Independence — where it appears
Named by 19 essays across 7 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.
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.
No two heaps alike
Welter's game is Nim with one extra clause — no two heaps may be the same size — and the clause is fatal to the nim-sum, which gives the right answer in none of the 120 three-coin positions. What replaces it is a function of pairs: ⟨a | b⟩ = (a ⊕ b) − 1, exact on all 55 two-coin positions, and nim-added over every pair it is exact on the whole board provided the number of coins is even.
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.
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.
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.
A compound of two different games
Every rule the compound theory has survives mixing exactly — the minimum-remoteness rule is right on all 5,184 mixed pairs and all 7,560 triples — and the reason is not that the rules are strong. It is that each of them reads one number per component, and a number does not remember which ruleset produced it. The thing mixing damages is the shortcut a reader carries instead.
What restores the theorem
Fibonacci Nim breaks the recipe every impartial game is supposed to obey: one number per heap, exclusive-ored, gets a quarter of two-heap sums wrong. Index the recursion on the pair of heap size and cap instead and the recipe is exact on every pair and every triple — and the number a heap of nine carries turns out to be five rather than one.
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 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.
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.
Two clauses and a third question
A component can carry its own rule when two things hold: its moves are a function of what it carries, and a move in it leaves every other component alone. Two rulesets built to fail one clause each are both caught on a named witness. The four real games sort exactly — every one the recipe gets right fails no clause, every one it gets wrong fails one — and the two clauses still miss something, because Fibonacci Nim and a held pass fail the same clause and only one of them can be repaired.
A wall that bends
On a NoGo strip, two empty stretches add when no group breathes into both — a wall of two stones of different colours does it, and the criterion explains nine of ninety-three boards and all nine that it covers. On a three-row board it explains none of 227, and not because it is less accurate. A wall across a board has to bend, a stone at the bend sees empty squares on both sides by itself, and every one of the 227 has a group breathing into both regions. The condition is unsatisfiable.
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.
A ko is won somewhere else
The rung below shows the ko rule buying finiteness by deleting one edge. What it buys with the same edge is a fight nobody can settle by looking at it — the prohibition forces a player to spend a threat, threats are counted on the rest of the board, and every decided cell of the sweep goes to whoever is ahead on a quantity that is not in the picture.
The second dimension is not the deep end
The rung below says a row of eight reaches every corner of the vocabulary and goes far into none of them, and that the narrowness is a fact about the board. So the obvious next move is a rectangle — and nine squares in a square hold twenty-five values where nine squares in a line hold fifty-eight. The geometry says why before any stone is placed.
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.
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 endgame theory arrives late
Every component the chain-and-loop theory names has coins of degree two, so a position it can read is one where every surviving coin holds exactly two strings. Over a six-box board that is 1,033 of the 28,028 positions with no free box on the table — 3.7 per cent — and more than half of them only after twelve of the board's seventeen strings have been cut.
Named alongside it
The objects these essays reach for when they reach for this one.
DecompositionExhaustive searchDisjunctive sumComponentRegionCounterexampleEnumerationBoardDomineeringGoGrundy valueOutcome class