Disjunctive sum — where it appears
Named by 136 essays across 9 fields — each of them below, with the objects they name alongside it.
The sum is the object
Real positions come apart into independent regions, and a move happens in exactly one of them. That operation — the disjunctive sum — is what the whole theory is built to survive, and it is the reason values exist at all.
Outcomes do not add
Knowing who wins each part of a position tells almost nothing about who wins the whole. Counted over every sum of two values born by day two, six of the outcome table's ten entries are settled and four are not — and every settled one is settled by the order rather than by anything about outcomes. Two first-player wins reach all four classes between them.
Two people, four years apart, one theorem
Roland Sprague proved it in 1935 and Patrick Michael Grundy proved it in 1939, neither knowing of the other. That looks like coincidence until the alternatives are examined — and the rule they both reached turns out to be the only one that can work at all.
The board falls apart, and the arithmetic changes
A 4×5 Domineering board with a wall down the middle has 2,916 positions in it, and that number is exactly 54 × 54 — the product of its two halves. Solving the halves separately costs 108. Decomposition is the one saving in this subject that turns a product into a sum.
Turn the board through a right angle
A two-by-four Domineering board is worth something no number can express, and Right is ahead on it. Turn a second board through a right angle, put the two side by side, and the total is exactly zero. Every position has an exact opposite, and that single fact is what makes subtraction — and therefore comparison — possible at all.
Two hot fights that add to a cold number
The mean of a sum is the sum of the means, every time. The temperature is not — it is bounded by the hottest part and is often far below it. Two positions each worth fighting over can add to a plain number that neither player wants to touch.
The same fight, eight times over
The mean is not roughly what a position is worth. It is the number that eight copies of the position stay close to — and the theorem is that the closeness does not decay as the copies pile up. The gap stops at the temperature, and stays there for ever.
One part that never ends
The game called `on` has one move and it is back to itself. Add anything to it — a star, a point, its own mirror image — and the whole board is drawn. So `off` is exactly the negative of `on` and their sum is not zero, which is the group law failing for a reason that has nothing to do with who is winning.
Three ways to add the same games
A move in exactly one component is a choice, not a law. Move in every component at once and the game is different; move in any set of them and it is different again. The same two positions, added three ways, give three different answers — and only one of the three has values that add.
A row of coins is already a sum
Everywhere else on this site a sum is several positions side by side. In a coin-turning game it is one row — each coin showing heads is a game in its own right, and the row is worth the exclusive or of them. The decomposition is inside a single picture.
Big is not the same as hot
A player sizes a move by how much changes hands when it is played, which is the number in every endgame book. The theory sizes it by temperature. On a plain switch the two agree exactly, so nothing shows; on a move with a follow-up they come apart, and the count gives up more than the guarantee the theory's rule carries.
Comparing two positions means playing a third
There is no way to look at two games and see which is better. The question "is G at least H?" is answered by building G − H and asking who wins it — so the most basic operation in the theory is a decision problem, and every canonical form is built out of them.
Counting at the end changes everything
Go is scored. So are Dots and Boxes, chess and almost everything anybody plays for money — and none of them is the kind of game this site's whole apparatus is built for. The simplest scoring game there is shows what that costs — the normal-play theory gives every position of it the same answer, and the answer is useless.
Misère play has no negatives
Put a position beside its own mirror image and answer every move with the mirror move. Under normal play the answerer wins and the sum is worth zero. Under misère the answerer still has every reply and loses because of it — so there is no zero, no subtraction, and no comparison, which is why the misère theory had to be rebuilt rather than adjusted.
The notation was the argument
Up, star and the brace form are not abbreviations for case analyses. They are the claim that these objects add — and the arithmetic they support is arithmetic that no table of outcomes could ever produce, because two positions with identical outcomes can have different sums.
A pawn ending is a sum
In a blocked pawn ending the material is level, the files never speak to each other, and whoever has to move is the one in trouble. Chess calls that mutual zugzwang; this site calls it a P-position; and the two vocabularies were built four decades and one subject apart to say the same thing.
A game where nobody can be ahead in moves
A blue stone beside a red one is a move for both players at once. So neither player can run out while the other still has something to do — and every value the game produces is smaller than every positive number, by the shape of the rule rather than by inspection.
"Hopeless" was a claim about a method
Misère analysis was declared intractable in the 1970s, and the verdict was correct about what was being attempted. Quotients did not refute it thirty years later — they changed the question from a value per position to a monoid per universe, and the computed sizes show why the first question has no good answer.
Sente is a fact about the rest of the board
A move that must be answered is called sente, and the word is used as though it described the local position. It does not. The same fight is answered while the rest of the board is quiet and ignored once it is busy, and the crossover — measured by solving the whole board at every temperature — sits at the follow-up's own temperature.
Where the impartial theory stops
Sprague–Grundy gives every impartial position one number, and the number is complete. The moment the two players have different moves no number works at all — not a harder one to compute, none — and three positions here are compared against every nimber to show it.
A green edge on a blue one
Blue over green and green over blue are the same two edges in the other order. One is worth 1∗ and the other ↑∗ — a number with a star on it against something smaller than every positive number — so a stalk with all three colours in it stops being a numeral and starts being a position whose value depends on what is underneath.
Independence is a claim
Splitting a position into parts and adding the values is the whole method of this subject, and the splitting step is a claim about the position rather than a fact about the drawing. Where it is false the two answers differ — and the failures that matter are the ones that keep the same winner and change the value, because nothing reports those.
