cover a circle — probability
- Two losing games with a memory — probability
- The subgroups of a surface group — topology
- Curves that cross a line and close — topology
- The error that predicts itself — geometry
- An electorate too large for a dictator — logic
- Four quantifiers, one of them blind — logic
21 October 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- The nearest point is not the best guess — algebra
- How often a table of random signs is flat — algebra
- Where to take the samples — analysis
- The sine of one degree — analysis
- A core is a rare thing to find at random — applied
- The element the judges are worst at — applied
- The rectangles a square can be cut into — computation
- A code built from an icosahedron — computation
- A number that is its own square — discrete
- A search for room that never ends — discrete
- How high an orbit climbs — dynamics
- Two constants at the flat point — dynamics
- The busiest point in a set — geometry
- Rounding to the nearest remainder — geometry
- The plane cut by a list — logic
- Syllogisms that say most — logic
- A small quotient for every denominator — number
- How close two sums of square roots come — number
- When to stop tossing a coin — probability
- The middle value has a bell of its own — probability
- How short a paper Möbius band can be — topology
- Many crossings and a small linking number — topology
19 October 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, topology and probability
- The class number is a degree — algebra
- How near nought a sum of roots can come — algebra
- A slope without a subtraction — analysis
- The zeros a partial sum cannot avoid — analysis
- A relaxation that answers in halves — applied
- A block in an ocean of small holders — applied
- The fewest points that meet every line — computation
- The fewest entries that fix a Latin square — computation
- Two arrows in and two out — discrete
- No graph is poorer in squares than chance — discrete
- The ash a random soup leaves — dynamics
- Doubling without an attractor — dynamics
- Five pyramids for every number — geometry
- A box with every face diagonal whole — geometry
- The second letter breaks the pattern — logic
- The sizes at which a sentence can come true — logic
- A constant that holds the primes — number
- Nine cubes for two numbers — number
- Counting the subgroups of a free group — topology
- Lines that leave half on each side — topology
- One random path down a search tree — probability
- A card game almost nobody wins — probability
14 October 2026
22 essays on probability, topology, number, applied, dynamics, logic, discrete, computation, algebra, analysis and geometry
- What not knowing the coin costs — probability
- Evidence that can be multiplied — probability
- How many ways a curve can cross itself — topology
- The lists no covering can have — topology
- Two neighbours share out their small primes — number
- The product raised to the twenty-fourth power — number
- Some good trades cannot be made — applied
- The cheapest tree costs ζ(3) — applied
- Fair in base two and not in base six — dynamics
- Two dusts that add up to a line — dynamics
- Thirty-one directions nobody can label — logic
- One type for every use — logic
- How many shapes a tree can take — discrete
- The forests a graph can hold — discrete
- A seed is a deck — computation
- Birthdays spaced too evenly — computation
- Chords that never cross — algebra
- Holding one still turns the other over — algebra
- A grid that packs the copies — analysis
- The route round that rates cannot avoid — analysis
- Valleys a search cannot count — geometry
- A chain of tetrahedra saves more — geometry
11 October 2026
44 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Three coordinates and a fourth power — algebra
- The magic squares are a space — algebra
- The length round an ellipse — analysis
- Turning a slow series geometric — analysis
- Sellers on a street stand back to back — applied
- A party that loses every issue can win — applied
- Dots that never repeat a step — computation
- A perfect coin and a biased shuffle — computation
- No triangle, and four colours — discrete
- How often a number can appear in Pascal's triangle — discrete
- The zero of a sandpile — dynamics
- A chaotic map that always comes back — dynamics
- A cube through a hole in a cube — geometry
- Where charges settle on a sphere — geometry
- The order that keeps elimination narrow — logic
- No way to hand out true and false — logic
- The largest prime in a typical number — number
- A partition piled into a corner — number
- How many new kinds the next sample will show — probability
- Two losing games that win together — probability
- How a random path winds round a point — topology
- Which knots a random loop ties — topology
- A rotation chosen at random — algebra
- Eigenvalues that keep their distance — algebra
- The corners of a random hull — analysis
- A wave known only at its ticks — analysis
- Haggling that ends at the largest product — applied
- Houses traded in cycles — applied
- The shortest string holding every ordering — computation
- What two points build in two rounds — computation
- The dots a ball catches — discrete
- Triangle-free, and two-sided eventually — discrete
- A cluster grown by random walkers — dynamics
- An ant that builds a highway — dynamics
- The areas a right triangle can have — geometry
- The largest polygon of diameter one — geometry
- Five axioms and fifteen logics — logic
- Truth tables up to renaming — logic
- The first number that is not a square — number
- Thirty-three as three cubes — number
- A threshold nobody can place — probability
- The best average is the worst bet — probability
- An odd number of squares on every polygon — topology
- The turns a tangent makes — topology
10 October 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- The fraction written on every bulb — algebra
- Two rays for every fraction — algebra
- What a missing string of digits leaves behind — analysis
- Where a random walk leaves the snowflake — analysis
- One, plus a quarter, plus a ninth — applied
- A best reply to the past — applied
- Half the chance for every silent set — computation
- Three hundred thousand outputs of nearly nothing — computation
- How far apart two triangulations can be — discrete
- Pairs of trees that count planar maps — discrete
- A long enough chain never locks — dynamics
- A ring of clocks settles into a twist — dynamics
- The numbers that come after the golden ratio — geometry
- The shape of a cell around a random point — geometry
- Resolving one letter away at a time — logic
- One block of quantifiers, read off an algebra — logic
- Real fields that factorise uniquely — number
- A walk on Gaussian primes stopped by a moat — number
- A result at five per cent that favours the null — probability
- Peeking until the answer is yes — probability
- A curve of pixels needs two kinds of touching — topology
- An instrument that computes with circles — topology
