An effect that changes sign

Room pulls two ways settled what sets the temperature of a shared Amazons region: the distance between the two amazons takes 30 per cent of the variation, the count of squares both can still reach takes another 17 on top of it, and more shared room makes a position colder rather than hotter. It closed on the refinement it had not made:

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.

Three in a line does not behave like three scattered — 0.63 against 2.51 — and a shared square beside a burnt one is worth more rather than less. The arrangement does about as much work again as the count. And unlike everything below it on this ladder, it cannot be said in a sentence, because its effect changes sign inside the population.

Six ways to say where #

The instrument is the rung below’s, unchanged: group the 2,088 shared $3 \times 3$ regions by a set of quantities and ask how much of the temperature’s variance the grouping removes, with a random grouping of the same size priced beside it so that a score bought by having more groups shows up as one.

What is new is the quantities. Six descriptions of the shared set, none of them a count of how large it is: how far apart the two furthest shared squares are, how many pairs of them touch edge to edge, how many touch at all, whether the middle square of the board is among them, whether they are collinear, and how many sit beside a burnt square.

Every one of them beats the count alone. Distance and count together take 47.5 per cent; adding one description takes it to between 51.8 and 56.3, against controls of about one per cent. All six together take 77.1, against a control of 2.9.

So the answer to the rung below’s first question is unambiguous. The arrangement of the shared squares carries roughly as much information about the temperature as their number does, and that information is not being bought with extra groups — 57 groups filled at random remove three per cent of the variance where these 57 remove seventy-seven.

One thing about the population needs saying, because it is what makes the grouping test honest here. A shared region is a $3 \times 3$ board with one amazon of each colour, at most two burnt squares, and every free square connected — 2,088 of them, each with its temperature computed from its own thermograph rather than estimated. The distance takes two values on such a board and the shared count takes six, so the rung below’s account had eleven groups; adding a description takes it to between twelve and twenty-nine. Those are small numbers, and small numbers are what makes the control meaningful: a grouping with 2,000 groups would explain everything and mean nothing, and none of these is remotely near that.

Three in a line #

The rung below’s own example is the sharpest single case in the sweep.

At a distance of two with exactly three shared squares, 48 regions have those three collinear and 208 do not. The collinear ones have a mean temperature of 0.63; the rest have 2.51. Four times, on a population where the count of shared squares — the quantity the rung below settled on — is identical throughout.

At a distance of one no region with three shared squares has them collinear, so the comparison cannot be made there, and the page says so rather than reporting the one side it has.

The mechanism is visible once the geometry is looked at. An amazon moves like a chess queen, so three squares in a line are three squares on one ray, and taking the near one blocks the two behind it. Three collinear shared squares are therefore worth far less than three shared squares in general position: they are nearly one square, contested once, rather than three contested separately. That the temperature falls to a quarter is a much stronger version of the rung below’s more shared room is colder — the squares that are collinear were never really separate room at all.

Beside a burnt square #

The rung below guessed that a shared square next to a burnt one would be worth less than one in the open, presumably on the reading that a burnt neighbour is a restriction and a restriction is a loss.

It is worth differently, and at a distance of one it is worth more. Regions with one shared square beside a burnt one have a mean temperature of 0.12; with two, 0.26; with three, 0.54. More burnt neighbours, hotter region.

The reason is the same reason more shared room is colder, read from the other end. A burnt square removes moves from both amazons, and what makes a position hot is that a move cannot be answered. Two amazons contesting squares hemmed in by burnt cells have few alternatives, and few alternatives is exactly the condition under which the first move decides something. The burnt squares are not restricting the value; they are removing the answers.

The sign that does not hold #

And then the finding that stops this from being a fifth term in an account.

Take how tightly the shared squares sit together, counted as pairs that touch. Hold the distance at one and the count at three: the mean temperature goes $-0{\cdot}11$, $0{\cdot}44$, $0{\cdot}72$ as the clumping rises. Hold the distance at one and the count at four: $-0{\cdot}43$, $0{\cdot}25$, $0{\cdot}36$. Clumping makes a region hotter.

Hold the distance at two and the count at four: $2{\cdot}13$, $1{\cdot}84$, $1{\cdot}08$. Clumping makes a region colder.

Same quantity, same board size, same census, opposite signs — and the switch happens between adjacent values of the distance, not at some far edge of the population.

That is a different kind of result from the four rungs below this one. The distance sets the temperature. More shared room is colder. A one-sided region is worth its count of free squares. Each of those is a sentence with a direction in it, and each is the reason its rung was worth writing. This one has no such sentence. Clumping predicts, and what it predicts depends on something else.

What a player is actually deciding #

It is worth stepping back to what any of this is for, because a temperature is not a move.

An Amazons board late in a game is a collection of independent regions, and a player choosing where to move is choosing which region to fight in. Playing the hottest is the standing heuristic — move in the hottest component — and it needs a temperature per component, which is what this ladder has been trying to supply without evaluating the component.

Read that way, the six descriptions are a proposal for what a player should look at when glancing at a region: how far apart the two amazons are, how many squares both can reach, and whether those squares are strung out on a ray or bunched. The first two are quick to see and the third is quicker — a line is a line. So the reading is cheap in the way a heuristic has to be, and the previous paragraph’s caveat is the one that bites: it is cheap to evaluate and expensive to state, because what to do with the answer depends on the distance.

Why that matters more than the percentage #

A quantity whose effect reverses inside the population cannot be a term in an additive account. Every reading on this ladder so far has been additive in shape — a base value from the distance, adjusted by the shared room — and a term that is positive on half the population and negative on the other half cannot be written that way at all.

