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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. +4 more
9 essays · applied Hamilton's method on 27 seats and 5 regions. A worksheet of populations, exact quotas, floors, remainders and the seats Hamilton's method awards to 5 regions.

Apportionment

  1. 1 The seat that vanishes when the house grows
  2. 2 Five rules and one dial
  3. 3 The rule with no favourites
  4. 4 Two out of three, and never all three
  5. 5 Choosing what unfair means
  6. +3 more
8 essays · applied 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. +3 more
8 essays · topology 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. +3 more
8 essays · computation 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. +3 more
8 essays · topology The Pythagorean theorem by dissection. Two squares of the same size. Each holds four copies of one right triangle. The space left over is a single tilted square on the left and two upright squares on the right.

Pythagoras

  1. 1 Two squares, four triangles, and no algebra
  2. 2 Euclid proves it without moving anything
  3. 3 Every triple, on one circle
  4. 4 The rope that squares a corner
  5. 5 Two right angles and the diagonal of a box
  6. +3 more
8 essays · geometry 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. +2 more
7 essays · probability Four conic sections from one cone. Circle, ellipse, parabola and hyperbola, produced by tilting a single cutting plane further and further.

Conic sections

  1. 1 One cone, four curves
  2. 2 Every ray comes back to the other focus
  3. 3 Aimed at one focus, turned towards the other
  4. 4 Two families that cross at right angles
  5. 5 One sign decides which curve
  6. +2 more
7 essays · geometry 3 rounds on chains of 4 and 5. Two chains of dots with pebbles placed in turn, and the transcript of a play: Spoiler picks an element of one chain, Duplicator answers in the other, and the pebbles must keep the same order.

Ehrenfeucht–Fraïssé games

  1. 1 A game that decides what can be said
  2. 2 Nearly always, or nearly never
  3. 3 The distance a sentence can see
  4. 4 The game the algorithm was playing
  5. 5 A language that can name a set
  6. +2 more
7 essays · logic Two squares of side 12 inside one of side 17. Two overlapping squares laid into opposite corners of a larger one, with the overlap and the two uncovered corners marked.

Irrationality

  1. 1 The square that cannot shrink
  2. 2 Which roots refuse to be fractions
  3. 3 A tail too small to be a whole number
  4. 4 An integral that cannot be a whole number
  5. 5 Approached too fast to be algebraic
  6. +2 more
7 essays · number the logistic map at 3.2, iterated from 0.2. A map drawn as a curve with the diagonal across it, and the staircase that iterating it produces.

Iteration

  1. 1 The staircase that shows the whole orbit
  2. 2 How fast the staircase arrives
  3. 3 The same map in different coordinates
  4. 4 The histogram an orbit leaves
  5. 5 The orbit a computer draws
  6. +2 more
7 essays · dynamics 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. +2 more
7 essays · computation Two points, and everything one round of compass and straightedge adds. Two starting points with the line and circles they permit, and the four points where those objects cross.

Constructible numbers

  1. 1 What two points can build
  2. 2 Every step is a square root
  3. 3 The cube that will not double
  4. 4 The angle that will not divide by three
  5. 5 Which polygons can be drawn
  6. +1 more
6 essays · computation A chord of x², and the curve under it. The curve x² with one chord drawn across it, the region between them shaded, and the midpoint heights of both marked. The comparison is computed at four hundred sample points.

Convexity

  1. 1 The curve of the average, and the average of the curve
  2. 2 A line under every point
  3. 3 The function seen from its tangents
  4. 4 Where the guarantee stops
  5. 5 Three points, however many there are
  6. +1 more
6 essays · analysis 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. +1 more
6 essays · topology The Fano plane, and the incidence table behind it. Seven points joined by six straight lines and one circle, beside the seven-by-seven table of which point lies on which line.

Finite geometry

  1. 1 Seven points, seven lines
  2. 2 A schedule where every pair meets once
  3. 3 More blocks than points
  4. 4 A plane in a list of numbers
  5. 5 A plane no field built
  6. +1 more
6 essays · computation the Koch curve, after 5 steps. the Koch curve drawn from its own rule: replace the middle third of every segment with two sides of a triangle.

Fractal dimension

  1. 1 A dimension that is not a whole number
  2. 2 Infinite on one side and nought on the other
  3. 3 A carpet with two dimensions
  4. 4 A dimension from the stretching rates
  5. 5 A dimension for every rate of crowding
  6. +1 more
6 essays · dynamics The dot product as a shadow. Two vectors and the shadow the first casts on the second. The shadow is 2.425 long and b is 4.123, so the dot product is 10.000.

Inner product

  1. 1 The dot product is a shadow
  2. 2 The square that cannot be negative
  3. 3 One subtraction clears a direction
  4. 4 The nearest point of a flat thing
  5. 5 Moving a map across a product
  6. +1 more
6 essays · algebra 3 consistent judges, and a majority that is not. A table of judges against three questions, every judge's row internally consistent, with the majority answer to each question underneath forming a combination no judge holds.

Judgement aggregation

  1. 1 The court that contradicts itself
  2. 2 No rule escapes the doctrinal paradox
  3. 3 Deciding the premises or the conclusion
  4. 4 Four ways out, and what each costs
  5. 5 The nearest consistent verdict
  6. +1 more
6 essays · applied Transversals of the cyclic square of order 6. A cyclic Latin square with a transversal marked if it has one, beside a count of transversals at neighbouring orders.

Latin squares

  1. 1 The thirty-six officers
  2. 2 A field's worth of squares
  3. 3 The plane hiding in the squares
  4. 4 Nine thousand four hundred and eight
  5. 5 Sixteen of five hundred and seventy-six
  6. +1 more
6 essays · computation

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