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A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
The value of a 2×3 zero-sum game, named from both sides. The row chooser's expected payoff against each column as a line over the mixing probability, with the lower envelope and its maximum, beside the same construction from the column chooser's side. Both give 19/15.

Equilibrium

  1. 1 The value from both sides
  2. 2 The road that makes everyone later
  3. 3 A signal both can see
  4. 4 The landscape nobody is looking at
  5. 5 Two equilibria and no way to choose
  6. +10 more
15 essays · applied A code on the 3-cube, and the balls around its words. The corners of a hypercube with the chosen codewords marked and the words within one error of each shaded.

Error-correcting codes

  1. 1 Distance is a picture
  2. 2 Sixteen spheres that fill a cube
  3. 3 Finding the error without reading the message
  4. 4 A polynomial through the gaps
  5. 5 The best a code can be
  6. +10 more
15 essays · computation The arithmetic of GF(4), and of the integers mod 4. Addition and multiplication tables of a finite field, optionally beside the table of a ring of the same kind of size.

Finite fields

  1. 1 The field with four elements
  2. 2 Every element is a power of one of them
  3. 3 Solutions that come in multiples of p
  4. 4 Give or take twice the square root
  5. 5 No set with a line in every direction is small
  6. +10 more
15 essays · computation 256 consecutive pairs from xₙ₊₁ = 137xₙ + 187 mod 256. Consecutive outputs of a linear congruential generator plotted as points of a square, falling on a small family of evenly spaced parallel lines.

Pseudorandomness

  1. 1 The planes a recurrence cannot leave
  2. 2 The test that ranks the generators
  3. 3 Four numbers and the rule is yours
  4. 4 Randomness that has to be earned
  5. 5 Nineteen thousand bits of state
  6. +10 more
15 essays · computation V − E + F = 2, five times. Vertices, edges and faces of the five regular solids, with the alternating sum. The edges are counted from the faces rather than listed, and the sum is 2 in every row.

Euler characteristic

  1. 1 Every corner pays for itself
  2. 2 Two trees, and every edge in exactly one of them
  3. 3 Seven hundred and twenty degrees of gap
  4. 4 The solid where the answer is not two
  5. 5 Twelve pentagons, whatever the hexagons
  6. +9 more
14 essays · topology Two factor trees of 360. The same number split two different ways, both ending in the same primes.

Unique factorisation

  1. 1 One way to factor, and no other
  2. 2 The shape of a number's divisors
  3. 3 A factorisation that hides its primes
  4. 4 Factoring uniquely with no way to divide
  5. 5 Counting the classes that break factorisation
  6. +8 more
13 essays · number A map of the interval must fix a point. a continuous map of the interval, drawn with the diagonal. Every continuous map of the interval into itself meets the diagonal somewhere; this one does so at x = 0.6944.

Fixed points

  1. 1 Something always stays put
  2. 2 Nothing on a sphere can be combed flat
  3. 3 A point that pulls, and a point that pushes
  4. 4 Three colours force a triangle
  5. 5 A map that shrinks everything
  6. +7 more
12 essays · topology 3 loops in one ring, and the number that separates them. Loops drawn in an annulus, each labelled with how many times it goes round the hole. Loops with different counts cannot be deformed into one another without leaving the ring.

Homotopy

  1. 1 A loop that cannot be pulled tight
  2. 2 The group a space has at a point
  3. 3 Every cover is a subgroup
  4. 4 Cutting a space to find its group
  5. 5 Why the second group commutes
  6. +7 more
12 essays · topology Euclid's construction on 2, 3, 5, 7. The product of the listed primes plus one, divided by each of them in turn; every division leaves one over.

Infinitude of primes

  1. 1 There is no last prime
  2. 2 Infinitely many of one kind
  3. 3 Which infinitudes are proved
  4. 4 Every class, and in equal shares
  5. 5 The sieve that cannot finish
  6. +7 more
12 essays · number (p ∨ q) ∧ ¬r, drawn on the cube of 8 assignments. The assignments as corners of a cube, joined when they differ in one variable, with the satisfying corners filled.

Truth functions

  1. 1 A formula is a corner of a cube
  2. 2 One connective is enough
  3. 3 The map that puts neighbours side by side
  4. 4 A plane through the cube
  5. 5 Half the cube and √n neighbours
  6. +7 more
12 essays · logic A table of shares written as a lottery over 3 whole assignments. A doubly stochastic table of shares, and beneath it the permutation matrices and weights that add up to it exactly, each drawn as a grid with one marked cell per row.