9 October 2026
22 essays on algebra, analysis, applied, computation, discrete, probability, number, logic, geometry, topology and dynamics
- Twenty-seven million cubics, sorted by their symmetry — algebra
- The quartics whose roots stay paired — algebra
- A sum of factorials that converges at every prime — analysis
- The slope a Gauss sum leaves behind — analysis
- A third of random games have no pure equilibrium — applied
- Every random game has an odd number of equilibria — applied
- A grid of bits whose rank stops at the state — computation
- When a press skips its own light — computation
- The roots every matching polynomial keeps real — discrete
- Log-concave with nothing to make it so — discrete
- Edges that crowd each other out — probability
- The longest thread between two random strings — probability
- When Euclid's number is itself prime — number
- The numbers that fool every base — number
- What a formula keeps when letters are fixed — logic
- Three models and never two — logic
- The sums a triangulation cannot change — geometry
- A cut that gets sharper as the map gets lazier — topology
- A crowd of clocks that falls into step — dynamics
- Stable cycles a hair from Hénon's chaos — dynamics
- The caustics only an ellipse keeps — geometry
- An octagon only the hyperbolic plane can hold — topology
7 October 2026
44 essays on algebra, analysis, applied, discrete, dynamics, geometry, computation, logic, number, probability and topology
- The quintics that have a formula — algebra
- Three real roots and no real radicals — algebra
- Settling in order settles everywhere at once — analysis
- Close in height and nowhere close in slope — analysis
- Which vote finds the truth more often — applied
- One glove too many — applied
- Half the neighbours forces a tour — discrete
- The boards a knight can tour — discrete
- Chaos on the line between two roots — dynamics
- Small Mandelbrot sets among the cubics — dynamics
- The five crescents that can be squared — geometry
- The point nearest in total to three corners — geometry
- The centre a compass finds in six circles — computation
- The arcs that make the centre cheap — computation
- The choice inside a countable union — logic
- A spanning tree for every graph — logic
- Euclid's proof run as a machine — number
- Which way a prime's two squares point — number
- A wait that ends and has no average — probability
- Two-thirds of a step past the line — probability
- Every loop of the butterfly links every other — topology
- A knot must turn twice, and a little more — topology
- Two random shuffles reach every shuffle — algebra
- Nineteen copies and no more — algebra
- The tangent counts zigzags — analysis
- Seven times half of π, and then not — analysis
- A market too large to bargain in — applied
- Every premise makes the verdict vote stricter — applied
- The best of thirty-four thousand scramblings — computation
- Every light on, from every light off — computation
- Signs that make a determinant count — discrete
- The diamond that freezes its corners — discrete
- The cosines of the powers of two — dynamics
- One constant for each shape of top — dynamics
- What a junction can save — geometry
- One determinant for every kaleidoscope — geometry
- What a proof about numbers can compute — logic
- One type up — logic
- The same bell on thinner sets — number
- Sectors that shrink with the primes — number
- The longest run has no limit — probability
- A share that depends on the average — probability
- Torus knots that fill the three-sphere — topology
- The pieces no formula counts — topology
5 October 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- The signs no completion can change — algebra
- Regression is a square completed halfway — algebra
- The sine of a figure eight — analysis
- The sine, rebuilt from its zeros — analysis
- A vote worth nothing until it was outnumbered — applied
- A third kind of member — applied
- Ten digits need a symmetry that does not commute — computation
- Square it and keep the middle — computation
- Every function is a tree with two marks — discrete
- Almost every tree can be turned over — discrete
- A threshold no average can see — dynamics
- Two chaotic orbits made to agree — dynamics
- Wings that cut a chord equally — geometry
- A total hung in a temple — geometry
- The zigzags a formula can draw — logic
- A contradiction that stays where it is — logic
- Abundant, and still not a sum of its parts — number
- Divisors that make every amount — number
- Four lists find a collision sooner — probability
- Counting a population by its repeats — probability
- What homology forgets about a loop — topology
- The room a projective space needs, read off Pascal's triangle — topology
3 October 2026
12 essays on algebra, analysis, applied, discrete, dynamics, geometry, logic, number, probability, topology and computation
- The area of the Mandelbrot set, from two sides — algebra
- Only one curve through the factorials bends the right way — analysis
- Two auctions that earn the same — applied
- A sum that forbids half the pairings — discrete
- A flow that doubles like a parabola — dynamics
- Three circles that touch and are not the largest — geometry
- Dominoes that no algorithm can match — logic
- Three unit fractions for every four over n — number
- Multiplying makes the digit one common — probability
- Pieces minus holes on a random surface — topology
- A rotation that hides the lattice — computation
- The boards on which every light goes out — computation
2 October 2026
15 essays on logic, dynamics, number, algebra, analysis, probability, topology, applied, discrete, geometry and computation
- The machines that need a reason never to stop — logic
- The column rule 30 will not explain — dynamics
- Is the partition count even half the time? — number
- The race that makes ζ(3) irrational — number
- The flattest polynomials of signs — algebra
- The best nodes have no formula — analysis
- An urn forgets its start only below one half — probability
- What a jump costs evenly spread points — probability
- Counting targets when the sensors make mistakes — topology
- What it costs to find the point that stays put — topology
- A rent nobody envies is not one rent — applied
- Matching when the arrivals are shuffled — discrete
- A skeleton that ignores small teeth — geometry
- How nearly equal an odd number of triangles can be — computation
- A mate is rare until order ten — computation
1 October 2026
11 essays on algebra, analysis, applied, discrete, dynamics, logic, geometry, probability, number and topology
- How rarely a walk on a group comes home — algebra
- When a table of moves came from steady rates — analysis
- Prices for things wanted only together — applied
- The symmetric graphs no tour can close — discrete
- The word a line spells in a cube — dynamics
- The formula that cannot share — logic
- Three sides and four are proved — geometry
- The trisectors on a sphere almost agree — geometry
