The fractions that were not there

A region one player owns established that an Amazons region with only one player’s amazons in it is worth exactly its count of free squares — on a strip, on a board, with one amazon and with two, without exception. It set that against the mixed case in a sentence: of the 2,088 three-by-three regions holding one amazon of each colour, “128 are integers and 1,452 are fractions”. And it closed by asking which:

A shared region is a fraction 70 per cent of the time and this ladder has never asked which fraction — whether the denominator is set by how many moves separate the two amazons … The 1,452 fractions are in hand with their positions, and grouping them by denominator is a filter over data already computed.

The filter was run. There are fifty-six fractions.

What the 1,452 were #

The rung below classified a value by testing its printed form. A value counted as an integer if the whole string was digits, and as a fraction if the string contained a slash anywhere. That second test is the one that went wrong, because a hot Amazons value looks like this:

$$\{4 \mid -2, \{0, \{0 \mid -1/2\} \mid -3\}\}$$

which has a slash in it, is a genuine fight, and is not a fraction by any reading. Thirty-two of the 2,088 regions carry exactly that value.

Asking instead whether the value is a number — the whole string a bare integer or a bare fraction — gives 128 integers, 56 fractions, 156 nimbers or numbers-with-a-star, and 1,748 hot positions. The 1,452 with a slash somewhere are mostly the last group.

This is worth stating plainly rather than tidying away, because the mistake had consequences beyond a wrong count. It made the shared region look like a number of a slightly awkward sort, and the natural question about a fraction is which fraction — so the closing paragraph asked for a denominator. The population has almost no denominators in it. It is a population of fights.

They really are fights #

Eighty-three per cent of the shared regions are hot, with a mean temperature of 1.74 and a mode at two and a half. On a nine-square board holding two amazons and one or two burnt squares there are five or six free squares in play, so a temperature of two and a half means that moving first in the region is worth nearly half of everything left in it.

That is a much stronger contrast with the one-sided result than the rung below drew. A one-sided region is an integer — a bank of moves with nothing to contest, which neither player ever wants to move in first because numbers avoid numbers. Put one amazon of the other colour into the same region and it stops being a bank and becomes the hottest kind of position the board has. The transition is not from an integer to a fraction; it is from a number to a fight.

What the separation does set #

The instinct behind the rung below’s question was that the distance between the two amazons ought to govern something. It does, and the something is the temperature.

On a three-by-three board two amazons are either touching — Chebyshev distance one, meaning one queen move would reach the other’s square — or two squares apart. With them touching, 72 per cent of the regions are hot and the mean temperature of the hot ones is 1.24. With them two apart, 97 per cent are hot and the mean is 2.20, three quarters again as high.

The mechanism is the one thing Amazons players talk about. An amazon shoots after it moves, and the shot is what closes space; two amazons standing next to each other have already spent most of the room between them, so what is left is a small number of moves that neither player can take from the other. Two amazons with space between them are each still able to shoot into that space, and every shot changes what the other one can reach. A region is hot exactly to the extent that the two amazons can still interfere with each other, and the distance between them is a direct reading of how much room there is to interfere in.

Why a string test is a trap here #

The mistake is easy to make and worth anatomising, because the conditions that produce it are common on this site.

A canonical form is printed as a nest of braces, and the printer writes every number inside it in the usual way — so a value four levels deep containing a single quarter carries a slash in its printed form. Testing for a slash therefore asks does a fraction appear anywhere in this position’s tree, which on a partizan game with any depth to it is nearly always yes. The test is not merely imprecise; it converges to “yes” as the positions get bigger.

The site’s own rule covers this and it was half followed. Deciding whether a game is a number by walking it is expensive, so reading the answer off the printed form is allowed and is what the ladders do — but the reading has to be anchored: a value is an integer when the whole string is digits, and a fraction when the whole string is a fraction. An unanchored test on a nested form asks a question about the subpositions and reports it as a question about the value.

Anchored, the two tests agree exactly. Over all 2,088 regions the anchored string test and the semantic one name the same 184 numbers — and there is a second trap in the semantic one, which is that it has to be handed the canonical form. On the raw form the evaluator returns, it calls not one of the 2,088 a number, including the eighty regions worth nought. The census here runs both and refuses to build if they ever disagree.

There is a smaller lesson in the numbers too. The rung below’s 128 integers is right: an anchored test and an unanchored one agree on integers, because a form with braces in it can never be all digits. Only the fraction test was loose, which is why nothing looked wrong — one column of the classification was exact and the other was three per cent right.

Where the fractions are, then #

The rung below’s question is not unanswerable — it just has a much smaller subject than it thought.

All 56 fractions have the amazons touching, and their denominators are 2 and 4 and nothing else: 32 halves and 24 quarters. Not one region with the amazons two apart is a fraction.

So the answer to which fraction is that the denominator is not indexed by the separation at all — it cannot be, because at the only separation that produces fractions there is one value of the separation. What the separation predicts is whether a region is a fraction at all: 56 of the 1,160 touching regions are, and none of the 928 separated ones is.

The reading is the same as the temperature’s, one step further along. A fraction is a cold position with a little left in it — a half or a quarter of a move — and a region only cools that far when the two amazons have almost finished shutting each other in. Halves and quarters are what is left of a fight, and there is no room for eighths on a board this small.

What the one-sided result looks like from here #

Reading the two rungs together gives a sharper statement than either makes alone, and it is worth setting out because the ladder has now measured both halves of the same census.

A one-sided region is an integer: worth its free-square count exactly, with no exception found in two dimensions or one. A shared region is, five times in six, not a number at all, and when it is one it is a small fraction sitting at the near end of the distance scale. There is no middle: the census contains no one-sided region that is a fight and almost no shared region that is a bank.

