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• A ball whose outside is not one
• A count that can say zero
• A covering is a permutation
• A curve that has area
• A disc sewn to a Möbius band
• A loop that cannot be pulled tight
• A map that offers a choice
• A polynomial behind the colourings
• Angles survive and areas do not
• As many cuts as colours
• Colours that count more than three
• Covering a surface multiplies its count
• Cutting a space to find its group
• Every corner pays for itself
• Every cover is a subgroup
• Every loop is a circle in disguise
• Every surface is a sphere with handles
• Every way to pair a polygon's edges
• Every word driven to a normal form
• Folding a graph until it decides
• Linked, and no two of them are
• Nothing on a sphere can be combed flat
• One chart is never enough
• One line that halves them both
• The surface with one side, and what happens when it is cut
• Opposite labels that have to meet
• Orientation is a sign
• Seven hundred and twenty degrees of gap
• Something always stays put
• A sphere is a plane plus one point
• The bottle that needs a fourth dimension
• The circles that fill a three-sphere
• The colours a circle forces
• The group a space has at a point
• The same loop, unrolled
• The solid where the answer is not two
• The sphere that complex numbers live on
• The subgroup that is freer than the group
• The surface a knot bounds
• The surface a random gluing makes
• The symmetries a cover has of its own
• The third number a surface needs
• Three moves, and what they cannot undo
• Twelve pentagons, whatever the hexagons
• Two loops and one number
• Two opposite points that agree twice
• Two pieces, in every dimension
• Two sheets over a one-sided surface
• Two trees, and every edge in exactly one of them
• What a branch point subtracts
• Where the fixed point escapes
• Which side of the line is inside
• Why the second group commutes
• Zero can mean two different things
• Zero in four dimensions
• The loops on a torus that never cross themselves
• A twist that carries one loop to another

• A bounce is a fold of the table
• A carpet with two dimensions
• A closer start buys only time
• A constant that does not care which map
• A difference too small to draw
• A dimension for every rate of crowding
• A dimension from the stretching rates
• A dimension that is not a whole number
• A matrix that counts the returns
• A point that pulls, and a point that pushes
• A road where nobody overtakes
• A room that cannot be lit
• A rotation in different coordinates
• A solvable chaos of every degree
• A table folded into a surface
• A twist that cannot avoid two points
• Almost every number comes down
• Almost every orbit is fair
• An area that never finishes
• Chaos on a set nobody lands on
• Covering rather than avoiding
• Eight rules and a triangle
• Every pattern happens exactly once
• How a lock comes apart
• How fast the staircase arrives
• How fast two orbits part
• How short a cycle could be
• Infinite on one side and nought on the other
• Neither a surface nor a solid
• No local rule can count the votes
• One c, one picture
• Sensitivity comes free
• Stretch, fold, and what is left
• The dark lines are one point's orbit
• The flow that is really a map
• The folds that measure chaos
• The heuristic that cannot be a proof
• The histogram an orbit leaves
• The obstacle that makes a table chaotic
• The orbit a computer draws
• The orbit that must come back
• The orbit written as a word
• The question nobody can answer
• The road paved with doublings
• The room a jagged graph takes up
• The rule that computes
• The same map in different coordinates
• The shape in every picture of itself
• The staircase that is flat almost everywhere
• The staircase that shows the whole orbit
• The triangle nobody can settle
• The window that opens with a stutter
• Three gaps and no more
• Two lobes and no cycle
• Where Newton's method goes instead
• A double root halves the error instead of squaring it
• A cubic method that is Newton's in disguise

• A centre is three weights
• A circle unrolled into a triangle
• Aimed at one focus, turned towards the other
• An angle that does not care where it stands
• Circles that are diamonds and squares
• Curvatures that stay whole
• Eight circles touching three
• Eighteen equilateral triangles
• Equal diagonals in a curved quadrilateral
• The oldest algorithm, drawn as a tiling
• Euclid proves it without moving anything
• Every flat graph is a pile of circles
• Every ray comes back to the other focus
• Every side measured by one diameter
• Every site in the middle of its own cell
• Half a circle against a wall
• The plane, divided by whoever is nearest
• Nearly the most means nearly round
• Nine points on one circle
• Every square is a stack of odd numbers
• One circle touching four
• One cone, four curves
• One dimension up, and the circles disappear
• One number for every chord through a point
• One sign decides which curve
• Pinned between two sequences
• Two squares, four triangles, and no algebra
• Round is not the only way to be the same width
• Seven pieces and an equilateral middle
• Six in four dimensions, and three forever after
• Six points on a conic, and the line they share
• Sums of powers, read off a staircase
• The fewest ordinary lines a polygon allows
• The five solids as three groups
• The four that are allowed to cross themselves
• The least area a width can hold
• The line with only two points on it
• The map that trades circles for lines
• The most area a fence can hold
• The number four points agree on
• The rectangle that eats itself
• The rope that squares a corner
• The same question in space
• The shape described from outside
• The slice that has to match
• The tree inside the triangulation
• The triangle that a globe gets wrong
• Thirteen more when one word is dropped
• Three ordinary lines from a count
• Three triangular numbers, and no fewer
• Three trisectors and a triangle nobody expected
• Two families that cross at right angles
• Two right angles and the diagonal of a box
• When the sites are not the same size
• Why the list of perfect solids stops at five
• The player who meets the first long run
• The remainders that count the roots