- A walk that follows its own footsteps — probability
- How many primes a typical number has — number
- Seven points split three ways — topology
30 September 2026
18 essays on probability, analysis, algebra, applied, computation, discrete, dynamics, geometry, logic, number and topology
- Certain of a coin that is not there — probability
- A series that waits on π — analysis
- The exponent a staircase shares with its set — analysis
- The shape a random ball grows into — algebra
- Five-eighths of the pairs, and no more — algebra
- What the agenda leaves standing — applied
- Three splits that answer every other — applied
- Every ordering once, around a cycle — computation
- Coins hidden in the roots — discrete
- The longest wait for a prime — discrete
- A growth rate no step contains — dynamics
- Fractions repeat and roots look random — dynamics
- How many pairs can be one apart — geometry
- The squares that answer every request — logic
- A letter the clauses already decide — logic
- Class groups drawn at random — number
- Seven graphs that must link — topology
- Four equal quarters with two lines — topology
29 September 2026
65 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Two graphs the eigenvalues cannot tell apart — algebra
- A hole is a cycle that bounds nothing — algebra
- A slope for a curve that is no function — analysis
- A slope can swing but never jump — analysis
- The corners are the orders of arrival — applied
- Each user pays for its own last link — applied
- The orders a plane cannot have — computation
- Every power of x that draws a hyperoval — computation
- The crossings a graph cannot avoid — discrete
- Every point at the average of its neighbours — discrete
- Every window is the whole diagram again — dynamics
- Windows counted like necklaces — dynamics
- A band of a sphere is a band of its cylinder — geometry
- The ball that is largest in five dimensions — geometry
- What counting can prove exists — logic
- Closed sets obey the continuum hypothesis — logic
- The row that proves a prime — number
- How often two generates every remainder — number
- When every value comes from the same hat — probability
- The rank that remembers every value — probability
- Two copies of the polygon, cross-matched — topology
- Every section of the band must vanish — topology
- The sums that obey a smaller equation — algebra
- The sums of primitive roots are always whole — algebra
- An area measured by walking round it — analysis
- No integrand sits on the border — analysis
- How few voters any majority needs — applied
- The fewest swaps to a winner — applied
- The spiral that measures its own circle — computation
- A third reached only in the limit — computation
- The fewest triangles an edge density allows — discrete
- As many points as two steps allow — discrete
- A jam that comes from nowhere — dynamics
- A rule that remembers one row back — dynamics
- Six numbers that need five pentagons — geometry
- Fifteen numbers decide every number — geometry
- A machine that carries one bit — logic
- There is a relation such that — logic
- When minus one can be reached — number
- A cube root that looks like chance — number
- Drawn without putting back — probability
- Charged for the variance, not the range — probability
- Six sticks tie a trefoil, and five cannot — topology
- Almost every long loop is knotted — topology
- Almost none of the roots are real — algebra
- Random roots crowd onto the circle — algebra
- Three sines tie a knot — analysis
- A dust that almost every line misses — analysis
- A bound that may be off by a third — applied
- Tours within half again of the best — applied
- The fewest moves to draw it — computation
- The longest code that survives every erasure — computation
- A labelling every tree seems to have — discrete
- Every edge on exactly two cycles — discrete
- How smooth the disguise is — dynamics
- Two numbers in one jagged record — dynamics
- What the signs allow — geometry
- Five where four were promised — geometry
- How many gates a truth table needs — logic
- A walk that beats trying everything — logic
- Euler's sixty-five convenient numbers — number
- Perfect but for one factor — number
- Two constants that do not walk at random — probability
- Every pair side by side, once — probability
- Seven points and a knot they cannot avoid — topology
28 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- The biggest box built from signs — algebra
- The count a fold cannot change — algebra
- The error that keeps coming back to its worst — analysis
- An ellipse, not a disc — analysis
- Prices at every corner — applied
- The cube that takes every corner — applied
- The centre a straightedge cannot find — computation
- The lengths dividers cannot reach — computation
- Every crowd holds a bowl or a dome — discrete
- The longest climb of a shuffle — discrete
- Where the time goes on an attractor — dynamics
- A region the orbit cannot leave — dynamics
- Every point has a partner across the bisectors — geometry
- A triangle that fits once fits everywhere — geometry
- Two lists that are one order — logic
- The graph that coin tosses always make — logic
- How evenly the fractions spread — number
- Counting the classes that break factorisation — number
- The narrowest door sets the pace — probability
- The forgetting that happens all at once — probability
- Every surface is sewn from pants — topology
- The fewest corners a surface needs — topology
27 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Thirty-one moves from solved — algebra
- Which graphs let the tokens go anywhere — algebra
- Two numbers decide the flow — analysis
- The matrix that has no logarithm — analysis
- Prices the bidders raise — applied
- The prices nobody can break away from — applied
- A rectangle made only of squares — computation
- No odd number of equal triangles — computation
- The order decides the colours — discrete
- How many colours the plane needs — discrete
- Where the Collatz map is a coin — dynamics
- Fourteen fractions that list the primes — dynamics
- The skeleton inside a shape — geometry
- Three mirrors make every solid — geometry
- The program that prints itself — logic
- No algorithm can read what a program does — logic
- Two halves a sieve cannot tell apart — number
- The patterns primes are allowed to make — number
- Points on a lattice that see almost nothing — probability
- An error bar for points that are not random — probability
- Every count a solid can have — topology
- Counting targets by their holes — topology
26 September 2026
22 essays on geometry, logic, number, topology, dynamics, probability, discrete, applied, algebra, analysis and computation
- At least as many lines as points — geometry
- Rows of three, planted on a cubic — geometry
- A contradiction that is only a sum — logic
- Two terms made equal, and no more — logic
- The wait for the next sum of two squares — number
- A fraction on the circle forces a whole point — number