What it can be written as is an interaction: the shape matters, and how it matters is a function of the distance. That is a perfectly respectable object and it is a much worse thing to have to explain to a player. Count the shared squares; if the amazons are adjacent, prefer them spread out; if they are two apart, prefer them clumped is not a rule anybody will carry.

It also explains something about the rung below, in retrospect. Distance and shared count together reached 47.5 per cent and the rung below reported that as a good account. Half the remaining variance is the arrangement, and the arrangement was invisible not because nobody had thought of it but because any average over the population would have found it to be nothing: the two signs cancel. A quantity that reverses is a quantity a correlation cannot see, and grouping tests can see it only because they never average across the groups.

That is the transferable half of this page. A threshold is a detection limit found a quantity that was measuring the instrument; this is a quantity that would measure as nothing under the wrong instrument and measures as a great deal under the right one, and the difference between the two instruments is whether they hold anything fixed.

What a picture of this would have to show #

Every figure on this page is a table of means, and it is worth saying what the missing picture is, because on this site it usually exists.

It would be two $3 \times 3$ boards side by side: same two amazons, same distance apart, same number of shared squares, one arrangement collinear and one not, with the two temperatures underneath. That figure is entirely drawable — the boards are nine squares — and it would carry the collinear result better than a table of means does, because a reader can see three squares on a ray and understand immediately why taking the near one ends the argument.

What it cannot carry is the reversal, and the reversal is the finding. A sign that flips between two values of the distance is a statement about two populations of a hundred boards each, and the honest picture of it is exactly the table printed here: a column of means with the direction changing halfway down. A region one player owns is where the boards themselves get drawn on this anchor, and it is the right page to look at one.

What this does not settle #

Three by three, with at most two burnt squares. The 2,088 regions are the rung below’s population and the census does not reach a board where an amazon has room to manoeuvre. Whether the sign reversal survives on a $4 \times 4$ region is the obvious question and it is a much larger sweep — and the reversal happening between distance one and distance two on a three-square board is a suspicious place for a boundary, because two is nearly the whole board.

The six descriptions are correlated with each other. Adding any one takes the account to roughly the same place, which is what happens when six quantities are six views of one thing. All six together reach 77 per cent, which is much more than any one, so they are not identical — but the page does not claim six independent effects and the sweep cannot separate them.

The burnt-square direction is measured at two cells. One at distance one and one at distance two, and the second is not monotone — 2.75, 0.25, 1.90, 1.48, 1.63 as the count of burnt neighbours rises, on cells holding as few as eight regions. The claim this page makes is about the distance-one column, where the sample is 288 regions and the trend is clean; the distance-two column is reported and not interpreted.

And 77 per cent is not a formula. Fifty-seven groups with a mean temperature in each is a lookup table with a good score, and there is no expression here. A region one player owns is what a formula looks like on this anchor — a one-sided region is worth its count of free squares, exactly — and nothing on this page is that.

The temperatures are exact and the means are not the point. Every region’s temperature comes from its own thermograph, computed by the ordinary recursion. What is being averaged is exact numbers; the averaging is the approximation, and it is why the direction rather than the size of each effect is what this page reports.

The collinear result is one distance. Forty-eight regions against 208 is a comfortable sample, and it is a sample at a single distance with a single shared count, because that is the only cell of the table where both kinds occur. A result measured in one cell is a result about that cell, and the mechanism offered for it — that three squares on a ray are nearly one square — predicts it should hold wherever the two kinds coexist, which is a prediction and not a measurement.

Normal play, one amazon each, one region. When a real board falls apart is the measurement of how often a played game produces a shared region at all, and it is what decides whether any of this is about Amazons or only about a census.

One number the ladder now has #

Four rungs of this anchor have been chipping at the same question — what a shared region’s temperature is a function of — and it is worth writing down where the account stands, because the fractions have moved a long way.

The distance alone reaches 30 per cent. The distance and the shared count reach 47.5. Everything measured here reaches 77.1, against a random control of 2.9. So a little under a quarter of the variation in a shared region’s temperature is still unaccounted for by anything anyone has thought to measure, and the remaining quarter is where a formula would have to come from.

That is a respectable place for an anchor to be after seven rungs, and it is worth contrasting with where it started: the fractions that were not there opened by discovering that 1,452 of the supposed fractions were an artefact and only 56 were real. The ladder began by correcting a census and has arrived at three quarters of a prediction, which is the ordinary shape of progress on a game nobody has solved.

Where the ladder goes next #

The amazons anchor has seven rungs: the game, Amazons on one line, the fractions that were not there, a region one player owns, what a real board falls apart into, which reading of the room decides how hot they get, and now what the room’s shape decides.

The rung above is the reversal itself. A quantity whose sign flips between two adjacent distances is either an artefact of a very small board or a real interaction, and the two are distinguishable: on a $4 \times 4$ region the distance runs to three, so there is a third point on the curve, and a sign that flips once and stays flipped is a different object from one that oscillates. That sweep is perhaps eight times this one and needs no new machinery — the same reachable sets, the same descriptors, the same grouping test — and it would say whether this page has found an interaction or a boundary effect.

Two neighbours are worth the trip. Room pulls two ways is where the count of shared squares became the second quantity and where more room is colder was established, and it is the sentence this page could not add a fifth clause to. And the fractions that were not there is where this census was cleaned up and where the temperature became the thing worth predicting, and it is worth reading beside a result about how much of that temperature is geometry.