Room pulls two ways

The fractions that were not there found the shared 3 × 3 Amazons regions to be fights rather than fractions, and the separation of the two amazons setting how hot they are. It closed by proposing a finer reading:

The rung above is the temperature as a function of the room. Distance is one reading of how much space the two amazons have to interfere in, and it is a coarse one — a pair two apart with a burnt square between them has less room than a pair two apart in the open. The finer quantity is the number of squares each amazon can still reach, or the size of the intersection of the two reachable sets … Whether either predicts the temperature better than the distance does is the same filter over the same 2,088 positions.

Neither does. And the reason neither does turns out to be more interesting than either doing.

The coarse reading wins #

The test is a grouping rather than a fit: hold a quantity fixed across the 2,088 regions and ask how much of the spread in temperature survives inside the groups. A grouping that removes all of it has found the quantity the temperature is a function of; a grouping that removes none has found a quantity the temperature ignores.

• the distance between the two amazons — 30.2 per cent, in two groups;
• the squares both amazons can reach — 10.2 per cent, in six;
• the squares either can reach — 8.7 per cent, in six;
• the squares one amazon can reach — 8.0 per cent, in eight.

So the answer to the question as asked is no, and it is not close. The rung below’s coarse reading, which takes only two values on this board, removes three times as much of the variation as the finest of the alternatives, which takes six.

Every one of those groupings is compared against a random grouping of the same size, and the random ones score between nought and half a per cent — so the scores are about the readings and not about how many groups each one has.

And together they do much better #

A quantity that loses to another can still be measuring something the other does not, and that is what is happening here.

Group the regions by both the distance and the shared reachable squares, and 47.5 per cent of the variation goes — against 30.2 for the distance alone and 10.2 for the intersection alone. The two together are worth far more than either, so the intersection is not a worse version of the distance; it is a different fact about the position.

Distance with the union of the reachable sets does less well, at 39.5, and the two reachable-set readings together do 19.7 — less than distance alone. Whatever the extra fact is, the intersection is where it lives.

Sharing more is colder #

Reading the two-way table is where the surprise is, because the direction is the opposite of the one the phrase room to interfere in suggests.

With the amazons touching, regions sharing one reachable square have a mean temperature of 1.53; sharing two, 1.03; sharing three, 0.23; sharing four, 0.06. With them two apart, sharing one gives 2.77, and sharing five gives 1.33. Monotone at both distances, on every class holding a hundred regions or more.

So the two halves of room pull opposite ways. Further apart is hotter. Sharing more is colder. A single quantity called room cannot do both, which is exactly why no single reading of the drawing was going to beat the distance and why the pair does.

Why shared room cools a position #

The mechanism follows from what a temperature is, once it is stated in the right order.

A position is hot when moving first is worth a lot, and moving first is worth a lot when the move cannot be answered. So the question is never how much is there but how replaceable is it.

Two amazons sharing five reachable squares are two amazons with alternatives. Whichever square one of them takes, the other has four more; whichever line one shoots down, the other still has somewhere to go. Nothing either player does is decisive, so the value of moving first is small, and the position is close to a number already — which is what a low temperature says.

Two amazons sharing one reachable square are two amazons contesting exactly one thing. Whoever takes it has taken all of it, and the other has nothing equivalent. That is a large first move and a hot position.

Distance works for a different reason, and the two do not overlap. Amazons far apart have not yet begun to interfere; each still has a whole neighbourhood to spend, so there is a great deal left to fight about. Amazons touching have already interfered — the fractions that were not there puts it as an amazon that can be reached in one move is an amazon that can be shut in — and much of the fight is over.

So the distance measures how much of the contest is still ahead, and the intersection measures how concentrated it is. A grouping needs both because a temperature is a size times a sharpness.

That reading is not local to Amazons. It depends on the company is the thread about temperature being a fact about alternatives rather than about a position, and this is the same fact one level down: alternatives inside a region cool it exactly as alternatives elsewhere on the board do.

Where the fractions sit in this #

The rung below found 56 genuine fractions among the 2,088 regions and reported them all as having the amazons adjacent. The two-way table says something sharper about them.

The coldest cell in the whole table is distance one with four shared squares, at a mean temperature of 0.06 — which is to say that most of the regions in it are numbers rather than fights. That is exactly where the fractions live: touching amazons with a great deal of shared room have nothing left to contest, so the recursion settles them to a number, and the number is a fraction because the two players’ remaining moves do not quite balance.

So adjacent was the right observation and not the whole one. Two amazons can touch and still be at temperature 1.53, if the only square they both want is one square. What makes a region a number is touching and sharing, and the second condition is the one the rung below could not see because it had only the distance.

That also explains the denominators. The fractions that were not there found them to be 2 and 4 and nothing else, and a fraction with a small denominator is a position a very short way from a whole number — one or two moves of imbalance. A region with four shared squares is a region whose two players have nearly interchangeable options, so whatever imbalance survives is small, and a small imbalance is a small denominator. The rung below had the fact and not the reason.

Why a grouping and not a fit #

The instrument here is worth a paragraph, because it is the second time this site has used it and the first time it was used to settle a different kind of question.

A fit asks is this the rule — take a formula, score it, and report how often it is right. A grouping asks are these the inputs — hold the candidate quantities fixed, vary everything else, and see whether the answer stops moving. The two numbers at the top introduced it on the temperature ladder to settle whether a crossover depends on anything below the top two temperatures, and the point there was the same as here: no amount of scoring a formula answers a question about what the formula is allowed to read.

