Find
Search
Titles, summaries, fields, series and the named objects of every essay, searched as you type.
Limits
- An outcome with no value behind it
- The closure that picks the nimbers
- What the class does not buy
- A function with no formula
- A product against a sum
- The wild side does not close
- Half a licence is nearly all of it
- One half multiplies, the other adds
- Wrong in one direction only
- A move that must be answered
- Start at the end and work backwards
- The clause that turns the class off
- The licence that weighs nothing
- A pass is not a move
- Equal in this company
- It ends, and nothing says when
- Loopy games
- Misère play has no negatives
- Misère play
- What survives misère play
- Nobody comes back
- Nobody has to move
- One part that never ends
- Tame and wild
- The company that is closed
- The condition the recursion rests on
- The genus of a sum
- The one outcome that adds
- The rule the symbols follow
- Three players and no answer
- Two misère outcomes are not enough
- Two ways to end with no bound
- What a component has to carry
- What a tame heap may be replaced by
- What a wider pool rescues
- What restores the theorem
- When never ending is a win
- The price of asking what the parts are
- A quotient that identifies nothing
- A stopper and how to find one
- Which games end at which level
- Two clauses and a third question
- Closing the wild side
- The auction never gets to the money
- A coin needs no tie-break
- Left always wins, and loses more often than not
- Every chance but a certainty
- The coldest position has the biggest swing
- The best chance is the wrong move
- Three heaps and a pass
- What a component would have to carry
Positions
- Maundy Cake
- How long a row a value needs
- Which shapes are worth fighting over
- The reading that survives too much
- Where the nimbers run out
- Counting the moves each side has
- The rows that are their own mirror
- How thick a wall has to be
- The moves a player can be talked out of
- A game with nothing at stake
- The criterion that cannot exist
- Two errors that cancel
- The size of a cake
- A numeral in the empty squares
- A region one player owns
- Half the difference in odd runs
- The cliff a cut invents
- Cut small unless you are behind
- The fractions that were not there
- The short side only says how many
- Read from the back forwards
- One domino every three cells
- Room pulls two ways
- Taking from the ends
- Squash every loop to a point
- Topple it from either end
- A sequence with a rule and no period
- The short side is not in the lemma
- Two strips that end the same way
- Where the runs meet
- An effect that changes sign
- A board that is a sum of its regions
- A green edge on a blue one
- A tree is still a number
- Amazons on one line
- Amazons, and when a position becomes a sum
- A game where nobody can be ahead in moves
- One board, two rules
- Cutcake, where every value is a whole number
- Domineering
- Every group must keep breathing
- Hackenbush is a numeral
- Nothing worth fighting over
- One row of Clobber
- One rule makes it cold, the other hot
- Sprouts, and the game that is not one
- The condition that survived the wider sweep
- The other way to move a row
- The same strip without the jump
- The strip nobody has a formula for
- The strip where every number is a whole one
- The values of every small board
- Toads and Frogs
- Two players, two lists
- When the bracket decides
- When the regions add
- The obvious cut is the wrong one
- A fraction does not reach
- Three distances too many
- The square that cannot be halved
- Where the numeral stops
- The third point on the curve
- A wall that bends
- A recipe instead of a census
Values
- How much a list of options can lose
- Fifty-two errors and seven sizes
- The cheapest way to show a value
- Which option the reduction keeps
- The birthday is a floor
- The bend is the condition
- How many moves are worth making
- The margin a count needs
- Half a follow-up out
- Wider costs less
- What a strategy has to remember
- The threshold was a fact about the census
- A second level of stops
- The entry fee was the cap
- Three rules and a tie-break
- The bend is in the stops
- The rate was the alphabet
- A threshold is a detection limit
- Seventy-two of them were not silence
- A bend that never reaches the surface
- The mirror was the floor
- A heuristic that becomes a theorem
- The price of taking the maximum
- A fight with no midpoint
- The class where nobody runs out first
- An option nobody would take
- Canonical form
- How old a value is
- How wide a form can get
- Infinitesimals
- Two hundred and fifty-six ways to write twenty-two things
- Nobody wants to move here
- Numbers avoid numbers
- Worth nothing, and worth fighting for
- The birthday of a sum
- The fight never runs backwards
- The day a number is born
- The numbers it is confused with
- The recursion this site cannot run
- The reduction that always shrinks
- The reduction that puts options back
- The same position, written once
- The simplest game above both
- The simplicity rule
- The switch a player is imagining
- The values nobody's game produces
- The weight that blunts the count
- Tiny, miny, and the sizes below every size