That makes the presence of a second colour a phase change rather than a complication, and it says something about what an Amazons player is doing. Shooting is the move that turns a shared region into two one-sided ones, and the moment it succeeds the region stops being worth playing in — it becomes a count that will be collected whenever the rest of the board runs out. So the whole of Amazons strategy inside a region is about the transition, and when a real board falls apart measures how often it happens on a played board.

It also explains why the site’s Amazons figures are so much more expensive than its Domineering ones. A number can be added to another number in constant time; a hot value has to be carried around as a form, and a board of several shared regions is a sum of several hot games. The board falls apart is where that cost is priced for Domineering, and Amazons is the same argument with the regions staying hot for longer.

Counting a printed form is measuring the printer #

The error corrected here is worth stating as a general hazard, because it is available to any census taken over rendered output and it looks exactly like a census taken over objects.

A value is a game. A printed value is a string produced by a function that has choices to make: which form to print, how to abbreviate, whether to write $\\{1 \\mid 0\\}$ or a name for it, whether a nested option is expanded or summarised. Counting a feature of the string counts a property of that function, and the function was written for legibility rather than for classification — which is the standing complaint against reading a value off its notation, arriving here as a measurement error rather than as a reader’s.

The specific failure here is a regular expression looking for a slash. A slash appears in a printed value whenever a fraction appears anywhere inside it, at any depth — so a hot position with a fraction three levels down is counted, and a position that is a fraction is counted the same way. The two are as different as a value can be from another, and the string does not distinguish them.

The repair is not a better pattern; it is asking the object. isNumber on the canonical form answers the question the count was for, in one call, with no dependence on how anything is written. Fifty-six against 1,452 is the size of the gap between the two methods on this one pool.

That is worth generalising into a rule for this site. A census must interrogate values and never their renderings, and a count whose implementation contains a regular expression over printed output is a count of something else. The printer is allowed to change; the values are not.

What this does not say #

Two amazons, not four. A real Amazons game has four of each colour, and a region holding two of one colour and one of the other is a different object again — with a shared region the count of amazons on each side is part of what the temperature depends on, and the sweep here fixes it at one apiece.

The board is three by three. Two amazons on nine squares can be one or two apart and no further, so distance has two values here and the trend across them is two points. On a five by five the distance runs to four and the same measurement would have a shape rather than a direction; that sweep is out of reach of an exact evaluator, since a five-by-five region with two amazons is a game tree of some millions of positions.

Two denominators is a fact about five free squares. A position with five or six free squares cannot carry a value with a large denominator: the denominator counts how finely the position can be split, and there is not much to split. A larger board would produce eighths and sixteenths and the question of what indexes them would become real again.

The reclassification does not overturn the rung below’s result. Its finding was about one-sided regions and is exact: every one of them is worth its free-square count, verified exhaustively. What is corrected here is a contrast drawn in its closing pages, and the correction strengthens the contrast rather than weakening it.

The temperature figure is a mean over the hot ones. A cold region has no temperature on this site — a number is given $-1$ so that a sum’s stack of temperatures holds only its genuinely hot parts — so a mean over everything would be a mean over two different kinds of quantity. What the distance table reports is the share that are hot and the mean temperature of those, kept apart, and both move in the same direction.

And a temperature is not a strategy. Knowing a shared region is hot at two and a half says a player should move there before moving in a cold one — which is the hottest-first rule, and that rule has a guarantee and is not exact. It says nothing about which of the region’s moves to make, and on a board where every move is also a shot the second question is much the harder one.

The convention, named #

Normal play throughout: a player who cannot move loses.

Amazons is played with queens that move as chess queens and then shoot an arrow, also along a queen line, from the square they land on; the arrow burns its square permanently. A region is a connected set of free squares; the sweep here takes only positions whose free squares form a single region, since a position in several regions is a sum and the point is what one region is worth.

A shared region holds one Left amazon and one Right amazon. The population is every three by three board with one of each, at most two burnt squares, and the free squares connected — 2,088 positions, which is the rung below’s population unchanged.

Distance is the Chebyshev distance between the two amazons: the number of queen moves one would need to reach the other’s square on an empty board, which is the larger of the row gap and the column gap. It is a property of the drawing and needs no evaluation.

A value is an integer or a fraction only if the whole of its printed form is one — the correction this page is about. Hot means a temperature above nought, which is to say the position is not a number and moving in it is worth something.

Where the ladder goes next #

The amazons anchor has five rungs to here, and this one has just replaced a count of denominators with a temperature. The rung above tries to predict that temperature from the board.

Room pulls two ways tests the two obvious refinements of distance — how many squares each amazon can reach, and how many both can — and neither beats distance on its own. Together they beat it by half as much again, and the reason is the finding: they pull in opposite directions. Further apart is hotter; sharing more reachable squares is colder.

That explains why either one alone does so badly. A single quantity that moves with both effects is measuring their difference, and their difference is close to nothing over much of the pool — so a predictor built from one of them is cancelling most of its own signal before it is used. Two quantities with opposite signs need to be entered separately or not at all.

It also says something about what a shared Amazons region is. The two amazons are competing for room, and room turns out to be two things rather than one: distance is how far apart the fight is spread, and shared reach is how much of it is genuinely contested. A big region that both players can reach everywhere is cold, because neither can gain; a big region they mostly cannot share is hot, because the boundary is worth fighting over.

Which is the reading that makes the temperature the right quantity rather than the denominator this page’s count was chasing. A denominator is a fact about a value’s arithmetic; a temperature is a fact about how much of the region is contested, and that is a fact the board shows.