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Geometry
- The rectangle that eats itself
- Why the list of perfect solids stops at five
- A circle unrolled into a triangle
- Round is not the only way to be the same width
- The oldest algorithm, drawn as a tiling
- Every square is a stack of odd numbers
- One cone, four curves
- Two squares, four triangles, and no algebra
- An angle that does not care where it stands
- The plane, divided by whoever is nearest
- Euclid proves it without moving anything
- The most area a fence can hold
- Nine points on one circle
- Every ray comes back to the other focus
- Three trisectors and a triangle nobody expected
- The rope that squares a corner
- Two right angles and the diagonal of a box
- The triangle that a globe gets wrong
- Circles that are diamonds and squares
- The map that trades circles for lines
- The line with only two points on it
- Thirteen more when one word is dropped
- The four that are allowed to cross themselves
- Six in four dimensions, and three forever after
- The five solids as three groups
- Eight circles touching three
- Curvatures that stay whole
- The number four points agree on
- Every flat graph is a pile of circles
- One dimension up, and the circles disappear
- Every site in the middle of its own cell
- When the sites are not the same size
- The tree inside the triangulation
- The shape described from outside
- The least area a width can hold
- The same question in space
- Aimed at one focus, turned towards the other
- Two families that cross at right angles
- One sign decides which curve
Discrete
- One sequence, counting everything
- Seven bridges, and the invention of throwing things away
- More things than boxes
- The primes on a spiral, and a pattern nobody ordered
- Pascal's triangle, in two colours
- Four colours, and a proof nobody can read
- Numbers that wrap
- Six people at a party
- A walk that changes one thing at a time
- Area by counting dots
- Two graphs that will not lie flat
- Sixteen trees on four points
- Five colours, and a chain that can be followed
- Counting the colourings
- Seven regions on a doughnut
- Three colours force a triangle
- One bottleneck and nothing else
- The edge that forces a triangle
- The bottleneck is the whole story
- The widest layer and the longest chain
- Eighteen people, and the seventeen that escape
- The colouring nobody has ever seen
- Three in a row on the number line
- The sequence that cannot avoid a staircase
- Everybody's share of the chains
- The cube cut into chains
- The largest family that always meets
- How many ways to sort it
- Counting the paths that go wrong
- One word, and four objects
- The equation a sequence satisfies
- The solid whose corners are triangulations
- The theorem that has no version in space
- A polynomial that counts
- Sixteen polygons with one dot inside
- The dots a circle catches
- The run that lands one place along
- Every entry counts the routes to it
- The carries decide the divisibility
Topology
- Every corner pays for itself
- A sphere is a plane plus one point
- Something always stays put
- The surface with one side, and what happens when it is cut
- Three moves, and what they cannot undo
- Nothing on a sphere can be combed flat
- Which side of the line is inside
- Every surface is a sphere with handles
- One line that halves them both
- A loop that cannot be pulled tight
- Colours that count more than three
- Two trees, and every edge in exactly one of them
- Seven hundred and twenty degrees of gap
- The solid where the answer is not two
- Where the fixed point escapes
- Two loops and one number
- The same loop, unrolled
- Zero can mean two different things
- The subgroup that is freer than the group
- Linked, and no two of them are
- The bottle that needs a fourth dimension
- A disc sewn to a Möbius band
- Two sheets over a one-sided surface
- Orientation is a sign
- A curve that has area
- Every loop is a circle in disguise
- Two pieces, in every dimension
- A ball whose outside is not one
- The group a space has at a point
- Every cover is a subgroup
- Cutting a space to find its group
- Why the second group commutes
- Angles survive and areas do not
- The sphere that complex numbers live on
- One chart is never enough
- The circles that fill a three-sphere
- The symmetries a cover has of its own
- A covering is a permutation
- Folding a graph until it decides
Algebra
- The dot product is a shadow
- The directions a map leaves alone
- A matrix is a picture of what happens to the grid
- Multiplying is turning
- Completing the square, by completing a square
- Eight ways to leave a square alone
- Colourings nobody can tell apart
- The number that says how much room is left
- A loop that cannot miss the middle
- The blocks a subgroup cuts out
- What a map throws away
- What the coefficients already know
- A tower whose degrees multiply
- A shared root, found without finding it
- The crossings that will not come out even
- A multiplication that remembers the order
- The group that will not come apart
- The polynomial whose roots are the stretches
- The same map in a better basis
- Symmetry forces a right angle