- What a move is worth to the player making it
- What a value costs to write down
- What a value leaves out
- When a switch is not a switch
- Where the fight stops
- Where the order and the sum disagree
- Which end of the interval is open
- Who moves last
- The easy case was not the reason
- Two measures bounded, and one not
- A factor, and not an overhead
- Close calls nothing resolves
- Three groups, and three yields
- The bend above the top
- A reduction that reads a graph
- One of four questions
Sums
- The number nobody needs
- A compound of two different games
- The thirty that cancel themselves
- How wrong a nearly-independent split is
- A self-negative value costs a day
- What a fight does to a fight
- A bound with one number too many
- At least five hundred and seventy-one
- Which end a sum lands at
- The option nothing names
- What identifies two subsets
- The same number in two currencies
- A mex with no impartial game in it
- Not a domination, in that order
- Where the value stops mattering
- The case that was supposed to be hard
- A side about to lose its move
- Add, then reduce again
- How many ups
- Comparing positions
- Comparing two positions means playing a third
- Confused is not the same as unknown
- Equal in every company
- Turn the board through a right angle
- How hot a background has to be
- How long it lasts
- How rare it is to be bigger
- Independence is a claim
- No fifth value
- Three ways to add the same games
- Outcomes do not add
- The other sum, the one that nests
- The sum is the object
- The values that are their own negatives
- What a number does to a fight
- What an infinitesimal does to a fight
- What can be struck out
- What is left when the small change is thrown away
- What the colon respects
- When the nested sum only sees the value
- When the ups add
- Where the impartial theory stops
- Which part to move in
- One expression proved, and one withdrawn
- A cross in the table
- The proof needs both reductions
- A floor, and not a decline
- One number, stated two ways
- Twenty-six other values
- The follower does the reversing
- The split slips one day deeper
- Twenty draws and a second recipe
Impartial
- The family the Fibonacci numbers belong to
- What the numerals knew
- The rule a smaller move breaks
- The wider move is the easier game
- The third digit
- A check in front of a search
- The count of odd heaps
- The only way to split into three
- The check that was not a check
- The parities, in size order
- A pairing, and the pairing
- Two counters, and one displaced term
- The parameter was the difference
- A symmetry that is not a pairing
- A pattern that has not started yet
- A period with a constant added
- A token on a graph
- Cram
- The move that gives counters back
- Grundy sequences, and where they stop being predictable
- Looking for the symmetry
- Nim, and the nim-sum
- No two heaps alike
- Naming a game with a number
- Splitting is a move
- Every impartial game is a Nim heap
- Take one, three or four
- Taking from several heaps at once
- The code names the move
- The heap is not the position
- The losing positions are a code
- The nimbers multiply
- The patch that generalised
- The period is small and the proof does not say so
- The strategy that is a symmetry
- The symmetry one move away
- The tartan theorem
- The values that keep arriving
- A row of coins is already a sum
- Wythoff's game, and the ratio nobody put there
- The family with two witnesses
- The dual was the value table
- One proof, and one wrong lemma
- The pairing the formula hides
- Four hundred and seventy steps
- The quantity that carried nothing
- A pairing that is not a symmetry
- The proof is sixteen cells
- One split is enough
- The formula is a limit
- A code that climbs by three
Complexity
- How often a board falls apart
- When the catalogue starts paying
- Where to stop building
- A catalogue that knows what it will meet
- The catalogue a strong player needs
- A catalogue that builds itself
- A rule that is never right and cannot be far wrong
- Four values, and the sequence is settled for ever
- "Left wins" has no short proof
- A puzzle asks once, a game asks alternately
- Finding the parts
- Nim is easy, in binary
- How hard is it
- Knowing who wins, and knowing what it is worth
- The game with the shortest rule is the hard one
- Hard, proved
- The class is named after memory, and that is not an accident
- The board falls apart, and the arithmetic changes
- The question in the middle
- A position reached eleven ways is one position
- What counts as the same position, and what that is worth
- The cost is in the closure, not in the positions
- Three different claims are all called solved
- The table that changes its mind
- A wall an amazon can walk through
- What it costs to notice a repetition
- A count that forgets
- One board, and recency still wins
- The order a solver tries the moves in
- A verdict that changes with the depth
- A key shorter than the position
- Where a search may stop
- A check bit halves the average and not the key
- Using up the edges instead
- Two graphs a rule cannot tell apart
- The bound names the hottest part and the cost does not
- The cheap fights make the rule cheaper
- A key is a code, and two squares come free
