Combinatorial Game Theory
who wins, and by how much
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The thread: The notation is not the position — page 4

A brace form, a binary numeral, an octal code and a thermograph are four ways of writing a position down, and each throws something away. Occasionally one of them turns out to be the argument.
Trees, and what each is worth. A row of blue-red Hackenbush trees with the value the recursion returns under each. Every one is a number, and none of them is the binary reading of anything a reader can see in the picture. Particular games

Where the numeral stops

A Hackenbush string is a numeral and a tree is a trunk with a forest on it, so the obvious next question is a graph with a cycle in it. Green Hackenbush answers that by fusing the cycle to a point. In blue and red the fusion is right on every three-edge cycle, on fewer than half of the six-edge ones, and the smallest thing it gets wrong has four edges.

 Every domino Left can topple in LRRL. A row of dominoes, blue for Left and red for Right, and each of the mover's options below it. Toppling a domino leftward removes it and everything to its left; rightward removes it and everything to its right. The value under each option is what the game recursion returns for the row that survives. Particular games

A recipe instead of a census

Counting a thousand values in seven dominoes suggests Toppling Dominoes reaches every short game, and a count is not a construction. The obvious construction — lay the two options either side of a Left domino and a Right one — is exact on day one, right on a third of day three, and cannot be applied to nine in ten values at all.

 The same game, written twice. A position as it arises and the same position reduced. Left would never move to 0 when 2 is available, so that option is dominated and can go. The two games are equal — checked, not assumed — and the second is the canonical form. Values

A reduction that reads a graph

The two reductions are defined as deletions from an option list, and the shared form has no option lists — a node is reached from several parents at once. Both restate as rewritings at a node, the rewriting is confluent, and its fixed point is the canonical form. What does not carry over is the sharing: four fifths of the shared nodes need a different answer under different parents.

 Three complete solutions, each asked about the others. Bouton's 1901 criterion for Nim, Wythoff's 1907 description of his own cold positions, Moore's 1910 rule for taking from several heaps, and the Grundy criterion that arrived thirty years later, each checked against the truth on every position of four games. Every one of the old criteria is exact about its own game and wrong about the others. The blanks matter more than the numbers: Wythoff's is a description of a pair and has no form for three heaps at all, and the Grundy criterion has no form for a game whose moves touch several heaps at once. How it was found

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.

 A vocabulary that is not closed under its own arithmetic. Every pair of named values added together, with the answer sorted by whether it has a name. The named vocabulary covers every game born by day two and one in twenty-three born by day three, and coverage is the wrong measurement: the notation exists so that positions can be added. A sixth of the sums of two named values at day three cannot be written without opening a brace, and the first one to escape is a sum of two of the symbols anybody learns first. How it was found

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.

 Two numbers instead of an expression. The mean and the temperature of a position, set against its brace expression. The pair is readable in a way the expression is not and is half its length, and it is not exact: most of the games born by day three share a pair with some other game. The cost is not abstract — two games written the same way here can be separated by adding an ordinary small position to each, which is exactly the test a table of outcomes fails one level down. How it was found

What two numbers cannot tell apart

A thermograph summarises a position in a mean and a temperature — nine characters against the brace form's twenty-two, and readable in a way the expression is not. It is also not exact: 1,454 of the 1,474 games born by day three share a pair with some other game, 291 of them share one pair, and adding a star to two of those gives different winners.

 The Bridg-It board of size 3, both players at once. A Bridg-It board of size 3: blue dots in 4 rows of 3, red dots in 3 rows of 4, interleaved. Every bridge blue can usefully build is drawn in blue and every bridge red can usefully build in red, and each blue bridge crosses exactly one red one. Blue's switching graph and its planar dual have the same numbers of points and links, because the dual is red's board turned a quarter. Out in the world

Cut is Short on another graph

Everything proved about the switching game is proved from Short's side, and Cut appears only as the player whose moves get enumerated. On a graph drawn without crossings Cut does not need a theory of its own: deleting a link is securing the link that crosses it in the dual, so Cut's game is Short's game on a different graph. Bridg-It is the board that is its own dual — one link short of two trees at every size, which is why its first player wins.

 Writing a board as a sum, and as the value it is. Every sum of two, three and four games born by day two, written as the parts joined by plus signs and as the single value the sum equals, with the number of distinct values, the share that can be written without a brace, the share longer as one value than as a sum, the middle length each way and the longest single value. The single value is shorter in the middle and far longer at the top, and needs a brace more often the more parts there are. How it was found

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.

 Every two-position loopy region, as two names. The 256 loopy regions of two positions, placed by the names of their onside and offside as identified against the 1,474 values born by day three. Ten names cover every side: 0, 1, −1, ∗, on, off, over, under, upon + ∗ and −upon + ∗. The largest groups are on & off with 94 regions and off & off and on & on with 53 each; 25 regions need only finite names. How it was found

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.