The tartan theorem
The nimbers are a field, with a multiplication defined by a mex-style rule that looks like an algebraist's amusement. Lay two coin-turning games on a grid and the Grundy value of each square is the nimber product of its two coordinates — which is the point at which the multiplication stops being a curiosity and starts computing answers.
Two misère outcomes are not enough
Knowing who wins each part does not say who wins the sum. Over 676 sums built from a pool of twenty-six positions, nine of the sixteen pairs of outcome classes settle the answer under normal play and not one of the sixteen settles it under misère — and the nine that work are theorems about a value being zero, which is exactly the thing misère play does not have.
A move that must be answered
Every argument on this site about sums assumes the parts are independent: a move in one leaves the others alone, and the reply may go anywhere. Top Entails denies it — take the top coin of a heap and the opponent must answer in that heap. The nim-sum then misreads 9 of 36 two-heap positions, and two heaps of two coins are a first-player win, which no impartial game the theory covers can be.
Confused is not the same as unknown
Two positions can be neither greater, nor smaller, nor equal. That is a fourth relation with its own symbol, it is a fact about the pair rather than a limit of the method, and it is what makes a game worth playing — a position is a first-player win exactly when it is confused with zero.
The first time it told somebody something
A theory earns its keep when it produces an answer nobody had. Temperature did that for Go endgames — the orthodox account gives a move order that is provably right and is not the one experience offers, and the position it is right about is small enough to check here completely.
The losing positions are a code
Turn over one, two or three coins, and the rows a player has already lost turn out to be closed under adding two of them together. That makes them a linear code — and on eight coins it is the extended Hamming code exactly, sixteen words with a weight enumerator of 1 + 14x⁴ + x⁸, produced by a move rule that knows nothing about codes.
When a real board falls apart
Amazons is played competitively, and late in a game the arrows have cut the board into regions no piece can cross. From that moment the position is a disjunctive sum — arrived at by the play rather than assumed — and the territory count every program uses can be measured against what the sum is actually worth.
An environment made of coupons
Beside the game sits a stack of coupons worth 4, 3, 2, 1, 0, and a player may always take the top one instead of moving. Play the whole thing out and two quantities the theory computes are measured instead: {4 | 0} comes out worth exactly 2, its mean value, and the coupons stop at 2, its temperature. For {10 | {9 | 1}}, whose temperature is 1, they stop at 7/2 — because what the stopping coupon measures is the hottest temperature anywhere in the tree.
The other sum, the one that nests
A move in one part wipes the other out entirely. That is the ordinal sum, it is what a Hackenbush stalk actually is — 1 : (−1) is a half, and 1 : (−1) : 1 is three quarters — and it is not an operation on values at all: three positions all worth zero give three different answers under it.
What a value leaves out
A value settles who wins, by how much, and what happens in every sum the position appears in. It says nothing about when. Seven positions here are worth exactly zero and interchangeable everywhere, and they run from two moves long to eighteen.
Maundy Cake
Cutcake with one word added: a cut must divide the piece into equal parts. The values are still whole numbers, and the rule this site has been repeating about them is false — over all 1,296 cakes to 36×36 the largest-odd-divisor account has 946 counterexamples. What survives is a count of prime factors, and it says who wins without saying by how much.
A rule that is never right and cannot be far wrong
Playing the hottest component is not optimal — over 440 measured lines it costs something on 17 of them. What makes it worth having is that the cost is bounded by the temperature, provably, and that the same test run with the ordering reversed breaks the bound on 54.
Equal in every company
Two games are equal when no third game can tell them apart — a quantifier over every position there is, discharged by one finite test. A search over 184 contexts separates all 5,790 unequal pairs it is handed and still calls two different games the same, which is exactly why G − H = 0 is a theorem and an exhaustive search is not.
Taking from several heaps at once
Moore's Nim lets a move take from as many as k heaps at a time, and the losing positions are still read off the binary columns — divisible by k + 1 rather than by two. The rule agrees with exhaustive search over 54,264 positions and never disagrees, and it decides every outcome while supplying no value at all: reading the same columns as a base-3 number gets the Grundy value right on 42 of 330 positions.
Where the fight stops
Keep taking the biggest thing on offer until somebody is left facing a number, and the number reached is a stop. Every position has two of them, one for each player moving first, and between them they say how a fight ends — except for the part of it no number can reach.
The one outcome that adds
Finite outcomes do not add: two first-player wins can sum to anything. Loopy play has seven outcome classes instead of four and adds even less — of the 28 cells in the table, eleven hold several answers. Two do not, and they are the two worth having: a second-player win added to anything leaves the outcome alone, and a draw added to a draw is a draw. A draw added to anything else is not.
The cost is in the closure, not in the positions
Under normal play, Dawson's chess needs four classes for every heap up to twelve, because its Grundy values stay at three or below there. Under misère play the same game needs six, then twelve, and the number rises with the universe rather than with the position — which is a different kind of expense entirely.
When the ups add
Atomic weight brackets do not add over a sum — they bound it. Over all 120 pairs from a fifteen-game family the sum's bracket came out exactly the sum of the parts' brackets 56 times, strictly narrower 64 times, and wider never; and the rule separating the two is one line long, because every one of the 54 pairs with a pinned part is exact and only 2 of the other 66 are.