The reason it is the right instrument for this rung is that the rung below’s question is about inputs. Does the intersection predict better than the distance is not a question about a formula; nobody has proposed a formula from either. It is a question about which property of a drawing the temperature is a function of, and a grouping is the direct test of that.

What a grouping cannot do is extrapolate or interpolate, and this page pays that price twice. It cannot say what the temperature would be at a distance of three, because no region on a 3 × 3 board has one. And it reports a monotone fall as eleven numbers rather than as a slope, because a slope would be a fit and there is no reason yet to think the relationship is linear.

What geometry is worth, priced #

A measurement of what a picture buys needs a number for what the picture is missing, so the panel includes a reading nobody could take off a board at a glance: how many legal moves the two players have between them, and how far apart those two counts are.

It removes 79.7 per cent of the variation, against a random grouping of its 116 groups at 5.7. Distance and the intersection together, on 11 groups, remove 47.5.

That is the honest ceiling. Roughly half of what decides a shared region’s temperature is visible in where the pieces are, and roughly a third more is visible only in how many moves each side has — which on an Amazons board means counting queen moves and arrow shots, an amount of work comparable to evaluating the position outright. The remaining fifth is in neither.

That is a familiar division of labour on this site. A bound instead of an answer is the standard a reading of a drawing is held to, and half of a temperature’s variation from two things a player can see at a glance is a good reading by that standard — better than most of what the Domineering ladder gets from counting packings. What it is not is a substitute for the recursion, and amazons is where the reason is set out: an Amazons position has a branching factor that makes every quantity about it expensive except the ones a picture shows.

Two predictors with opposite signs #

The finding that neither refinement works alone and both work together is the standard signature of a confound, and it is worth stating in general terms because it is a hazard for every single-quantity predictor on this site.

Suppose a target depends on two quantities with opposite signs — further apart is hotter, more shared reach is colder — and the two quantities are themselves correlated, as they must be here, since amazons that are far apart on a small board share less of it.

A predictor built from one of them is then measuring a mixture. The direct effect and the effect inherited through the correlation partly cancel, so the predictor’s apparent strength is the difference between two real effects rather than either of them. It can look weak, look absent, or even come out with the wrong sign, and none of those readings is evidence that the quantity does not matter.

That is why the two together beat either by half as much again rather than by a little. Entering both separates the effects, each recovers its own coefficient, and the cancellation stops.

The diagnostic is cheap and worth running before abandoning a candidate. If a quantity that ought to matter has no predictive power, look for a second quantity that moves with it and pushes the other way — and enter both. A predictor that is useless alone and strong in company is not a weak predictor; it is one that was being asked to carry two jobs with opposite signs.

What this does not say #

Two thousand and eighty-eight regions on one board. Every position here is 3 × 3 with one amazon of each colour and at most two burnt squares, which is the rung below’s population kept unchanged so the two censuses compare. Distance takes only two values on such a board, which is a real limitation: a reading with two groups explaining thirty per cent may be doing better or worse than it would with five.

Grouping is not fitting, and that cuts both ways. A grouping test says whether a quantity’s levels have different temperatures; it says nothing about the shape of the relationship, and it cannot extrapolate. The monotone fall reported here is read off eleven group means and is not a fitted curve.

The intersection is a count, not a set. Two regions sharing three squares share three different squares in different places, and nothing here distinguishes them. That is the obvious next refinement and it is the one this page does not make.

And a mean temperature is a summary of a distribution with real width. The class at distance two sharing two squares has a mean of 2.24 and a spread of 1.11, so individual regions in it range from cold to very hot. Every claim here is about group means and none is about a particular position.

The convention, named #

Normal play throughout: a player who cannot move loses.

Amazons is played with queens that move like chess queens and then shoot an arrow, along a queen’s line from the square they land on; the arrow burns its square permanently. A region here is a connected component of free squares holding one amazon of each colour.

The distance between the two amazons is the Chebyshev distance — the number of queen moves one would need to reach the other’s square on an empty board — which is 1 when they are on touching squares and 2 when they are not.

An amazon’s reachable set is the squares it can move to in one move: the free squares along its eight lines, up to the first obstruction. The intersection is the squares in both reachable sets and the union the squares in either.

A grouping’s score is one minus the mean within-group variance of the temperature over its total variance — the share of the variation the grouping removes. It is reported with the number of groups and with the score of a random grouping of that size, because a grouping with more groups removes more for nothing.

A number has no temperature in this site’s convention and is recorded as $-1$, which is what keeps a sum’s stack of temperatures holding only its genuinely hot parts. Those positions are included in every mean here, so a class with many numbers in it reads cold, which is what it is.

Where the ladder goes next #

The amazons anchor has six rungs: the game, one line of it, a region one player owns, what a real board falls apart into, what the shared regions actually are, and now which reading of the room decides how hot they get.

The rung above is where the shared squares are. The intersection is a count and this page has shown a count is worth having; what it has not asked is whether which squares are shared matters — whether three squares in a line behave like three squares scattered, and whether a shared square adjacent to a burnt one is worth less than one in the open. That is a refinement of a quantity already computed for all 2,088 regions and it is the first reading of an Amazons region that would be about the shape of a set rather than its size. Amazons on one line is the case where that question has a trivial answer — a line has only one arrangement — which is why the two-dimensional board is where it has to be asked.

Two neighbours are worth the trip. A region one player owns is the case where the answer is exact — a one-sided region is worth its count of free squares — and reading it beside this page is the clearest statement of what a second amazon costs. And when a real board falls apart says how often a played game produces a shared region at all, which is what decides whether any of this is about Amazons or only about a census.