• A coin in front of every term
• A curve with a corner at every point
• A denominator that reaches past the radius
• A limit can jump at every fraction
• A limit that forgets to be continuous
• A line under every point
• A map that shrinks everything
• A rectangle cut by a curve
• A rectangle grown on two sides
• A series that converges nowhere
• A set that has no size at all
• A staircase with no steps
• A sum read from inside
• A sum whose terms vanish and whose total does not
• A wall between two bodies
• Adding up rectangles until they stop being rectangles
• Almost none of it left, and still uncountably many
• An endless region with a finite area
• An error with an unknown in it
• Area is the undoing of slope
• A sine wave is a circle seen from the side
• Covering a set from outside
• No interval in it, and length to spare
• One point's worth of information
• The same terms, in a different order, adding to whatever is asked
• A square wave built entirely out of round ones
• The angle that is really an area
• The area that names the number
• The centre is a choice
• The constant that counts what does not happen
• The corners go first
• The curve of the average, and the average of the curve
• The curve that is its own slope
• The equation with only one answer
• The exponential of a square
• The flat map that fits closest
• The function seen from its tangents
• The length belongs to the journey
• The points that ruin the fit
• The repair at the boundary
• The series everything else is measured against
• The slope of a single point
• The slope of the mirror image
• The staircase that is not the diagonal
• The subsequence that has to exist
• The sum that fits in one square
• The sum that steps over every whole number
• Three points, however many there are
• Uniform, except on a small set
• When the period grows without bound
• When two circular motions come home
• Where the coefficients come from
• Where the guarantee stops
• Which curves have a length at all
• Which functions can be added up
• The length the derivative never sees
• A length counted by the lines that cross it

• A collision that finds a factor
• A room where nobody is alone
• A round table with no couple together
• A sum stopped early still says something
• A walk that always comes home, until it does not
• A walk that samples a distribution
• Add the odds from the end
• An average that never settles
• Any unevenness brings the match sooner
• Bayes' theorem is a picture of a square
• A bell curve assembled out of coin flips
• Finding a threshold with two moments
• Fourteen people within a day
• Giving up on the best
• Half of what an oracle takes
• Half the time is the rarest answer
• How far from the average a thing can be
• How fast the bell arrives
• How long until every one turns up
• How long until it forgets
• How many get their own hat
• No single input can move it far
• Nobody gets their own hat
• One coin, counted by runs and by wakings
• Getting pi by dropping needles on the floor
• Points too even to be random
• Sampling where the answer lives
• Sharp, or merely a threshold
• The average settles and the wobble does not
• The bound is the answer to a search
• The cells a permutation must miss
• The chain that runs the same backwards
• The chain that stops
• The door that was not opened
• The error that does not care how many dimensions
• The moment a giant appears
• The moment everything joins up
• The one that hardly ever comes up
• The path folded at its first touch
• The rule that forgets where it came from
• The shape that averaging leaves alone
• The tail is not a bell
• The thresholds that nest
• The time spent and the share held
• The walk that becomes a curve
• The window where the giant is born
• Twenty-three people
• Two barriers and a fair game
• Two children and the sentence about one of them
• Two patterns, one chance, different waits
• Two thresholds, not one
• When the numbers are shown
• When the whole histogram deviates
• When to stop looking
• Where the shares have nowhere to go
• The ground a walk covers
• A walk that may not step where it has been