Amazons on one line
A board one square high is small enough to evaluate completely: every strip from two to ten squares with one amazon a side is 37,886 positions taking 81 distinct values, and every one of them is an integer, a switch, a number plus a star, or a bare star. Not one is a fraction — and forcing the arrow onto the square just vacated, which takes a freedom away rather than adding one, produces 1,196 that are.
The values that are their own negatives
Every game satisfies G + (−G) = 0, so a game equal to its own negative satisfies G + G = 0 — it has order two in a group whose elements otherwise have infinite order. The nimbers do. So does ±1, on sight. Over the 1,474 values born by day three there are 30 of them and only four are nimbers, every one of the 900 sums of two is another, and the equality test and a symmetry of the written form agree 1,474 times out of 1,474.
A pass is not a move
Put a single pass token on a Nim board and one clause decides everything. If it may be taken at any time — including as the move that ends the game — the value of the whole is the nim-sum with a one added, in all 120 positions swept: the pass is a heap of one. Forbid it as the final move and the value stops being a function of the nim-sum at all, and 3 and 1 + 2 come apart.
Two games in one environment
A coupon stack measures a position: play the whole board out and the coupon the players stop at is the temperature, the score is the mean. Put a second position beside the first and one of the two measurements stops working. Over 36 ordered pairs the contributions still add to the means every time, and the coupon a fight is entered at moves on 13 of them — without either position changing.
When the nested sum only sees the value
The ordinal sum reads the form and not the value: three positions all worth zero, placed under a star, give three different answers. On impartial games it reads the value after all — 72 substitutions of an equal-valued heap from a different game, and every ordinal sum comes back unchanged. That difference is the whole reason a green Hackenbush tree can be collapsed one branch at a time.
A rule with a guarantee
Evaluating a sum of a dozen fights is impossible; following a rule is not. Move where the stake is largest, and over 220 sums of three hot components the rule scores exactly what perfect play scores in 196 of them, is never more than one point behind, and never ends more than the largest single stake below the mean. The rule that is supposed to be different — answer the threat — chose differently in none of the 220.
Nobody has to move
Every convention here rests on one sentence nobody examines — the players move alternately. Replace it with an auction and a position stops having an outcome class and starts having a number: the share of the money the second player needs. The 22 values born by day two collapse to seven of those numbers, eight of them landing on exactly a half; the new number respects the game order on all 179 comparable pairs, and is not determined by the parts under addition on 14 of 49.
What a number does to a fight
Adding a number to a position moves everything and changes nothing: over three thousand translations the temperature never once shifted and both stops moved by exactly the number added, every time. What the number decides is whether the fight is worth having — and the interval where the two players are confused is exactly the open interval between the negated stops, right in all 2,890 cases away from its endpoints and wrong in 110 that are all on them.
Cooling adds and heating does not
The two operators are presented as a pair, and they are not one. Cooling a sum is the same as cooling the parts and adding, on every one of the 1,768 pairs tried, at two taxes and on two pools. Heating fails on 263 — and not for the obvious reason: in every failure neither part and not the sum is a number, so the clause exempting numbers never fires at the top. It fires two levels down, where an option of a sum is one part's option plus the whole of the other.
Equal in this company
Equality quantifies over every game there is, and the quantifier can be made smaller. Restricted to a company of nine numbers, the twenty-two values born by day two collapse to seventeen; restricted to four games — nought, one, minus one and star — they stay twenty-two, and no three of the four will do. The company that decides equality is tiny, and it has to contain a star.
Splitting is a move
Add to Nim a move that removes nothing — break a heap in two — and the Grundy sequence gets simpler, not harder. Lasker's Nim has a closed form with one clause per residue modulo four, exact on all 2,001 heaps checked: the identity with every fourth pair transposed. Kayles is the same kind of game with the taking bounded instead of the splitting, and it has no closed form at all, settling into a period of twelve only from heap 71 with fourteen values outside it for ever.
What is left when the small change is thrown away
Canonical form answers a demanding question: which positions are interchangeable inside every sum whatever. A player with a hot board does not have every sum — an infinitesimal difference cannot decide anything against a genuine fight — so there is a coarser question with an exact answer. The reduced canonical form takes the 1,474 values born by day three to 61, with 292 of them collapsing to zero, and it is a homomorphism on all 8,100 pairs tested only when a second pass is made.
The heap is not the position
Fibonacci Nim bounds a move by twice the previous move, which puts the state outside the board: a heap of six with a cap of two and a heap of six with a cap of five are different games. So there is nothing to add and no Grundy value to compute — and the game is completely solved anyway. The opener loses on exactly the nine Fibonacci numbers up to 120, and the smallest term of the Zeckendorf numeral is a winning move in all 110 winnable heaps.
What can be struck out
From G + X = H + X it follows that G = H, in one line, by adding −X to both sides. It is the shortest theorem here and the most used: it is what makes comparing two boards region by region legitimate. Over 10,648 triples the hypothesis fires 484 times and the conclusion holds 484 times — and the licence expires in three separate directions, each of which loses the same axiom in a different way.
A rule with no promise at all
Playing in a hottest component comes with a bound: never more than the largest single temperature below the mean of the board. Over 220 sums the bound holds 220 times — and so does the bound for a rule with nothing behind it, which scores exactly what perfect play scores on 205 sums against the hottest rule's 196. The control that shows the bound is doing work is the rule that plays the coldest component, which breaks it 74 times and loses up to eleven points.