- A whole number split into two that are not — topology
- Six points in space and a pair that must link — topology
- Counting in a base that is not a whole number — dynamics
- Where a base-β orbit spends its time — dynamics
- Evidence measured in decibans — probability
- The envelope that always looks better — probability
- Enough partners in every finite group — discrete
- Matching as they arrive — discrete
- Envy that any single item would cure — applied
- A rent nobody envies — applied
- How a polynomial breaks modulo the primes — algebra
- Units that form a lattice — algebra
- A geometric series whose ratio is a matrix — analysis
- A series that converges to minus one — analysis
- A curve that divides any angle — computation
- Two weak sources make one fair bit — computation
25 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, number, probability, logic and topology
- What is left when the middle is taken out — algebra
- The polygon a lattice becomes from far away — algebra
- A coin in front of every power — analysis
- Two sign patterns that land together — analysis
- One extra person on one side — applied
- The matching in the middle — applied
- Twenty cards with no set among them — computation
- The polynomial that bounds the caps — computation
- Three places cut apart — discrete
- The cheapest way to send — discrete
- A whole interval of speeds — dynamics
- The last circle to break — dynamics
- The least wall for equal rooms — geometry
- Every chord slid to the middle — geometry
- Divisors that add to three times the number — number
- A ratio nobody else has — number
- A coin that lets the first player win — probability
- Three patterns in a circle — probability
- A proof that says which half — logic
- No table of truth values is enough — logic
- How many changes undo a knot — topology
- Several colours on every vertex — topology
24 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Discs that fence in the eigenvalues — algebra
- The highest point on the sphere is an eigenvalue — algebra
- A plucked string keeps its corners — analysis
- The ripples that make a series run away — analysis
- How often the majority goes in a circle — applied
- A majority wiser than its members — applied
- A Latin square with boxes — computation
- Orthogonal squares are a code — computation
- Multiplying every number on the dial at once — discrete
- A root lifted one digit at a time — discrete
- The ball that stays outside the table — dynamics
- The word a straight line spells — dynamics
- The diagonal no unit measures — geometry
- Tiles that never repeat — geometry
- The word that cannot describe itself — logic
- Arithmetic with addition alone — logic
- The shape a random partition takes — number
- Two counts that agree for no visible reason — number
- The median of many small averages — probability
- A sphere that is nearly all equator — probability
- A polynomial that tells left from right — topology
- The crossings an alternating knot cannot lose — topology
23 September 2026
22 essays on number, geometry, logic, algebra, analysis, dynamics, probability, discrete, computation, applied and topology
- Two families of solutions, and a box that holds both — number
- Sixty needs two digits and sixty-one needs ten — number
- The player who meets the first long run — geometry
- The remainders that count the roots — geometry
- One thing in each region is enough — logic
- The conclusion is what survives the erasing — logic
- The sums of roots of unity that add to nothing — algebra
- Three ways to pair four roots — algebra
- The length the derivative never sees — analysis
- A length counted by the lines that cross it — analysis
- A double root halves the error instead of squaring it — dynamics
- A cubic method that is Newton's in disguise — dynamics
- The ground a walk covers — probability
- A walk that may not step where it has been — probability
- Cars that park, and trees that grow — discrete
- A random tree is one part in e leaves — discrete
- No set with a line in every direction is small — computation
- A sum of two sets modulo a prime cannot be small — computation
- What a missing input is worth — applied
- Cutting a link costs both of its ends the same — applied
- The loops on a torus that never cross themselves — topology
- A twist that carries one loop to another — topology
22 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Twenty-four ways to set a cube down — algebra
- Necklaces made of symmetries — algebra
- A limit can jump at every fraction — analysis
- Uniform, except on a small set — analysis
- The nearest consistent verdict — applied
- Agendas that cannot contradict themselves — applied
- The rate a noisy channel allows — computation
- Erasures a code can see — computation
- Every place changes back — discrete
- The walk through the middle levels — discrete
- Sensitivity comes free — dynamics
- Chaos on a set nobody lands on — dynamics
- Seven pieces and an equilateral middle — geometry
- Eighteen equilateral triangles — geometry
- A plane through the cube — logic
- Half the cube and √n neighbours — logic
- An order that proves a prime — number
- Two primes where Fermat holds twice — number
- Fourteen people within a day — probability
- A room where nobody is alone — probability
- Twelve pentagons, whatever the hexagons — topology
- Zero in four dimensions — topology
21 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Every power sum, from the coefficients alone — algebra
- The roots of the slope stay inside — algebra
- The sum that steps over every whole number — analysis
- A coin in front of every term — analysis
- When one of the two numbers is missing — applied
- The lines the optimum lies under — applied
- The price of a construction — computation
- One cell short of a transversal — computation
- A walk that splices in its own detours — discrete
- The streets a postman walks twice — discrete
- Almost every orbit is fair — dynamics
- A solvable chaos of every degree — dynamics
- One number for every chord through a point — geometry
- Every side measured by one diameter — geometry
- Six sentences from two quantifiers — logic
- A quantifier is a shadow — logic
- A factorisation that hides its primes — number
- Factoring uniquely with no way to divide — number
- The cells a permutation must miss — probability
- A round table with no couple together — probability
- Opposite labels that have to meet — topology
- The colours a circle forces — topology
18 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Where two roots run into each other — algebra
- One number under every bell — algebra
- An endless region with a finite area — analysis
- A rectangle cut by a curve — analysis
- The people every stable answer leaves out — applied
- A ring that no pairing can break — applied
- Fair bits from an unfair coin — computation
- A filter that changes only the spread — computation
- Moves that only ever add edges — discrete
- The densest graph without a square — discrete
