Combinatorial Game Theory
who wins, and by how much
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The thread: The notation is not the position — page 3

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.
No fifth value. Forms of day-three values built by adding day-three gift horses, and the ordinal sums they give. Over eighteen thousand forms and four followers, no value's forms disagree. Sums and comparison

No fifth value

The colon reads a form rather than a value, and the rung below found the forms of a value disagreeing at exactly four of them — the values born by day one. It could only check forms whose options came from day two. Built one day deeper, by adding day-three gift horses to day-three values, eighteen thousand forms give no disagreement at all, while the same treatment still splits nought four ways. The class is about the width of the base's form and not the depth of its options.

 The error is half the follow-up. Each bent value's true temperature, the temperature the stop reading gives it, the difference, and half the temperature of its hot option. The last two columns agree on every value. Values

Half a follow-up out

The rung below settled which values the stop reading is wrong about — the ones whose walls bend — and left the size of the error unmeasured. It is not bounded by anything readable off the diagram; it equals something readable off the diagram. On all thirty-two, the mean and the temperature are each out by exactly half the follow-up's temperature, and the temperature is always read too low.

 The trend, running backwards. The excess grouped by the width of the value's canonical form. Values with wide option lists are exhibited nearer their birthdays than narrow ones, which is the opposite of what the proposal predicted. Values

Wider costs less

The rung below found the cheapest exhibit of a value never smaller than its birthday, exactly equal on two thirds, and the ruleset explaining 40 per cent of the rest. The variable it proposed for the remainder was the width of the form. Width and excess correlate at −0.39: the wider the value, the closer to its birthday it is exhibited, and inside a ruleset the relation cannot even agree on a sign.

 A numeral in the empty squares. Runs of coins with one to five empty squares in front, and the value of each. Every row is a binary expansion converging on a fraction the colours determine. Particular games

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 correction on eleven times the pool. The stop reading and the corrected stop reading scored against the true temperature over every non-number value born by day three. The correction was measured on thirty-two values and holds on three hundred and forty-eight. Values

A second level of stops

The rung below found the stop reading's error to be exactly half the follow-up's temperature and asked whether the correction survives a wider pool, survives two bends, and can be stated without a thermograph. It survives eleven times the pool, missing two values in 1,459. It needs no thermograph — the follow-up's temperature is half its own stop gap. And it does not survive two bends, because day three contains no value with two of them.

 The excess is not a flat fee. The excess fitted against the birthday inside each ruleset with enough values to fit a line. A fee would have a slope of nought; every slope here but one is negative, so the excess is largest on the values born earliest. Values

The entry fee was the cap

Two rungs measured how much bigger a position has to be than the value it exhibits, and attributed what was left to the ruleset — Toads and Frogs paying 2.25 squares on everything, green Hackenbush paying nothing. Neither number is a property of the rules. Inside every ruleset the excess falls as the birthday rises, because the sweep's size cap censors exactly the values that would pay most — and three squares past the cap, Toads and Frogs exhibits values born later than the strip is long.

 The digits they share. The condition satisfied by eighteen of the nineteen codes that climb by three. It says that splitting a heap into three is available on exactly one take and buys nothing else. Impartial games

The only way to split into three

Nineteen three-digit hexadecimal codes climb by three, and the rung below asked whether they share a form and what digits they have in common. The digits are exact: on eighteen of them the only way to split a heap into three is by taking exactly three counters, and taking three counters can do nothing else. The form is not shared — the eighteen carry four distinct sequences, and exactly one of the four counts in base three.

 Running products, and where to stop. Six Maundy Cakes with the prime factors of the longer side, the running products those primes make, and the value the sum of them gives. Particular games

The short side only says how many

The rung below settled which cut to make in a Maundy Cake and left the value open. With the cut settled the recursion is a walk, the walk unrolls, and what it unrolls into is the running products of the long side's prime factors, largest first. The short side never enters the products at all — it decides how many of them there are and nothing else, so sixty-two different short sides give one value.

 Every gap dies at the last run's rate. Three-run Push strips with each gap widened in turn, and which of four candidate rates the convergence matches. The rearmost run's rate wins every family and the compound rate wins none. Particular games

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 wall as an envelope. A thermograph with each option's contribution to its wall drawn over it, built from that option's two stops alone. The wall is the envelope of those contributions and it bends where the envelope has a corner. Values

The bend is in the stops

The rung below reduced the whole stop reading to one question — does this wall bend? — and asked whether that could be answered from the options' stops instead of from a diagram. It can, in four lines, and it gives more than the bend: on all 1,459 non-number values born by day three the options' stops determine the entire thermograph. One day deeper it breaks, and every failure is a value with a bent-walled option.

 Two orders of magnitude. The share of positions of a size whose value is one no smaller position exhibits. Every Hackenbush string is a new value and fewer than one Clobber row in a hundred is. Values

The rate was the alphabet

The rung below asked for a quantity a size cap cannot censor and proposed the rate: how many new values a ruleset produces per extra square. The rate is honest and it measures the notation — every ruleset grows at close to the number of symbols its positions are written in, and the seven span less than a factor of two. What separates them is the yield, which spans a hundred and nineteen.