A tree is still a number
A Hackenbush string spells its own value in binary. Put a fork in it and the numeral has nothing to read — there is no leftmost anything. The value is still a number, in all 10,066 forests up to six edges; it is still computable, by the ordinal sum, in all 3,238 single-trunk trees; and the reading is right on 762 of them, of which 126 are the strings it was written for.
How long it lasts
Move in every component at once and the game ends the moment any one of them does. Grundy values say nothing about that game; what decides it is the remoteness, a second number computed from the same tree that measures how long a component can be made to last. Over 2,268 positions the rule is right every time, and the two numbers determine each other in neither direction.
The genus of a sum
A genus symbol is meant to be carried one per heap, so that a solver never has to look at the heap again. That is a claim that the pair of symbols determines the sum's, and across nine games and 405 pairs it holds without exception — while the bases alone determine it in only 38 of 50 cases and the superscripts alone in 70 of 74. Both halves of the symbol are load-bearing, and two wild heaps can add to a tame sum.
What an infinitesimal does to a fight
Adding a number moves both stops by exactly itself. Adding something smaller than every number moves neither — across 10,318 additions to a whole day of values, not once — and the outcome class changes anyway, 2,622 times. It changes at exactly one kind of position: the ones with a stop sitting on zero, which is where the numbers have run out of things to say.
How cold a sum of hot games can be
The temperature of a sum is at most the largest temperature in it, and the bound leaves the whole interval below it open. Over 1,035 pairs the sums do not use that interval: 864 sit exactly at the maximum, 160 are frozen outright, and eleven land anywhere in between — every one of them with a component whose wall bends.
Every group must keep breathing
NoGo is Go with no captures at all: a stone may be placed only if, afterwards, every group on the board still has a liberty. That makes a move's legality a fact about the whole board rather than about the squares it occupies — and a board therefore almost never breaks into independent parts. Of 117 boards here whose empty points fall into two regions, 24 are the sum of their regions and 93 are not.
What a move nobody makes is worth
If Right's move in a fight has to be answered, then Left's move in the same fight prevents an exchange Right was going to get for nothing. That is a reverse sente, and pricing it is the awkward case: over sixty measurements the gain matches the local temperature once and the follow-up's swing twice, and the band over which the move is worth taking is not the band over which the move it reverses is sente.
A board that is a sum of its regions
A table of rectangles is a table about the openings. A partly played Domineering board is not a rectangle, and evaluating one means splitting it into pieces no domino can straddle, looking each piece up and adding. The catalogue of 104 shapes does it correctly on every one of the 3,227 positions of a 3×4 board it covers — and among the shapes are two worth an up and a down, which no rectangle ever is.
What is left when the copies pair off
A pile of n copies stays within a bounded distance of n times the mean, and the distance never grows. The difference is a game rather than a number, and what it actually is has a much better answer: for a plain switch it alternates between one fight and nothing at all, and for a fight with a follow-up it is different every time — bounded in size and unbounded in complexity.
The birthday of a sum
Two values born by days m and n have a sum born by day m + n at the latest, which is the bound that stops a board made of many small parts from being unboundedly complicated. Over 231 pairs of day-two values the bound holds every time and is exact 163 times — and every pair it misses by three days or more has a sum that is a number or a nimber, so the slack is not noise but a measure of how much cancelled.
When two thermographs can be added
The temperature of a sum is not the sum of the temperatures, and the natural repair is to add the whole diagrams instead. Over every pair of hot values born by day two the added walls always bound the true ones and the mast always comes out right — and the whole diagram is right exactly when at most one of the two components is hot, which is precisely the case a reader has no use for.
A pool built to punish greed
The rung below found a rule with no theorem behind it beating the rule with one, and predicted that a pool of deliberate traps would reverse the result. It does not. The traps miss, because the rule called greedy scores a move by the stop it leaves and a stop already contains the follow-up — and the rule the traps do catch, losing sixteen points where the guaranteed rule loses five, had to be written to make the point.
What a wider pool rescues
The misère outcome table has sixteen cells, and over a pool of ten positions fifteen of them hold fewer than four outcomes — which looks like structure and might be a shortage of positions. Thirteen values further on there is nothing left: every pair of outcome classes takes every outcome, so the near-misses were the pool, and the prediction the rung below made was right.
What a component has to carry
Three impartial games on this site break the sum, and they break it for the same reason: a component cannot say what its own legal moves are. Measured with one instrument — one number per part, exclusive-ored — the failure rate runs from a quarter to nearly half, against a control where the same recipe is a theorem and is never wrong.
When the regions add
The rung below described the NoGo boards whose regions add as the ones with symmetric walls, and said the description was a guess made from six examples. It is wrong: fourteen symmetric boards do not add and sixteen that add are not symmetric. What replaces it is a criterion about liberties — sound on all 117 boards, provable in a line, and complete on only nine of the twenty-four.
The number nobody needs
The compound theory carries a third quantity — the suspense number — computed by the remoteness recursion with both preferences reversed, for the compound that stops as soon as any component stops. It governs that compound correctly. So does remoteness, so does the plain Grundy value, and the shortening does not change the winner on any of 1,176 positions.
Where the order and the sum disagree
Day two is a lattice, and day two is a group, and it is not a lattice-ordered group. The one identity that would join the two structures — the join plus the meet equals the pair — holds on exactly the 201 pairs where it cannot fail and on none of the other 52, and the errors split thirteen high, thirteen low and twenty-six confused.