- A road where nobody overtakes — dynamics
- No local rule can count the votes — dynamics
- Three ordinary lines from a count — geometry
- The fewest ordinary lines a polygon allows — geometry
- A function that adds and is nowhere a line — logic
- Infinitely many guessers, finitely many wrong — logic
- The fraction Lambert built for the tangent — number
- The pattern in e's continued fraction — number
- Two children and the sentence about one of them — probability
- One coin, counted by runs and by wakings — probability
- Every way to pair a polygon's edges — topology
- The surface a random gluing makes — topology
16 September 2026
44 essays on discrete, applied, analysis, number, probability, topology, geometry, algebra, computation, dynamics and logic
- What the search has when it fails — discrete
- The piece that cannot pair off — discrete
- Patience instead of a contract — applied
- The reading that is almost right — applied
- The series everything else is measured against — analysis
- The repair at the boundary — analysis
- The two squares actually produced — number
- Almost no number is one — number
- The one that hardly ever comes up — probability
- Two patterns, one chance, different waits — probability
- A count that can say zero — topology
- A map that offers a choice — topology
- Pinned between two sequences — geometry
- The slice that has to match — geometry
- Counted across and counted down — algebra
- The cycles and the cuts — algebra
- More blocks than points — computation
- A plane in a list of numbers — computation
- An area that never finishes — dynamics
- Covering rather than avoiding — dynamics
- One gadget defeats every refinement — logic
- The boundary at three variables — logic
- Any unevenness brings the match sooner — probability
- A collision that finds a factor — probability
- The sum of the parts, taken again — number
- Why a quarter of numbers overshoot — number
- A polynomial behind the colourings — topology
- The surface a knot bounds — topology
- The dark lines are one point's orbit — dynamics
- The window that opens with a stutter — dynamics
- The angle that is really an area — analysis
- When two circular motions come home — analysis
- Sums of powers, read off a staircase — geometry
- Three triangular numbers, and no fewer — geometry
- A number larger than every number — logic
- A countable field that passes for the line — logic
- How many cuts a fair share costs — applied
- The product that makes a division fair — applied
- Seven powers in a space of six — algebra
- The integers a field contains — algebra
- A plane no field built — computation
- The curve that no three points in line define — computation
- Five spokes squeezed into K5 — discrete
- The few points that cut a flat graph — discrete
14 September 2026
22 essays on discrete, number, applied, analysis, algebra, dynamics, probability, topology, geometry, logic and computation
- The product that deals the labels — discrete
- The coefficient that is a polynomial — discrete
- One residue whose powers are all of them — number
- The exponent that is smaller than Euler's — number
- Five weighings and the question is closed — applied
- An objection one player makes to another — applied
- Which curves have a length at all — analysis
- The length belongs to the journey — analysis
- How fast the ball fills — algebra
- The edge that is as big as the ball — algebra
- How a lock comes apart — dynamics
- A rotation in different coordinates — dynamics
- The bound is the answer to a search — probability
- No single input can move it far — probability
- Every word driven to a normal form — topology
- The third number a surface needs — topology
- One circle touching four — geometry
- A centre is three weights — geometry
- Not one step but a continuum — logic
- Refutable in something small — logic
- Two instruments with one reach — computation
- A quintic a sliding mark reaches — computation
13 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Every third coefficient — algebra
- On the circle and never home — algebra
- The subsequence that has to exist — analysis
- A sum read from inside — analysis
- The table inside every quota — applied
- Where the rounding runs out — applied
- Solutions that come in multiples of p — computation
- Give or take twice the square root — computation
- When several pairs share the roads — discrete
- One tree for every cut — discrete
- A matrix that counts the returns — dynamics
- The folds that measure chaos — dynamics
- Half a circle against a wall — geometry
- Nearly the most means nearly round — geometry
- Two literals make an arrow — logic
- A failed search is a proof — logic
- A method that is allowed to miss — number
- Why the expansion has to repeat — number
- A sum stopped early still says something — probability
- How many get their own hat — probability
- Two opposite points that agree twice — topology
- As many cuts as colours — topology
12 September 2026
22 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- Moving a map across a product — algebra
- A denominator that reaches past the radius — analysis
- A series that converges nowhere — analysis
- A plane disguised as an arrow — algebra
- A mixture that is a population — applied
- Worth more for being seen first — applied
- The obstruction that was the only one — computation
- Equal area on a sphere, without a rectangle — computation
- A remainder read two digits at a time — discrete
- Averaging down the triangle — discrete
- A dimension for every rate of crowding — dynamics
- The room a jagged graph takes up — dynamics
- Six points on a conic, and the line they share — geometry
- Equal diagonals in a curved quadrilateral — geometry
- Every derivation is a term — logic
- The sentence between a premise and its consequence — logic
- The arcs a line crosses on its way to a number — number
- The function that sends fractions to binary — number
- The thresholds that nest — probability
- Add the odds from the end — probability
- Covering a surface multiplies its count — topology
- What a branch point subtracts — topology
9 September 2026
33 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- The square that cannot be negative — algebra
- One subtraction clears a direction — algebra
- The nearest point of a flat thing — algebra
- An error with an unknown in it — analysis
- The centre is a choice — analysis
- The points that ruin the fit — analysis
- A signal both can see — applied
- The landscape nobody is looking at — applied
- Two equilibria and no way to choose — applied
- A dissection that never comes apart — computation
- Slid, but never turned — computation
- Finitely many, and nobody says how many — computation
- The run that lands one place along — discrete
- Every entry counts the routes to it — discrete
- The carries decide the divisibility — discrete
- Infinite on one side and nought on the other — dynamics
- A carpet with two dimensions — dynamics
- A dimension from the stretching rates — dynamics