- Search on in pairs of moves
- Twelve turns, and three different prices
- Eleven moves and one decision
- Proving a loss means answering everything
- The opponent stops choosing
- A turn is not a bit
- A misère sum is searched, not added
- Two heaps of testing are enough
- Twelve classes, seven questions
- A staircase, not a slope
Temperature
- A schedule instead of a number
- What the halving is a function of
- When to leave the environment
- How big the answer is
- What a game actually produces
- One fight makes a board a fight
- A rule that beats the hottest
- The two numbers at the top
- The worst value in its own interval
- Eight squares, and no hotter
- The premises an induction would need
- A pool built to have an answer
- Which top is the top
- A number and a fight
- A pool built to punish greed
- A rule with a guarantee
- A rule with no promise at all
- A subtraction, not a factor
- A thermograph is built from its options'
- A thermograph with two bends
- An environment made of coupons
- Below zero
- Big is not the same as hot
- Cooling adds and heating does not
- Cooling
- Cooling by exactly one
- Double sente is not a property of the position
- Half of the smaller temperature
- How cold a sum of hot games can be
- How hot a day gets
- How hot a real position is
- The same fight, eight times over
- Playing the hottest
- Reading a thermograph
- Sente is a fact about the rest of the board
- Two hot fights that add to a cold number
- The answer that starts another fight
- The ceiling was a plateau
- The endgame, accounted for
- The obstacle was the catalogue
- The operator chosen for one game
- The operator that puts the star back
- The quantity that does not order a board
- Two games in one environment
- What a move nobody makes is worth
- What is at stake
- What is left when the copies pair off
- When two thermographs can be added
- A description, and not a detector
- The four paragraphs prove something else
- A second pool, designed differently
- The residues as a sequence
- An environment instead of a stack
- The second bend is the boundary
History
- A chess problem that turned out to be an octal game
- A conjecture from hand play
- A golden ratio thirty years early
- "Hopeless" was a claim about a method
- The first theorem, and the winner it declines to name
- The first time it told somebody something
- The notation was the argument
- The numbers came out of the game
- The sequence nobody has settled
- The theorem that needed none of the theory
- Two people, four years apart, one theorem
- What computing further has bought
- Where the braces stop
- Three bits of rule
- The paper was about how long
- The gap between two answers
- Every play ends and no round settles
- What the arithmetic cost in 1956
- Three complete solutions in nine years
- A set with a short description
- The sentence that solved the other convention
- Two names that add to nothing nameable
- What two numbers cannot tell apart
- A board is written as a sum
- What the play keeps coming back to
- One bit of memory
- The capture that has to be made
- The convention Dawson actually used
- A wall the pawns cannot cross and the rule can
- The rule decides who has to remember
- A difference the rows cannot predict
- Two things to hold at once, or three
- A loop is written with two names
- The step nobody took for thirty-four years
- The picture Bouton's proof leaves behind
- Four thousand nine hundred regions with no name
- The names are not built out of the old ones
- Four positions, sampled
Applied
- A game older than the theory
- A pawn ending is a sum
- A winning strategy that is a spanning tree
- Counting at the end changes everything
- The chains decide it before the boxes do
- The digits say which move wins
- The game in every exercise book
- The game that is a number system
- The rule that makes Go a finite game
- The theorem that names a winner and no move
- When a real board falls apart
- A set with three descriptions, and a function with none
- The parts are worth nothing and the sum is not
- A parity with a first exception
- What has to break before a pawn is worth a number
- One king, and two files to be in
- A position with no value, and the rule that gives it one
- A ko is won somewhere else
- Two ways to count a finished board
- The second dimension is not the deep end
- What a pass is worth to a theory
- A hypothesis has to hold all the way down
- Nothing to subtract with
- The two moves that are not captures
- The first move is a link that is not there
- Cut is Short on another graph
- A point with three neighbours
- Where the needle has a sentence
- A board one column wider
- Every move closes the largest gap
- A potential that names every move
- The pairing removes moves it cannot name
- The winning reply is the fourth choice
- A shortlist with nothing at the top
- The restriction that buys the most
- A move whose every reply is struck
- The endgame theory arrives late
- A thousand positions and no exception
- A coin with three strings is worth something
- Four boxes for every chain after the first
- Two and four are not conventions
- Cancelling is not pairing
- A cancelling pair is a zero
- Even rows always reward the move