The question in the middle
Between knowing who wins each part and knowing what each part is worth sits the question a player actually has: who wins the board. Priced on sums of two it lands between the other two on seven of eight, cheaper than the values by up to eight times. On sums of three, with nothing repeated, it is dearer than the values on five of six — because a component multiplies a search and only adds to a value.
How thick a wall has to be
A single stone between two empty stretches of a NoGo board couples them, and the obvious repair is a thicker wall. Over 590 walled strips a thicker wall does help — and splitting the same 590 by the colour of the stones shows that thickness was never the variable. A wall of four one colour couples the sides exactly as one stone does.
How often a board falls apart
A decomposition turns a product into a sum, so a solver wants to know how often one arrives. Over every position of a 4 × 4 Domineering board the answer is 47 per cent — nought for the first two moves, three fifths in the middle, and nought again at the end. What one decomposition is worth is the other half of the answer and it is a factor of 1.8.
A schedule instead of a number
Playing in the hottest component loses at most the largest temperature on the board — the classical guarantee, stated against one number. Sorting the temperatures and reading the guarantee one step further down gives a promise 47 per cent smaller that is never breached over 1,734 sums. Two steps down it fails 120 times, so the schedule has exactly one step of slack in it.
How wrong a nearly-independent split is
Treating a connected board as a sum of two halves is a claim, and the rung below counted how often it fails. This one prices it: over every vertical cut of every small Domineering rectangle the error is a game rather than a number, it is never in Right's favour, and it is bounded below by twice the height of the cut — a bound the height alone does not supply.
The rule a smaller move breaks
Moore's Nim lets a player take from at most k heaps, and its winning condition is the binary columns summed modulo k + 1. Cap the amount as well and the obvious repair — reduce each heap modulo the cap plus one, then read the columns — is exact at every cap when k is one and wrong at every cap when k is two or three. The reason is stronger than a broken rule: at k ≥ 2 the residues do not determine the outcome at all, so nothing of that shape can work.
The closure that picks the nimbers
Closure under addition lets a sum be rewritten and turned out to admit companies that are not nimbers at all. Closure under forming options lets a subposition be rewritten, and it pulls the other way: every company this site computes in has it and none has the first, and among the seven finite addition-closed companies, keeping every option is exactly being a group of nimbers — four of seven, both directions, no exception. Demand both at once and nineteen of twenty-two day-two values generate nothing finite.
When the catalogue starts paying
The rung below priced two questions — who wins one board, and what it is worth — and named the third: a program pays for a family of regions once and answers every board over them by addition. The crossover is between five boards and two hundred, depending on how far the catalogue reaches, and it falls as the board grows. The whole catalogue of every region to eight squares costs one part in seventy-six of one undecomposed five-by-five board.
What a fight does to a fight
A number added to a position shifts both its stops by itself; an infinitesimal moves neither. A hot game does neither: over 720 sums the two stops add on 330 and are wrong on the rest. What survives is the mean, which adds on every one of the 720 — and the failure has a bound, since no stop is ever out by more than twice the smaller of the two temperatures, a bound 222 of the sums attain exactly.
What a game actually produces
Fifty-three per cent of the Domineering regions of at most eight squares are hot. Of the components eleven hundred random games actually produce, sixteen per cent are — and ten per cent once single squares are counted. The figure is the same on three sizes of board, so it is a property of play rather than of the board, and it says that every temperature census this site has taken over a catalogue overstates how hot the game is by a factor of three.
What the class does not buy
Dead-ending is the hypothesis several modern misère results are stated under, and the rung below sorted this site's games into it without running the comparison those results are about. Running it: a quarter to a half of ordered pairs compare inside a ruleset's own universe, which is a great deal — and the ruleset that is not dead-ending sits in the middle of that range. Ten comparisons are lost when a universe is enlarged, and every one is lost to a dead-ending company.
One fight makes a board a fight
The rung below found 16 per cent of the components a played game produces to be hot, against 53 per cent of the catalogue they are drawn from, and predicted that the share of hot boards would be much larger. Taking the same play-outs and tallying at the board gives 32 per cent — twice the piece figure and not ten times it, because a Domineering board carries only 1.68 pieces and the hot ones cluster on the same boards.
A bound with one number too many
The rung below bounded how far a hot addend can drag a stop — twice the smaller of the two temperatures — over sums whose addends were all plain switches, and conjectured that an addend with a follow-up would need twice the smaller of three numbers. Over 1,440 sums with bent addends the two-number bound holds everywhere and is attained 358 times, and the three-number version fails on 66.
Two errors that cancel
Replacing the packing count with an interval left a doubt that the pessimistic half would add across a board. It adds, for a one-line reason. What is worth measuring is what the reading is then worth: over boards of one to four regions the count decays from exact on 45 per cent to exact on 11, and the interval's containment does not decay at all — it rises from 67 per cent to 74, because the interval's width adds and its error does not.
The quantity that does not order a board
The rung below found the players leaving an environment at the larger of a position's two temperatures, and proposed that a board should therefore be played in the order of that quantity. Over 220 boards of three components it plays exactly on 124 against playing-in-the-hottest's 196, loses 85 of the 97 disagreements, breaks Hotstrat's guarantee on six boards, and costs nine points on its worst one.