- What a map does to a circle
- The only function that behaves like a volume
- One point in every big enough shape
- A determinant that counts trees
- The same sum without its minus signs
- A rotation of four-space takes two of them
- The identity that multiplies sums of squares
- The integers among the quaternions
- What is lost at eight
- Two hundred and forty directions
- The only bit that survives
- The polygon an equation forces
- The lattice that runs the other way
- The group drawn as a map
- The puzzle that is exactly half solvable
- When the label may be a matrix
- The square that cannot be negative
- One subtraction clears a direction
- The nearest point of a flat thing
Probability
- Getting pi by dropping needles on the floor
- A bell curve assembled out of coin flips
- Twenty-three people
- Bayes' theorem is a picture of a square
- A walk that always comes home, until it does not
- The door that was not opened
- Nobody gets their own hat
- How long until every one turns up
- The rule that forgets where it came from
- How far from the average a thing can be
- The average settles and the wobble does not
- When to stop looking
- The path folded at its first touch
- Half the time is the rarest answer
- Two barriers and a fair game
- The walk that becomes a curve
- The shape that averaging leaves alone
- An average that never settles
- How fast the bell arrives
- The tail is not a bell
- When the whole histogram deviates
- The moment everything joins up
- The error that does not care how many dimensions
- Sampling where the answer lives
- A walk that samples a distribution
- Points too even to be random
- The chain that stops
- The chain that runs the same backwards
- The time spent and the share held
- Where the shares have nowhere to go
- How long until it forgets
- The moment a giant appears
- Two thresholds, not one
- Finding a threshold with two moments
- The window where the giant is born
- Sharp, or merely a threshold
- When the numbers are shown
- Giving up on the best
- Half of what an oracle takes
Analysis
- A sine wave is a circle seen from the side
- The curve that is its own slope
- The slope of a single point
- One point's worth of information
- A sum whose terms vanish and whose total does not
- Adding up rectangles until they stop being rectangles
- A square wave built entirely out of round ones
- The same terms, in a different order, adding to whatever is asked
- Area is the undoing of slope
- The sum that fits in one square
- The staircase that is not the diagonal
- A curve with a corner at every point
- Almost none of it left, and still uncountably many
- Where the coefficients come from
- When the period grows without bound
- The corners go first
- A map that shrinks everything
- A rectangle grown on two sides
- The flat map that fits closest
- The slope of the mirror image
- The curve of the average, and the average of the curve
- A limit that forgets to be continuous
- The area that names the number
- The equation with only one answer
- The exponential of a square
- The constant that counts what does not happen
- No interval in it, and length to spare
- Covering a set from outside
- Which functions can be added up
- A set that has no size at all
- A staircase with no steps
- A line under every point
- The function seen from its tangents
- Where the guarantee stops
- Three points, however many there are
- A wall between two bodies
- An error with an unknown in it
- The centre is a choice
- The points that ruin the fit
Number
- The primes are what is left over
- A fraction that never closes
- One way to factor, and no other
- Numbers that are their own parts
- Every triple, on one circle
- Two dials at once
- The square that cannot shrink
- The shape of a number's divisors
- Necklaces that prove a theorem
- Counting one rectangle, twice
- A diagram turned on its side
- Every fraction, exactly once
- A tree that holds every triple
- Counting what has no formula
- Always one before the double
- The sieve written as a product
- One solution that makes all the others
- Which roots refuse to be fractions
- A tail too small to be a whole number
- An integral that cannot be a whole number
- Approached too fast to be algebraic
- Every partition, hidden in a product
- The terms that cancel almost everything
- The size of a number with no formula
- Every fifth one divides
- The two supplements, and where the eight comes from
- The symbol is the sign of a shuffle
- One sum, squared two ways
- Which primes a form takes
- Infinitely many of one kind
- Which infinitudes are proved
- There is no last prime
- Two squares, and a lattice
- How close a fraction can get
- Every class, and in equal shares
- The sieve that cannot finish
- Two matrices that generate the tree
- Every rational in one sequence
- The fractions that beat every smaller one
Dynamics
- The staircase that shows the whole orbit
- A point that pulls, and a point that pushes
- The road paved with doublings
- A difference too small to draw
- How fast two orbits part
- The shape in every picture of itself
- Where Newton's method goes instead
- Eight rules and a triangle