Assignment

  1. 1 A lottery over whole assignments
  2. 2 The corners are whole assignments
  3. 3 A price for every person and task
  4. 4 One table, two lotteries
  5. 5 Where the corners stop being whole
  6. +6 more
11 essays · applied Elementary cellular automaton, rule 90. A row of cells evolving downward, each cell decided by the three above it.

Cellular automata

  1. 1 Eight rules and a triangle
  2. 2 The rule that computes
  3. 3 A road where nobody overtakes
  4. 4 No local rule can count the votes
  5. 5 A jam that comes from nowhere
  6. +6 more
11 essays · dynamics A Galton board after 600 balls. 600 balls fall through 12 rows of pegs, each bouncing left or right at random, and pile up in a bell-shaped heap.

Central limit

  1. 1 A bell curve assembled out of coin flips
  2. 2 The average settles and the wobble does not
  3. 3 The shape that averaging leaves alone
  4. 4 An average that never settles
  5. 5 How fast the bell arrives
  6. +6 more
11 essays · probability The line spiralling over the circle. A circle with a helix drawn above it: the helix is the real line, and the map that sends each of its points straight down onto the circle covers the circle once per turn. Above one marked point sits a column of points, one per turn.

Covering spaces

  1. 1 The same loop, unrolled
  2. 2 The subgroup that is freer than the group
  3. 3 The symmetries a cover has of its own
  4. 4 A covering is a permutation
  5. 5 Folding a graph until it decides
  6. +6 more
11 essays · topology Waiting for all 6 kinds. One bar per new kind: the expected number of draws needed to see a kind not yet seen, rising as fewer of them are left, and adding to 14.70 draws in total.

Expectation

  1. 1 How long until every one turns up
  2. 2 The one that hardly ever comes up
  3. 3 Two patterns, one chance, different waits
  4. 4 A coin that lets the first player win
  5. 5 Three patterns in a circle
  6. +6 more
11 essays · probability the trefoil. the trefoil, drawn as a closed curve with 3 crossings. At each crossing the strand passing underneath is broken, which is the only information the flat picture carries that the curve alone does not.

Knots

  1. 1 Three moves, and what they cannot undo
  2. 2 Colours that count more than three
  3. 3 A polynomial behind the colourings
  4. 4 The surface a knot bounds
  5. 5 A polynomial that tells left from right
  6. +6 more
11 essays · topology Nine walks, and the square root. 9 independent walks of 400 steps, each step one place left or right. The dashed curves are ±√n: the walks stay near them, spill past them, and come back — which is what a typical distance means as opposed to a limit.

Random walk

  1. 1 A walk that always comes home, until it does not
  2. 2 The path folded at its first touch
  3. 3 Half the time is the rarest answer
  4. 4 Two barriers and a fair game
  5. 5 The walk that becomes a curve
  6. +6 more
11 essays · probability The five Platonic solids. Tetrahedron, cube, octahedron, dodecahedron and icosahedron, drawn at a common scale.

Regular polyhedra

  1. 1 Why the list of perfect solids stops at five
  2. 2 Thirteen more when one word is dropped
  3. 3 The four that are allowed to cross themselves
  4. 4 Six in four dimensions, and three forever after
  5. 5 The five solids as three groups
  6. +6 more
11 essays · geometry A triangle cut into three pieces that make a rectangle. A triangle sliced at half its height and again down the altitude of the small triangle, beside the rectangle the same three pieces make when each top piece is turned a half turn.

Scissors congruence

  1. 1 Equal area is enough, and equal volume is not
  2. 2 A dissection that never comes apart
  3. 3 Slid, but never turned
  4. 4 Finitely many, and nobody says how many
  5. 5 The obstruction that was the only one
  6. +6 more
11 essays · computation Two orbits of the logistic map at 3.9, started 0.0001 apart. Two sequences from almost the same starting point, plotted together against the step number.

Sensitive dependence

  1. 1 A difference too small to draw
  2. 2 How fast two orbits part
  3. 3 Sensitivity comes free
  4. 4 Chaos on a set nobody lands on
  5. 5 A growth rate no step contains
  6. +6 more
11 essays · dynamics

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