- Aimed at one focus, turned towards the other — geometry
- Two families that cross at right angles — geometry
- One sign decides which curve — geometry
- The assumption a proof pays back — logic
- A lemma, and the proof that never mentions one — logic
- The instance that has to be guessed — logic
- Two matrices that generate the tree — number
- Every rational in one sequence — number
- The fractions that beat every smaller one — number
- When the numbers are shown — probability
- Giving up on the best — probability
- Half of what an oracle takes — probability
- The symmetries a cover has of its own — topology
- A covering is a permutation — topology
- Folding a graph until it decides — topology
8 September 2026
21 essays on algebra, analysis, applied, computation, discrete, dynamics, geometry, logic, number, probability and topology
- The puzzle that is exactly half solvable — algebra
- When the label may be a matrix — algebra
- Three points, however many there are — analysis
- A wall between two bodies — analysis
- Four ways out, and what each costs — applied
- The best a code can be — computation
- Past half the distance — computation
- Sixteen polygons with one dot inside — discrete
- The dots a circle catches — discrete
- How short a cycle could be — dynamics
- Every pattern happens exactly once — dynamics
- The least area a width can hold — geometry
- The same question in space — geometry
- The game the algorithm was playing — logic
- A language that can name a set — logic
- Every class, and in equal shares — number
- The sieve that cannot finish — number
- The window where the giant is born — probability
- Sharp, or merely a threshold — probability
- One chart is never enough — topology
- The circles that fill a three-sphere — topology
5 September 2026
50 essays on geometry, discrete, topology, dynamics, computation, applied, probability, analysis, logic, number and algebra
- One dimension up, and the circles disappear — geometry
- Every site in the middle of its own cell — geometry
- When the sites are not the same size — geometry
- The tree inside the triangulation — geometry
- The shape described from outside — geometry
- Counting the paths that go wrong — discrete
- One word, and four objects — discrete
- The equation a sequence satisfies — discrete
- The solid whose corners are triangulations — discrete
- The theorem that has no version in space — discrete
- The group a space has at a point — topology
- Every cover is a subgroup — topology
- Cutting a space to find its group — topology
- Why the second group commutes — topology
- Angles survive and areas do not — topology
- The sphere that complex numbers live on — topology
- The flow that is really a map — dynamics
- A closer start buys only time — dynamics
- Stretch, fold, and what is left — dynamics
- Neither a surface nor a solid — dynamics
- Almost every number comes down — dynamics
- The heuristic that cannot be a proof — dynamics
- The test that ranks the generators — computation
- Four numbers and the rule is yours — computation
- Randomness that has to be earned — computation
- Nineteen thousand bits of state — computation
- One circle, and a straightedge — computation
- The compass that will not open — computation
- Seats to parties and places at once — applied
- The corners are whole assignments — applied
- A price for every person and task — applied
- One table, two lotteries — applied
- Where the corners stop being whole — applied
- No rule escapes the doctrinal paradox — applied
- Deciding the premises or the conclusion — applied
- How long until it forgets — probability
- The moment a giant appears — probability
- Two thresholds, not one — probability
- Finding a threshold with two moments — probability
- A staircase with no steps — analysis
- A line under every point — analysis
- The function seen from its tangents — analysis
- Where the guarantee stops — analysis
- How many worlds a formula can need — logic
- Nearly always, or nearly never — logic
- The distance a sentence can see — logic
- Infinitely many of one kind — number
- Which infinitudes are proved — number
- Two hundred and forty directions — algebra
- The only bit that survives — algebra
3 September 2026
20 essays on number, algebra, analysis, probability and logic
- The two supplements, and where the eight comes from — number
- The symbol is the sign of a shuffle — number
- One sum, squared two ways — number
- Which primes a form takes — number
- A rotation of four-space takes two of them — algebra
- The identity that multiplies sums of squares — algebra
- The integers among the quaternions — algebra
- What is lost at eight — algebra
- No interval in it, and length to spare — analysis
- Covering a set from outside — analysis
- Which functions can be added up — analysis
- A set that has no size at all — analysis
- The chain that stops — probability
- The chain that runs the same backwards — probability
- The time spent and the share held — probability
- Where the shares have nowhere to go — probability
- Two diagrams the language cannot tell apart — logic
- Worlds built out of sentences — logic
- The axiom with no property of the arrows — logic
- Necessity that means provable — logic
1 September 2026
20 essays on topology, computation, applied, geometry and discrete
- A curve that has area — topology
- Every loop is a circle in disguise — topology
- Two pieces, in every dimension — topology
- A ball whose outside is not one — topology
- Every necklace, in order — computation
- A memory of four bits — computation
- A page that knows where it is — computation
- A cycle for every pair — computation
- Five rules and one dial — applied
- The rule with no favourites — applied
- Two out of three, and never all three — applied
- Choosing what unfair means — applied
- Eight circles touching three — geometry
- Curvatures that stay whole — geometry
- The number four points agree on — geometry
- Every flat graph is a pile of circles — geometry
- Everybody's share of the chains — discrete
- The cube cut into chains — discrete
- The largest family that always meets — discrete
- How many ways to sort it — discrete
30 August 2026
20 essays on computation, logic, algebra, number and dynamics
- A field's worth of squares — computation
- The plane hiding in the squares — computation
- Nine thousand four hundred and eight — computation
- Sixteen of five hundred and seventy-six — computation
- One step in front of infinitely many — logic
- Every ordinal in base omega — logic
- Reached from below, or not at all — logic
- An ordinal as a growth rate — logic
- The only function that behaves like a volume — algebra
- One point in every big enough shape — algebra
- A determinant that counts trees — algebra
- The same sum without its minus signs — algebra
- Every partition, hidden in a product — number
- The terms that cancel almost everything — number
- The size of a number with no formula — number
- Every fifth one divides — number
- A table folded into a surface — dynamics