 Thirty-two words, four of them lost. Every position of five heaps grouped by the parities of its heaps in decreasing order of size. Each word is uniform, and four of the thirty-two are losing. Impartial games

The parities, in size order

The rung below settled four of six parity classes in bounded Moore's Nim and asked whether the sizes pick out the losing positions in the two it could not. They do — but only through the order they put the parities in. Sort the heaps largest first, read off their parities, and that five-bit word settles the whole game at every width of move, with the losing words forming a subspace.

 Two counters, not one. The four periods of the odd-saltus class against the two base-three counters, with which each follows. Impartial games

Two counters, and one displaced term

The rung below found four Grundy sequences in the odd-saltus class and asked which term each displaces and whether the digits predict it. They do — but there are two base-three counters and not one, chosen by whether a heap of one can be taken away. And there are three sequences rather than four: the fourth is the third with three isolated values, and was counted separately because its period had not settled.

 How many levels, and how often. Every value in both pools by the number of levels of the recursion its thermograph needs before the stops suffice. Values

A bend that never reaches the surface

How many levels of the recursion a thermograph needs before its stops suffice is a number attached to a position, and the rung below conjectured it was the depth of the deepest bend in the tree. It is not: on 124 values a bend one level down costs nothing at all. What the number counts is the longest unbroken chain of bends running down from the top, exact on 2,400 of 2,403.

 Same runs, different values. Three groups of Domineering regions sharing a run-length multiset, with the value of each member. Particular games

Where the runs meet

A Domineering region's value interval is a function of its run lengths and its value is not — twenty-one groups of shapes share a multiset and disagree. The crossing count separates none of them, and neither do ten other local statistics, fifteen sets of which agree on everything and differ in value. What separates nineteen of the twenty-one is where along its runs each crossing sits.

 Not rare at all. How many hexadecimal codes whose sequence settles have a stretch of heaps before the pattern begins. Impartial games

A pattern that has not started yet

A pre-period was supposed to be rarer in this family than a defect. Two hexadecimal codes in five have one, 321 have a pre-period longer than their own period, and the code the rung below found slow takes fifty-four heaps to settle rather than two blocks — which is also the account of three defects the rung below recorded and could not explain.

 Two claims read as one. The proposed geometry separated into the claim the rung below established and the claim it needs but did not. Temperature

The four paragraphs prove something else

Three rungs earned the right to write the crossover law's proof as two straight walls meeting where the law says. The walls are straight — one of them everywhere, for a trivial reason, and the other only above the answer's own temperature. The crossover sits below that height, so the geometry holds nowhere the law is about, and where it does hold it proves the temperature instead.

 The dual is the value table's span. The dual code against the span of the bit-planes of the one-coin Grundy values, on every game measured. Impartial games

The dual was the value table

A coin-turning game's losing rows form a linear code, and a code has a dual that nothing in the game appeared to read. It reads it constantly: the dual is spanned by the bit-planes of the Grundy values — the parity checks are the value table stood on end — and on Mock Turtles over eight coins the losing rows are exactly the span of the table that decides them.

 Two statements, two routes. The two measured identities the rung below left unproved, with the argument each was expected to need. Impartial games

One proof, and one wrong lemma

Two measured identities were left for a proof: the move rule by induction, the gap condition from the reply bound. The induction is exact on 31,731 heaps at eight factors. The reply bound holds at c = 2 and on one index pair in twenty-seven at c = 3 — and the inequality that does the work is a third one nobody proposed.

 Four readings, one game. The closed form, the strict mating, the cancelling matching and the parity term, with how much of the game each accounts for. Impartial games

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.

 Two of three are subadditive. The birthday, the option count and the written length, each tested for subadditivity under the disjunctive sum. Values

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.

 A factor, and it grows. The ratio between reducing and deciding, by the number of options the form carries. Values

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.

 One set, described three ways that share no arithmetic. Wythoff's cold positions can be stated as the Beatty pairs of the golden ratio, as a greedy construction over the integers that mentions no constant, and as a condition on Fibonacci numerals. None of the three consults the game. The fourth column is the game — a mex table over the moves — and all four name the same set of cold pairs over the whole square, which is what the figure counts. Out in the world

A set with three descriptions, and a function with none

Wythoff's cold positions can be written three ways that share no arithmetic — an irrational constant, a greedy rule, a condition on Fibonacci digits — and all three are exact. The same game's Grundy values have no closed form at all. Both facts are about one table, and the gap between them is the subject.

 The offsets, and what they separate. The junction descriptor of each member of two split groups, beside the value each holds. Particular games

Three distances too many

The junction descriptor records how far a crossing sits from four ends, and the rung below asked what the value does when one crossing slides along its run. It reads one bit — the offset's parity — and only when the run has odd length. The other three distances reach the value not at all.

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