- The rule that computes
- Three gaps and no more
- The orbit that must come back
- Two lobes and no cycle
- The question nobody can answer
- A bounce is a fold of the table
- A twist that cannot avoid two points
- The staircase that is flat almost everywhere
- A dimension that is not a whole number
- The orbit written as a word
- The same map in different coordinates
- The histogram an orbit leaves
- The orbit a computer draws
- How fast the staircase arrives
- A table folded into a surface
- A room that cannot be lit
- The obstacle that makes a table chaotic
- The triangle nobody can settle
- A constant that does not care which map
- One c, one picture
- The flow that is really a map
- A closer start buys only time
- Stretch, fold, and what is left
- Neither a surface nor a solid
- Almost every number comes down
- The heuristic that cannot be a proof
- Every pattern happens exactly once
- How short a cycle could be
- Infinite on one side and nought on the other
- A carpet with two dimensions
- A dimension from the stretching rates
Logic
- A formula is a corner of a cube
- One connective is enough
- The map that puts neighbours side by side
- Twenty-four out of two hundred and fifty-six
- Every row, or one column
- The tree that closes
- Two worlds that both obey the rules
- A list that cannot contain itself
- The sentence that says it has no proof
- Two injections make a bijection
- A sequence that explodes and still stops
- The middle that is not excluded
- Four circles cannot do it
- A game that decides what can be said
- A proof with one rule
- The axiom is the shape of the graph
- The choice nobody can write down
- A line with as many points as a square
- Countable, and everywhere
- The size that cannot be pinned down
- The arithmetic that loses subtraction
- One step in front of infinitely many
- Every ordinal in base omega
- Reached from below, or not at all
- An ordinal as a growth rate
- Two diagrams the language cannot tell apart
- Worlds built out of sentences
- The axiom with no property of the arrows
- Necessity that means provable
- How many worlds a formula can need
- Nearly always, or nearly never
- The distance a sentence can see
- The row that is not on the list
- An infinite tree has an infinite path
- The game the algorithm was playing
- A language that can name a set
- The assumption a proof pays back
- A lemma, and the proof that never mentions one
- The instance that has to be guessed
Computation
- What two points can build
- Every step is a square root
- The cube that will not double
- The angle that will not divide by three
- Which polygons can be drawn
- The circle that will not square
- Distance is a picture
- Sixteen spheres that fill a cube
- Finding the error without reading the message
- A polynomial through the gaps
- The field with four elements
- Every element is a power of one of them
- Seven points, seven lines
- A schedule where every pair meets once
- The thirty-six officers
- Every word once, around a cycle
- Equal area is enough, and equal volume is not
- The straightedge buys nothing
- The planes a recurrence cannot leave
- The mark that changes what is reachable
- A field's worth of squares
- The plane hiding in the squares
- Nine thousand four hundred and eight
- Sixteen of five hundred and seventy-six
- A memory of four bits
- A page that knows where it is
- A cycle for every pair
- Every necklace, in order
- The test that ranks the generators
- Four numbers and the rule is yours
- Randomness that has to be earned
- Nineteen thousand bits of state
- One circle, and a straightedge
- The compass that will not open
- The best a code can be
- Past half the distance
- A dissection that never comes apart
- Slid, but never turned
- Finitely many, and nobody says how many
Applied
- One cuts and the other chooses
- Envy-free, up to one item
- Nobody has a reason to run away
- The side that proposes wins
- The seat that vanishes when the house grows
- Two numbers that have to meet
- No stable rule is safe from a lie
- Three people and a trimmed piece
- Four conditions, and no rule that has all of them
- The road that makes everyone later
- The majority that goes in a circle
- A lie that pays
- Five rules and five winners
- The value from both sides
- A lottery over whole assignments
- The order everybody arrives in
- A split nobody can walk away from
- The court that contradicts itself
- The objection nobody can make louder
- None of the four conditions is spare
- A share of the votes is not a share of the power
- Too many orders to list
- Sharing a cost that is not the sum of its parts
- Five rules and one dial
- The rule with no favourites
- Two out of three, and never all three
- Choosing what unfair means
- Seats to parties and places at once
- The corners are whole assignments
- A price for every person and task
- One table, two lotteries
- Where the corners stop being whole
- No rule escapes the doctrinal paradox
- Deciding the premises or the conclusion
- What a constraint is worth
- Four ways out, and what each costs
- A signal both can see
- The landscape nobody is looking at
- Two equilibria and no way to choose