Add, then reduce again
The homomorphism promises that a sum's reduced form can be computed from its parts', and says nothing about what the operation is. It is addition followed by a second reduction — needed on 431 of 3,600 pairs of day-three values, and on not one of the 1,751 pairs with a cold part. What the second pass removes is an option that only becomes dominated once the two fights are side by side.
A function with no formula
The rung below's composition rule is exact on tame pairs and wrong on all fourteen wild ones, which looked like an exact boundary. Two heaps further it is wrong on 34 of 35 and right on one — Kayles' five and nine — so the boundary was a boundary of the pool. What survives is stronger and stranger: the pair of symbols still determines the sum on the wild side, and no rule of that shape describes it.
A product against a sum
A company closed under both addition and options licenses a solver to rewrite any subposition, and the rung below found that exactly the nimber groups have both closures. Priced on Cram boards, that licence is the difference between walking a product of position sets and walking their sum — four to twenty-five times on two components, twenty-four to a hundred and sixty-one on three — and it is available to impartial games because their class representative is a heap rather than a form.
A numeral in the empty squares
The rung below ruled out a quantitative criterion for Push and asked for a numeral over the coins combined with a count over the gaps. The two ingredients are the right way round: the colours pick a fraction — −1, −1/3, −1/7, −1/15 — and the empty squares give the binary precision, so a run of k coins before one of the other colour with g gaps is worth exactly (1 − 2^(−kg)) ÷ (2^k − 1). And it does not compose: a strip of two runs is not the sum of them, on any pair tried.
The cliff a cut invents
The rung below asked for a correction term in the gap between two Push runs. There is none, because the gap's contribution vanishes: widen it and the strip's value converges geometrically, at a rate set by the back run's length alone, to a limit that is not the sum. And Shove — whose reading is exact everywhere — fails at the same cut, which says the broken thing is the cut and not the game.
A rule that beats the hottest
The rung below proposed the reverse of the rule that had just failed — discount a component by its answer's temperature rather than promoting it — and predicted, before the sweep, that it would not beat playing in the hottest component. It does. It plays exactly on 201 of 220 three-component boards against 196, wins two thirds of the boards where the two disagree, keeps inside a guarantee proved for the other rule, and the gap widens as the board grows.
Which end a sum lands at
The rung below found the errors in a translated stop clustered at the two ends of the range its bound allows — 660 at nought and 358 exactly on the bound — and asked for a rule saying which end a given pair lands at. There is one, in four lines, exact on all 1,440 sums. Three of the four cases are decided by the value being translated alone, and the property that decides them is the bend the switches ladder found for a different question entirely.
The option nothing names
The rung below found the arithmetic on reduced forms to be add and reduce again, needing the second pass on 431 of its sums, and asked whether the option that pass deletes can be named from the parts. Eight rules were scored and the best reaches four in five — and on a pool closed under negation it falls to under half, which says the near-miss is a property of the population. What the second pass does have is a shape and a cheap test that rules it out.
What identifies two subsets
Every subset of a day gives a self-negative value by mirroring it, and 1,793 subsets of day two give thirty values. The collapse happens in two stages with different characters: domination takes the 1,793 to 96 antichains and is a theorem, and the rest of the reduction takes 96 to 30 and is concentrated almost entirely on two values — nought, which has an exact description, and star, which has none.
The wild side does not close
The rung below asked for the wild composition table and for two things about it: whether the wild genus symbols form a small closed set, and whether that set is a misère quotient in disguise. Building the table needed a wider sweep — nine counters a heap gives a diagonal rather than a table — and both answers are no. Not one of the twelve entries is a symbol any wild heap carries, and two wild heaps added together are tame two thirds of the time.
Half a licence is nearly all of it
The rung below priced the substitution licence a restricted universe gives a solver and asked what half of one is worth — the licence to rewrite components but not subpositions. It is worth nearly the whole saving. Rewriting components collapses a million and a half states to three thousand six hundred; rewriting subpositions collapses those to eight hundred and eighty-four, and splitting the pieces takes it to fourteen.
Read from the back forwards
The rung below found a two-run Push strip converging at a rate set by the back run and asked what a third run does — whether the rate is still the rearmost run's, or whether the rates compound. It is the rearmost run's, and for every gap: widen the front gap of a three-run strip, two whole runs away, and the value still dies at the last run's rate. Shove, the game one clause away, compounds.
The same number in two currencies
The rung below found bent-walled values falling strictly inside the translation bound and asked how far. The shortfall is the value's own hottest follow-up's temperature — exactly, on 400 of 408 pairs, and twice it on the other eight — which makes the whole error one expression. And it is the switches ladder's constant: a half there and a whole here, because a temperature is half a stop gap.
Not a domination, in that order
The rung below asked which pair the second reduction acts on, taking for granted that the operation is a domination. It is not: on none of the 525 deletions is a surviving option greater than or equal to the deleted one. In the order the reduced form actually works in — both stops at least as good — every deletion with a survivor is a domination, the dominator is unique on all but twelve, and it always comes from the other part.
The short side is not in the lemma
The closed form for a two-sided Maundy Cake rested on one unproved statement: that no divisor beats the largest prime. Written out, that statement never mentions the short side — it is an inequality between a multiset of primes and a term count — and once it is stated that way it has a two-line proof, term by term. The ladder ends in a theorem rather than a grid.