• A countable field that passes for the line
• A failed search is a proof
• A formula is a corner of a cube
• A function that adds and is nowhere a line
• A game that decides what can be said
• A language that can name a set
• A lemma, and the proof that never mentions one
• A line with as many points as a square
• A list that cannot contain itself
• A number larger than every number
• A plane through the cube
• A proof with one rule
• A quantifier is a shadow
• A sequence that explodes and still stops
• An infinite tree has an infinite path
• An ordinal as a growth rate
• Countable, and everywhere
• Every derivation is a term
• Every ordinal in base omega
• Every row, or one column
• Four circles cannot do it
• Half the cube and √n neighbours
• How many worlds a formula can need
• Infinitely many guessers, finitely many wrong
• Nearly always, or nearly never
• Necessity that means provable
• Not one step but a continuum
• One connective is enough
• One gadget defeats every refinement
• One step in front of infinitely many
• Reached from below, or not at all
• Refutable in something small
• Six sentences from two quantifiers
• The arithmetic that loses subtraction
• The assumption a proof pays back
• The axiom is the shape of the graph
• The axiom with no property of the arrows
• The boundary at three variables
• The choice nobody can write down
• The distance a sentence can see
• The game the algorithm was playing
• The instance that has to be guessed
• The map that puts neighbours side by side
• The middle that is not excluded
• The row that is not on the list
• The sentence between a premise and its consequence
• The sentence that says it has no proof
• The size that cannot be pinned down
• The tree that closes
• Twenty-four out of two hundred and fifty-six
• Two diagrams the language cannot tell apart
• Two injections make a bijection
• Two literals make an arrow
• Two worlds that both obey the rules
• Worlds built out of sentences
• The conclusion is what survives the erasing
• One thing in each region is enough

• A cycle for every pair
• A dissection that never comes apart
• A field's worth of squares
• A filter that changes only the spread
• A memory of four bits
• A page that knows where it is
• A plane in a list of numbers
• A plane no field built
• A polynomial through the gaps
• A quintic a sliding mark reaches
• A schedule where every pair meets once
• Distance is a picture
• Equal area is enough, and equal volume is not
• Equal area on a sphere, without a rectangle
• Erasures a code can see
• Every element is a power of one of them
• Every necklace, in order
• Every step is a square root
• Every word once, around a cycle
• Fair bits from an unfair coin
• Finding the error without reading the message
• Finitely many, and nobody says how many
• Four numbers and the rule is yours
• Give or take twice the square root
• More blocks than points
• Nine thousand four hundred and eight
• Nineteen thousand bits of state
• One cell short of a transversal
• One circle, and a straightedge
• Past half the distance
• Randomness that has to be earned
• Seven points, seven lines
• Sixteen of five hundred and seventy-six
• Sixteen spheres that fill a cube
• Slid, but never turned
• Solutions that come in multiples of p
• The angle that will not divide by three
• The best a code can be
• The circle that will not square
• The compass that will not open
• The cube that will not double
• The curve that no three points in line define
• The field with four elements
• The mark that changes what is reachable
• The obstruction that was the only one
• The plane hiding in the squares
• The planes a recurrence cannot leave
• The price of a construction
• The rate a noisy channel allows
• The straightedge buys nothing
• The test that ranks the generators
• The thirty-six officers
• Two instruments with one reach
• What two points can build
• Which polygons can be drawn
• A sum of two sets modulo a prime cannot be small
• No set with a line in every direction is small

• A determinant that counts trees
• A loop that cannot miss the middle
• A multiplication that remembers the order
• A plane disguised as an arrow
• A rotation of four-space takes two of them
• A shared root, found without finding it
• A tower whose degrees multiply
• Colourings nobody can tell apart
• Counted across and counted down
• Eight ways to leave a square alone
• Every power sum, from the coefficients alone
• Every third coefficient
• How fast the ball fills
• Completing the square, by completing a square
• A matrix is a picture of what happens to the grid
• Moving a map across a product
• Multiplying is turning
• Necklaces made of symmetries
• On the circle and never home
• One number under every bell
• One point in every big enough shape
• One subtraction clears a direction
• Seven powers in a space of six
• Symmetry forces a right angle
• The blocks a subgroup cuts out
• The crossings that will not come out even
• The cycles and the cuts
• The directions a map leaves alone
• The dot product is a shadow
• The edge that is as big as the ball
• The group drawn as a map
• The group that will not come apart
• The identity that multiplies sums of squares
• The integers a field contains
• The integers among the quaternions
• The lattice that runs the other way
• The nearest point of a flat thing
• The number that says how much room is left
• The only bit that survives
• The only function that behaves like a volume
• The polygon an equation forces
• The polynomial whose roots are the stretches
• The puzzle that is exactly half solvable
• The roots of the slope stay inside
• The same map in a better basis
• The same sum without its minus signs
• The square that cannot be negative
• Twenty-four ways to set a cube down
• Two hundred and forty directions
• What a map does to a circle
• What a map throws away
• What is lost at eight
• What the coefficients already know
• When the label may be a matrix
• Where two roots run into each other
• The sums of roots of unity that add to nothing
• Three ways to pair four roots