 A turn is not a bit. The number of turns a game lasts, beside the number of quantified bits those turns amount to. Each ply is measured over the positions actually reachable at it rather than along one line, and the bits are the logarithm of the branching, which is what a quantifier prefix would need one of. What it costs

A turn is not a bit

The prefix a game is read as gives each player one quantifier a turn, and a turn on a board is a choice among however many moves there are. Nim on heaps of 3, 4 and 5 lasts twelve moves and carries 23.6 bits of choice; a Toads and Frogs strip lasts eleven and carries two. Corrected for that, the model predicts a strategy 539 times too large on one board and 67 times too small on another, and the two failures have different causes.

 Every chance the coin gives, by day 2. The probabilities the coin produces over all the values born by a given day, drawn on the unit interval. They fall on a grid of dyadic fractions, every interior point of it is reached, and the two ends never are — so no position is ever a certainty under random turns. Where it stops

Every chance but a certainty

The coin's number lands on a grid of dyadic fractions, and which points of that grid arrive is a count rather than a guess. Over the 1,474 values born by day three it reaches every one of the fifteen interior sixteenths and neither end — no position is ever certain. The groups sharing a chance run 1, 2, 4, 8 on the small pool, which looks like doubling, and 1, 2, 4, 20 on the large one, which is not.

 Every region of three positions, counted. The 262,144 graphs on three positions reduced to the regions that are genuinely three positions with a cycle in them, and then split by whether the two-position vocabulary has a name for both of their sides. How it was found

Four thousand nine hundred regions with no name

Two positions give 256 regions and ten names cover every side of all of them. Three positions give 262,144 graphs, 110,934 genuine loopy regions — and 4,931 of those have a side that no name in the two-position vocabulary reproduces, with 3,990 of them named on one side and blank on the other. The count the earlier essay left open comes back in the affirmative.

 The guess, and what it covered. Two attempts to name the leftover sides out of the old vocabulary: every pair of the six stoppers, and every two-position region that is a stopper, each with small finite games added. Both cover nothing, and the count of distinct leftovers is what remains. How it was found

The names are not built out of the old ones

The guess was that a three-position region's missing names would be sums of two loopy ones — on plus over, and that family. Built and tried, every pair of the six stoppers covers none of the 4,931 regions that need one, and so does every two-position stopper there is, all seventy-nine of them with small games added. Thirteen names have to be invented, and forty-eight cover the whole census against ten at two positions.

 The formula as the limit of periodic games. Lasker's Nim above eight versions of it with the number of counters that may be taken bounded at one to eight. Each bounded game is periodic and agrees with Lasker's formula on its first few heaps; the region of agreement grows with the bound. Impartial games

The formula is a limit

Cap the take in Lasker's Nim at k counters and the game is a finite rule table, 4.33…3, whose Grundy sequence repeats with period k + 1 rounded up to even and follows Lasker's formula until the cap bites. The formula is what those periods converge to. And the same column of codes, with a free split in front, holds Kayles itself: the rule 4.4 on a heap of n + 1 is Kayles on a row of n.

 The reversing move is the follower's. For each follower under which some gift horse escapes domination, the number of escapes, how many are reversed by Right's move inside the follower, and how many by a Right move in the base part. The follower's move reverses every one. Sums and comparison

The follower does the reversing

The gift-horse theorem for the ordinal sum needs two cases, and the second — the added option is reversible — was counted and not described. Recorded move by move, the reversing answer is always Right's move inside the follower: on all 410 escapes under five followers, and on every one of the 2,628 gift horses under every follower that gives Right a move at all. The case split is by follower, not by horse.

 What survived, and what did not. The gift-horse theorem, the two-case proof and the one-line description of the reversal case, each scored on the day-three sweep and its mirror and on the day-four sweep and its mirror. Sums and comparison

The split slips one day deeper

The reversal case of the gift-horse theorem was described in one line — the follower's own move reverses every gift horse, whenever the follower has one — and tested only where it was found. In the mirror it holds exactly, with 1 and −1 trading places. One day deeper it fails: under ↑ and ½, three gift horses on built day-four bases are not reversed by the follower's move. All three are dominated, so the theorem stands; the clean split by follower does not.

 Four positions, sampled. Three samples of three thousand loopy regions on four positions, drawn with each possible move present at a chance of one half, about a third and a quarter. For each: how many regions have both sides named by the thirty-five names two-position regions use, by those together with the thirteen invented for three positions, and how many need a new name. How it was found

Four positions, sampled

Ten names write both sides of every loopy region of two positions, and forty-eight every region of three. Four positions are over four billion graphs and cannot be counted, but they can be drawn. Three thousand regions at each of three densities: the forty-eight names cover between 95.9 and 99.5 per cent, the thirteen names invented for three positions come back at four almost all of them, and the sparsest sample meets thirty-five sides nothing earlier reproduces — a floor of eighty-three names, and a curve that grows by accretion rather than collapse.

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