Two strips that end the same way
If a Push strip's sensitivity is governed by its last run, then two strips agreeing at the far end should behave the same however different their fronts. On 78 of 80 tails they do, exactly. On two of them a single empty square in the prefix reaches across a gap that grows without bound and halves the rate — and the run reading turns out to be sound in one direction only.
A pool built to have an answer
The coefficient in the rule score a component by t − λa scored identically for every λ in the unit interval, because the rule reads an ordering and that pool's orderings changed at three places. A pool designed to have twelve crossings turns the interval into thirteen different rules, and all three board sizes agree on one cell: between a quarter and a third.
Where the value stops mattering
Fourteen straight-walled pairs missed the bound on a translated stop and had only a threshold to explain them. Their stops move by exactly the addend's temperature — a formula with the value nowhere in it — which turns the threshold into the boundary between two lines and closes a census of 1,440 pairs that has been open for four rungs.
The case that was supposed to be hard
The mex rule for the mirror construction was to be proved by induction, and the step flagged as needing care was the one where an option is incomparable with the nimber. There is no induction: the argument is four lines, and incomparability is what makes two thirds of the cases go through — because a fuzzy sum is a first-player win and the first player is the opponent.
A side about to lose its move
A fifth of the second reduction's work removes the last option a player had on a side, and no rule on the ladder had looked at one — because a deletion with no survivor has no pair in it. The recognisable object is not which option goes but whether the side is one an option can go from, and two comparisons on the parts decide it on all 525.
The licence that weighs nothing
The third substitution licence is constant in the number of components, exactly as predicted, and it saves under two times where the first saves seventy-six million. Priced by its table instead of by its saving it is the only one of the three whose cost does not run away — which reverses the order three rungs of this anchor have put them in.
The table that changes its mind
The advice that ten entries chosen by use serve nine lookups in ten was untested: it describes a table sorted after the fact rather than a solver that only ever held ten. A solver that only ever held ten gets 94.2 per cent — beating the best ten chosen with the whole run in view, because there is no best ten.
The easy case was not the reason
The rung below found the mobility rule reaching a failure rate of exactly nought near the endgame and named what a proof would need: that a decomposed board's comparable options are ordered by reply count. That statement is false on all five boards, at margins up to two — and split positions go exact two squares of depth before whole ones, so decomposition is the easy case rather than the cause.
The price of asking what the parts are
The third licence lets a solver look up a region rather than a position, and the rung below priced it by the entries it stores. Priced by the work it costs, it saves between a third and two thirds of the expansions and pays for them with a flood fill at every node — six times the total. A square would have to be ten times cheaper than a table probe before it broke even.
One expression proved, and one withdrawn
The census closed with three expressions exact on 1,440 pairs, and the rung above asked for derivations. The cold one has a four-line proof. The straight one has a threshold the census cannot determine — any constant between 4/3 and 3/2 fits it — and eight more addends of the same family break it on 38 pairs while leaving the bound above it untouched.
The pairing the formula hides
Welter's closed form sums a function over every pair of coins and needs an extra term when the count is odd, which the rung below called a surprise. Read as a matching it is not: an odd number of coins cannot be paired, the left-over coin contributes its own square, and some matching gives the value on every position measured.
A cross in the table
Which pairs need a second reduction had never been asked. Sorted by the two temperatures the answer is a cross — the whole gap-of-a-quarter column and the whole colder-is-three-quarters row — and it is exactly necessary on all 431 with no exception, and wrong 477 times in the other direction.
Two measures bounded, and one not
A sum is born no later than its parts' birthdays together, and it has no more options than they have between them — a bound nobody had checked, and it is attained. What runs away is the length of the written form: 27 pairs of 231 exceed it, the worst by 29 characters, on a sum with exactly as many options as it was entitled to.
One number, stated two ways
Twice the height of the cut held and was loose; the height alone failed. The smallest true constant is three halves — exact and attained as a bound on how far the value can fall, and an infimum attained nowhere as a bound on the value. The gap between the two is one move.
A factor, and not an overhead
Deciding who wins a form searches the form's own tree. Reducing it to canonical form searches a difference game for every comparison, and a difference game is a sum. Over 256 forms the reduction expands 5.46 times as many positions — and the ratio runs from 0.58 at one option to 9.80 at eight.
The square that cannot be halved
Every number in hopless Toads and Frogs is a whole number, which the rung below measured on seven thousand strips and could not explain. The reason is that every empty square is either one player's alone or split evenly between them — except one, and that one is where the numbers stop.
A second pool, designed differently
One designed pool put the rule's best coefficient between a quarter and a third, and all three board sizes agreed. A second pool, built by the identical greedy criterion from different material, has no cell that is best at every size — so the coefficient is a property of the pool and there is no number to find.
The residues as a sequence
Four of the eight residue sequences never repeat, and a recurrence is not a description. There is a closed form and it is not for the game: the stops and the temperature of the n-th residue are periodic with period one, two or four on every sequence in the pool, while three of them produce a different game at every n.
An environment instead of a stack
The guarantee behind playing the hottest component survives one step down a board's sorted temperatures and fails at two. Against a coupon environment as hot as the board it survives all of them — because the rule stops being approximate and starts being optimal, on every sum in the pool built to punish it.
A stopper and how to find one
The class a value theory for loopy games would need is the ones with no infinite alternating run, and the qualifier does the work: seventy-nine of the two hundred and fifty-six two-node loopy games qualify and seventy-two of them have a cycle. Every one has a decided outcome, and under eight tests the seventy-nine collapse to six.