• A diagram turned on its side
• A factorisation that hides its primes
• A fraction that never closes
• A method that is allowed to miss
• A tail too small to be a whole number
• A tree that holds every triple
• Almost no number is one
• Always one before the double
• An integral that cannot be a whole number
• An order that proves a prime
• Approached too fast to be algebraic
• Counting one rectangle, twice
• Counting what has no formula
• The primes are what is left over
• Every class, and in equal shares
• Every fifth one divides
• Every fraction, exactly once
• Every partition, hidden in a product
• Every rational in one sequence
• Every triple, on one circle
• Factoring uniquely with no way to divide
• How close a fraction can get
• Infinitely many of one kind
• Necklaces that prove a theorem
• There is no last prime
• Numbers that are their own parts
• One residue whose powers are all of them
• One solution that makes all the others
• One sum, squared two ways
• One way to factor, and no other
• The arcs a line crosses on its way to a number
• The exponent that is smaller than Euler's
• The fraction Lambert built for the tangent
• The fractions that beat every smaller one
• The function that sends fractions to binary
• The pattern in e's continued fraction
• The shape of a number's divisors
• The sieve that cannot finish
• The sieve written as a product
• The size of a number with no formula
• The square that cannot shrink
• The sum of the parts, taken again
• The symbol is the sign of a shuffle
• The terms that cancel almost everything
• The two squares actually produced
• The two supplements, and where the eight comes from
• Two dials at once
• Two matrices that generate the tree
• Two primes where Fermat holds twice
• Two squares, and a lattice
• Which infinitudes are proved
• Which primes a form takes
• Which roots refuse to be fractions
• Why a quarter of numbers overshoot
• Why the expansion has to repeat
• Two families of solutions, and a box that holds both
• Sixty needs two digits and sixty-one needs ten

• A lie that pays
• A lottery over whole assignments
• A mixture that is a population
• A price for every person and task
• A ring that no pairing can break
• A share of the votes is not a share of the power
• A signal both can see
• A split nobody can walk away from
• Agendas that cannot contradict themselves
• An objection one player makes to another
• Choosing what unfair means
• Deciding the premises or the conclusion
• Envy-free, up to one item
• Five rules and five winners
• Five rules and one dial
• Five weighings and the question is closed
• Four conditions, and no rule that has all of them
• Four ways out, and what each costs
• How many cuts a fair share costs
• No rule escapes the doctrinal paradox
• No stable rule is safe from a lie
• Nobody has a reason to run away
• None of the four conditions is spare
• One cuts and the other chooses
• One table, two lotteries
• Patience instead of a contract
• Seats to parties and places at once
• Sharing a cost that is not the sum of its parts
• The corners are whole assignments
• The court that contradicts itself
• The landscape nobody is looking at
• The lines the optimum lies under
• The majority that goes in a circle
• The nearest consistent verdict
• The objection nobody can make louder
• The order everybody arrives in
• The people every stable answer leaves out
• The product that makes a division fair
• The reading that is almost right
• The road that makes everyone later
• The rule with no favourites
• The seat that vanishes when the house grows
• The side that proposes wins
• The table inside every quota
• The value from both sides
• Three people and a trimmed piece
• Too many orders to list
• Two equilibria and no way to choose
• Two numbers that have to meet
• Two out of three, and never all three
• What a constraint is worth
• When one of the two numbers is missing
• Where the corners stop being whole
• Where the rounding runs out
• Worth more for being seen first
• Cutting a link costs both of its ends the same
• What a missing input is worth

• A polynomial that counts
• A remainder read two digits at a time
• A walk that changes one thing at a time
• A walk that splices in its own detours
• Area by counting dots
• Averaging down the triangle
• Counting the colourings
• Counting the paths that go wrong
• Eighteen people, and the seventeen that escape
• Every entry counts the routes to it
• Every place changes back
• Everybody's share of the chains
• Five colours, and a chain that can be followed
• Five spokes squeezed into K5
• Four colours, and a proof nobody can read
• How many ways to sort it
• More things than boxes
• Moves that only ever add edges
• Numbers that wrap
• One bottleneck and nothing else
• One sequence, counting everything
• One tree for every cut
• One word, and four objects
• Pascal's triangle, in two colours
• The primes on a spiral, and a pattern nobody ordered
• Seven bridges, and the invention of throwing things away
• Seven regions on a doughnut
• Six people at a party
• Sixteen polygons with one dot inside
• Sixteen trees on four points
• The bottleneck is the whole story
• The carries decide the divisibility
• The coefficient that is a polynomial
• The colouring nobody has ever seen
• The cube cut into chains
• The densest graph without a square
• The dots a circle catches
• The edge that forces a triangle
• The equation a sequence satisfies
• The few points that cut a flat graph
• The largest family that always meets
• The piece that cannot pair off
• The product that deals the labels
• The run that lands one place along
• The sequence that cannot avoid a staircase
• The solid whose corners are triangulations
• The streets a postman walks twice
• The theorem that has no version in space
• The walk through the middle levels
• The widest layer and the longest chain
• Three colours force a triangle
• Three in a row on the number line
• Two graphs that will not lie flat
• What the search has when it fails
• When several pairs share the roads
• Cars that park, and trees that grow
• A random tree is one part in e leaves