- A room that cannot be lit — dynamics
- The obstacle that makes a table chaotic — dynamics
- The triangle nobody can settle — dynamics
29 August 2026
20 essays on topology, number, logic, dynamics and applied
- The bottle that needs a fourth dimension — topology
- A disc sewn to a Möbius band — topology
- Two sheets over a one-sided surface — topology
- Orientation is a sign — topology
- Which roots refuse to be fractions — number
- A tail too small to be a whole number — number
- An integral that cannot be a whole number — number
- Approached too fast to be algebraic — number
- The arithmetic that loses subtraction — logic
- A line with as many points as a square — logic
- Countable, and everywhere — logic
- The size that cannot be pinned down — logic
- How fast the staircase arrives — dynamics
- The same map in different coordinates — dynamics
- The histogram an orbit leaves — dynamics
- The orbit a computer draws — dynamics
- None of the four conditions is spare — applied
- A share of the votes is not a share of the power — applied
- Too many orders to list — applied
- Sharing a cost that is not the sum of its parts — applied
28 August 2026
20 essays on algebra, analysis, geometry, probability and discrete
- The polynomial whose roots are the stretches — algebra
- The same map in a better basis — algebra
- Symmetry forces a right angle — algebra
- What a map does to a circle — algebra
- The area that names the number — analysis
- The equation with only one answer — analysis
- The exponential of a square — analysis
- The constant that counts what does not happen — analysis
- Thirteen more when one word is dropped — geometry
- The four that are allowed to cross themselves — geometry
- Six in four dimensions, and three forever after — geometry
- The five solids as three groups — geometry
- The error that does not care how many dimensions — probability
- Sampling where the answer lives — probability
- A walk that samples a distribution — probability
- Points too even to be random — probability
- Eighteen people, and the seventeen that escape — discrete
- The colouring nobody has ever seen — discrete
- Three in a row on the number line — discrete
- The sequence that cannot avoid a staircase — discrete
26 August 2026
15 essays on geometry, analysis, discrete, probability, dynamics, number, logic, computation, topology, algebra and applied
- The line with only two points on it — geometry
- The curve of the average, and the average of the curve — analysis
- A limit that forgets to be continuous — analysis
- The edge that forces a triangle — discrete
- The bottleneck is the whole story — discrete
- The widest layer and the longest chain — discrete
- The moment everything joins up — probability
- A dimension that is not a whole number — dynamics
- The orbit written as a word — dynamics
- One solution that makes all the others — number
- The choice nobody can write down — logic
- The mark that changes what is reachable — computation
- Linked, and no two of them are — topology
- The group that will not come apart — algebra
- The objection nobody can make louder — applied
25 August 2026
15 essays on topology, probability, algebra, applied, computation, logic and dynamics
- Zero can mean two different things — topology
- The subgroup that is freer than the group — topology
- When the whole histogram deviates — probability
- A shared root, found without finding it — algebra
- The crossings that will not come out even — algebra
- The lattice that runs the other way — algebra
- A multiplication that remembers the order — algebra
- The order everybody arrives in — applied
- A split nobody can walk away from — applied
- The court that contradicts itself — applied
- Equal area is enough, and equal volume is not — computation
- The straightedge buys nothing — computation
- The planes a recurrence cannot leave — computation
- The axiom is the shape of the graph — logic
- The staircase that is flat almost everywhere — dynamics
23 August 2026
15 essays on geometry, discrete, topology, dynamics, probability, logic, computation, applied and algebra
- The map that trades circles for lines — geometry
- One bottleneck and nothing else — discrete
- A polynomial that counts — discrete
- Two loops and one number — topology
- The same loop, unrolled — topology
- A bounce is a fold of the table — dynamics
- A twist that cannot avoid two points — dynamics
- The tail is not a bell — probability
- A game that decides what can be said — logic
- A proof with one rule — logic
- Every word once, around a cycle — computation
- A lottery over whole assignments — applied
- The polygon an equation forces — algebra
- The group drawn as a map — algebra
- A tower whose degrees multiply — algebra
22 August 2026
15 essays on number, geometry, probability, discrete, analysis and topology
- A tree that holds every triple — number
- Circles that are diamonds and squares — geometry
- The walk that becomes a curve — probability
- Three colours force a triangle — discrete
- A map that shrinks everything — analysis
- Where the fixed point escapes — topology
- Counting what has no formula — number
- Always one before the double — number
- The sieve written as a product — number
- The shape that averaging leaves alone — probability
- An average that never settles — probability
- How fast the bell arrives — probability
- A rectangle grown on two sides — analysis
- The flat map that fits closest — analysis
- The slope of the mirror image — analysis
21 August 2026
15 essays on geometry, topology, probability, analysis and discrete
- The rope that squares a corner — geometry
- Two right angles and the diagonal of a box — geometry
- The triangle that a globe gets wrong — geometry
- Two trees, and every edge in exactly one of them — topology
- Seven hundred and twenty degrees of gap — topology
- The solid where the answer is not two — topology
- The path folded at its first touch — probability
- Half the time is the rarest answer — probability
- Two barriers and a fair game — probability
- Where the coefficients come from — analysis
- When the period grows without bound — analysis
- The corners go first — analysis
- Five colours, and a chain that can be followed — discrete
- Counting the colourings — discrete
- Seven regions on a doughnut — discrete
20 August 2026
15 essays on topology, algebra, probability, discrete, analysis and geometry
- One line that halves them both — topology
- A loop that cannot be pulled tight — topology
- Colours that count more than three — topology
- The blocks a subgroup cuts out — algebra
- What a map throws away — algebra
- What the coefficients already know — algebra
- How far from the average a thing can be — probability
- The average settles and the wobble does not — probability
- When to stop looking — probability
- Two graphs that will not lie flat — discrete
- Sixteen trees on four points — discrete
- A curve with a corner at every point — analysis