A wall an amazon can walk through
An arrow burns a square for good, so an Amazons board that has fallen into pieces should stay in pieces. Over 127,583 positions it does not: fifty-one thousand moves put two regions back together. Every one of them is a single diagonal step, and what is wrong is not the game but the rule used to find the regions — which was borrowed from a game whose pieces lie along the board's own lines.
Where the braces stop
The brace notation names every game exactly — 1,474 games born by day three, 1,474 different expressions, no two alike. It also gets long: the middle one is twenty-two characters and the abbreviations everybody actually writes cover one game in twenty-three. And it has two hard edges. A game with a cycle in it has no finite expression at all, and the equation the minus sign encodes — that a game and its negative cancel — is false under misère play on every one of those 1,474.
The second bend is the boundary
Adding two thermographs wall by wall gives a diagram that is right at the mast and wrong below it. Over every pair of hot values born by day two, the added walls sit outside the true ones at every height — an outer envelope with the truth somewhere inside — and the two pictures separate at exactly the lower of the two temperatures, on all twenty-eight pairs. Above that height both components are still fights and the addition is exact; one sixteenth below it, every pair has parted.
One king, and two files to be in
The whole apparatus needs the files to be independent, and a king is what makes them not. With the kings unable to move the sum of the parts is exact on every configuration; give each king a single waiting move and the sum names the wrong winner on one configuration in six, and on a hundred and twenty-six of two hundred and forty-three with three files.
Three complete solutions in nine years
Bouton in 1901, Wythoff in 1907, Moore in 1910 — three airtight solutions of three games, all published before there was any theory of games at all. Asked about each other's games they all fail, and two of them fail by being wrong while one fails by having no form for the question. Only the last kind of failure decides anything.
Two names that add to nothing nameable
The special symbols reach one game in twenty-three at day three. Coverage is the wrong measurement. The notation exists so that positions can be added, and a sixth of the sums of two named values at day three cannot be written without opening a brace — starting with a sum of two of the six symbols anybody learns first.
A board is written as a sum
Every measurement of the brace notation so far has been of a single position, and nobody writes a single position. A board is several parts, and it can be written as the parts joined by plus signs or as the one value they add up to. Over every sum of up to four games born by day two, the one value is usually the shorter — and the share of boards that need a brace climbs with every part added, until the longest value is four times its sum.
A wall the pawns cannot cross and the rule can
Two rows of Dawson's diagram separated by a file with no pawn on it: 1,616 moves were examined and not one crosses the gap. With captures optional the rows add on every diagram checked. With captures compulsory they do not, because the compulsion is a rule about the whole board — and the game that is a sum is the one ·137 does not describe.
The bound names the hottest part and the cost does not
Moving in the hottest component costs at most the largest temperature on the board, and that bound is attained: 100 lines of 4,240 pay exactly it. It is still the wrong quantity. Across four pools and boards of two, three and four parts the cost is nothing on 90.8% of lines and otherwise takes one of two values — half a point or one — on boards whose largest temperature runs to three, and it exceeds the coolest component on 13 lines and twice it on none.
A difference the rows cannot predict
The diagrams that are not the sum of their rows have been counted and never priced. Priced over 50 diagrams and 63,408,981 positions, the difference takes three values and is a function of nothing a reader can see: seven diagrams whose rows are worth ∗ and ∗ split five to two on it, the third value arrives only at the ninth file, and the one rule that survives is a parity — all twenty-one diagrams of three, five and seven rows add, and every failure carries an even number of rows.
The cheap fights make the rule cheaper
A conjecture stands that playing the hottest part costs at most the coolest temperature times the number of parts sharing it — proposed on a range where that number never exceeds two. Swept to five-part boards over 10,410 lines it is false, and false the other way round: every line costing more than the coolest part has one or two parts at that temperature, and over the 3,230 lines with three or more, not one does.
A loop is written with two names
A region with a cycle in it has no brace expression, and every one of the 256 regions of two positions can be written anyway — as two names, the game it is when a play that never ends goes to Left and the game it is when it goes to Right. Checked against all 1,474 values born by day three, ten names cover every side, 25 regions need only finite ones, and the pair predicts every sum with a finite game, draws included: a draw arrives exactly where the two names disagree.
What a component would have to carry
For a held pass to be decided by a summary of each component, the summary must separate every pair of components some company tells apart. The Grundy value does not — Nim 1 and Kayles 8 are equal games that a held pass separates beside a single Nim heap of two. Nor does the Grundy value with the component's own held-pass value: Kayles 3 and Kayles 6 agree on both and are split by a company of two Nim heaps. Over twenty-four components, fifteen classes against fourteen pairs, and the gap widens as the pool grows.
A misère sum is searched, not added
Under normal play the outcome of a sum of heaps is a nim-sum of numbers already known: twenty stored values decide every sum of Dawson's chess with heaps up to nine, however many heaps it has. Under misère play each sum is a new position to search. One outcome costs six positions for a single heap, two hundred for four heaps and over five thousand for eight, and a table of every eight-heap outcome costs a hundred thousand. The misère quotient is the only thing that brings the price back down.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchEnumerationTemperatureCanonical formOutcome classDecompositionMean valueCounterexampleThermographAdditivityGrundy valueComparison