- Almost none of it left, and still uncountably many — analysis
- Every ray comes back to the other focus — geometry
- Three trisectors and a triangle nobody expected — geometry
19 August 2026
15 essays on algebra, probability, topology, discrete, analysis and geometry
- Eight ways to leave a square alone — algebra
- Colourings nobody can tell apart — algebra
- The number that says how much room is left — algebra
- A loop that cannot miss the middle — algebra
- Nobody gets their own hat — probability
- How long until every one turns up — probability
- The rule that forgets where it came from — probability
- Which side of the line is inside — topology
- Every surface is a sphere with handles — topology
- A walk that changes one thing at a time — discrete
- Area by counting dots — discrete
- The sum that fits in one square — analysis
- The staircase that is not the diagonal — analysis
- The most area a fence can hold — geometry
- Nine points on one circle — geometry
17 August 2026
15 essays on applied
- The majority that goes in a circle — applied
- Five rules and five winners — applied
- Four conditions, and no rule that has all of them — applied
- A lie that pays — applied
- The seat that vanishes when the house grows — applied
- One cuts and the other chooses — applied
- Three people and a trimmed piece — applied
- Envy-free, up to one item — applied
- Nobody has a reason to run away — applied
- The side that proposes wins — applied
- No stable rule is safe from a lie — applied
- Two numbers that have to meet — applied
- What a constraint is worth — applied
- The value from both sides — applied
- The road that makes everyone later — applied
15 August 2026
15 essays on computation
- What two points can build — computation
- Every step is a square root — computation
- The cube that will not double — computation
- The angle that will not divide by three — computation
- Which polygons can be drawn — computation
- The circle that will not square — computation
- Distance is a picture — computation
- Sixteen spheres that fill a cube — computation
- Finding the error without reading the message — computation
- A polynomial through the gaps — computation
- The field with four elements — computation
- Every element is a power of one of them — computation
- Seven points, seven lines — computation
- A schedule where every pair meets once — computation
- The thirty-six officers — computation
14 August 2026
15 essays on logic
- A formula is a corner of a cube — logic
- One connective is enough — logic
- The map that puts neighbours side by side — logic
- Four circles cannot do it — logic
- Twenty-four out of two hundred and fifty-six — logic
- Every row, or one column — logic
- The tree that closes — logic
- Two worlds that both obey the rules — logic
- An infinite tree has an infinite path — logic
- The row that is not on the list — logic
- A list that cannot contain itself — logic
- The sentence that says it has no proof — logic
- Two injections make a bijection — logic
- A sequence that explodes and still stops — logic
- The middle that is not excluded — logic
12 August 2026
15 essays on dynamics
- The staircase that shows the whole orbit — dynamics
- A point that pulls, and a point that pushes — dynamics
- The road paved with doublings — dynamics
- A constant that does not care which map — dynamics
- A difference too small to draw — dynamics
- How fast two orbits part — dynamics
- The shape in every picture of itself — dynamics
- One c, one picture — dynamics
- Where Newton's method goes instead — dynamics
- Eight rules and a triangle — dynamics
- The rule that computes — dynamics
- Three gaps and no more — dynamics
- The orbit that must come back — dynamics
- Two lobes and no cycle — dynamics
- The question nobody can answer — dynamics
11 August 2026
15 essays on number
- The primes are what is left over — number
- There is no last prime — number
- A fraction that never closes — number
- Every fraction, exactly once — number
- How close a fraction can get — number
- One way to factor, and no other — number
- The shape of a number's divisors — number
- Numbers that are their own parts — number
- Two squares, and a lattice — number
- Every triple, on one circle — number
- Necklaces that prove a theorem — number
- Two dials at once — number
- Counting one rectangle, twice — number
- The square that cannot shrink — number
- A diagram turned on its side — number
10 August 2026
12 essays on geometry, analysis, algebra, discrete, topology and probability
- An angle that does not care where it stands — geometry
- The plane, divided by whoever is nearest — geometry
- Euclid proves it without moving anything — geometry
- The same terms, in a different order, adding to whatever is asked — analysis
- Area is the undoing of slope — analysis
- Completing the square, by completing a square — algebra
- Numbers that wrap — discrete
- Six people at a party — discrete
- Three moves, and what they cannot undo — topology
- Nothing on a sphere can be combed flat — topology
- A walk that always comes home, until it does not — probability
- The door that was not opened — probability
6 August 2026
15 essays on geometry, analysis, algebra, discrete, topology and probability
- The rectangle that eats itself — geometry
- A circle unrolled into a triangle — geometry
- Round is not the only way to be the same width — geometry
- A sum whose terms vanish and whose total does not — analysis
- One point's worth of information — analysis
- The slope of a single point — analysis
- The dot product is a shadow — algebra
- The directions a map leaves alone — algebra
- More things than boxes — discrete
- Four colours, and a proof nobody can read — discrete
- One sequence, counting everything — discrete
- Every corner pays for itself — topology
- Something always stays put — topology
- Twenty-three people — probability
- Bayes' theorem is a picture of a square — probability
2 August 2026
18 essays on geometry, analysis, algebra, discrete, topology and probability
- Two squares, four triangles, and no algebra — geometry
- Every square is a stack of odd numbers — geometry
- The oldest algorithm, drawn as a tiling — geometry
- Why the list of perfect solids stops at five — geometry
- One cone, four curves — geometry
- A sine wave is a circle seen from the side — analysis
- Adding up rectangles until they stop being rectangles — analysis
- A square wave built entirely out of round ones — analysis
- The curve that is its own slope — analysis
- A matrix is a picture of what happens to the grid — algebra
- Multiplying is turning — algebra
- Seven bridges, and the invention of throwing things away — discrete
- Pascal's triangle, in two colours — discrete
- The primes on a spiral, and a pattern nobody ordered — discrete
- The surface with one side, and what happens when it is cut — topology
- A sphere is a plane plus one point — topology
- A bell curve assembled out of coin flips — probability
- Getting pi by dropping needles on the floor — probability