The third point on the curve

There is a curve running through four rungs of this ladder and it has two points on it.

Amazons on one line establishes the population; a region one player owns settles the easy half of it; room pulls two ways finds that the Chebyshev distance between two amazons — the number of queen moves one needs to reach the other — sets a shared region’s temperature better than any measure of room does; and an effect that changes sign finds that which squares are shared matters as much as how many, and that the effect of clumping is hotter at one distance and colder at the next.

All four are measured on a 3 × 3 board, where the distance between two squares is 1 or 2 and there is nothing else it can be.

Two points give a direction. Further apart is hotter is the finding, and it is consistent with a rise that keeps rising, a rise that levels off, and a rise that turns round and comes back. Nothing on a 3 × 3 board distinguishes those, and the last of the four rungs found an effect changing sign between the only two distances it had — which is precisely the observation that makes the shape of the curve worth knowing.

Distance 1: mean temperature 0.098, and 43 per cent of the regions hot at all. Distance 2: 1.428, and 87 per cent hot. Distance 3: 1.912, and 92 per cent.

The rise continues. It does not turn.

That is the headline and it is the least interesting part of what three points give, because the direction was the one thing two points could already say. What follows is about the shape.

It is worth being precise about the quantity on the horizontal axis, since it is the one thing all five rungs share. The Chebyshev distance between two squares is the number of queen moves needed to get from one to the other on an empty board — the larger of the row difference and the column difference. It is a property of the drawing rather than of the value: it can be read off a picture without evaluating anything, which is exactly what makes it worth setting a temperature against. A quantity that had to be computed from the game would explain nothing.

On a 3 × 3 board that number is 1 or 2. On a 4 × 4 it is 1, 2 or 3. That is the entire difference between the rungs below and this one, and it is the reason a board with seven more squares was worth the trouble.

What the third point cost #

The board that has a third distance cannot be evaluated, and that is worth stating before anything is read off it.

A 4 × 4 Amazons position with two amazons and nothing burnt exhausts an eight-gigabyte heap. So does one with nine free squares. Six free squares takes two tenths of a second and five takes one tenth, and the gap between those is where the wall is.

So the population here is not “4 × 4 regions”. It is 4 × 4 regions with at most five free squares, which is a restriction and is the reason this rung took a phase to arrive. There are 315,068 of them, read at a stride of 26 to give 12,000 evaluated regions, and every one of the 12,000 is a full canonical-form evaluation and a full thermograph rather than an estimate.

That restriction is not neutral and the essay is not going to pretend it is. Regions of five free squares are small, and small regions are cold — 43 per cent hot at distance 1 against a 3 × 3 sweep where more are. What the restriction cannot do is manufacture a trend across distances, because it applies equally at all three, and the check for that is the third figure below.

There is a second restriction hidden in the first and it should be said out loud. Capping the free squares at five means the amazons have at most five squares between and around them, and two amazons three queen moves apart with five free squares are in a thin, stretched region — a corridor rather than a room. So the distance-3 column is not a random sample of distance-3 positions; it is a sample of the distance-3 positions that fit in the budget, and those are systematically narrow.

That is a real limitation on how far the conclusion generalises and it does not damage the comparison, because the same budget applies at every distance. Distance 1 with five free squares is also cramped. What the curve compares is three distances under one constraint, which is the comparison a curve needs; what it cannot say is what the temperature would be at distance 3 in a region with room to spare, and nothing on this site can say that.

A rise that is running out #

The direction was already known. What a third point buys is a rate.

From distance 1 to distance 2 the mean temperature rises by 1.330. From 2 to 3 it rises by 0.484. The second step is 36 per cent of the first.

That is the finding, and it is a third possibility neither of the two the rungs below could distinguish. The sign does not flip. The curve does not oscillate. It rises and it decelerates, hard, in one step.

The 3 × 3 row underneath is the check that the two boards are reading the same curve. Their levels differ — 0.749 and 2.123 against 0.098 and 1.428 — and they should, because the populations are different: a 3 × 3 sweep allows at most two burnt squares and this one allows eleven, so its regions are larger and hotter throughout. What has to agree is the direction, and it does. The sweep refuses to return a result if it does not, because two populations disagreeing about the sign between distances 1 and 2 would mean they are not two readings of one thing and the third point would be a third point on a different curve.

What forty-three per cent means #

The hot-share column runs 43, 87, 92 per cent and it is the column to read if the mean temperature looks like a strange average.

A region with temperature nought is a number — nobody is in a hurry to move in it, because moving costs a player exactly what waiting does. Fifty-seven per cent of the distance-1 regions are like that. Two amazons standing next to each other on a board with five free squares mostly produce a position that is worth a number and has no fight in it: whatever there is to do, both players can do it in either order.

Move them one square further apart and 87 per cent of the regions become hot. That is the whole of the first step, and it is a change in kind rather than in degree — the mean temperature going from 0.10 to 1.43 is mostly the hot share going from four in ten to nine in ten, not the hot regions getting hotter.

The second step is the other thing. From distance 2 to 3 the hot share moves from 87 to 92 per cent, five points, and the mean moves by 0.48. There is very little conversion left to do — nearly everything is already hot — so the second step is the hot regions getting hotter, which is a much smaller effect.

So the deceleration has a mechanism and it is not mysterious. The first step converts cold regions into hot ones and there are plenty to convert; the second finds almost none left. A curve made of a conversion that saturates is a curve that flattens, and it would flatten whether or not distance kept buying anything.

The confound #

There is one obvious objection and it has to be answered with a column rather than a sentence.

Two amazons three queen moves apart need somewhere to be three moves apart. So a region at distance 3 might simply be a larger region, and the curve might be a curve against size wearing a distance label.

It is not. The mean free-square count runs 4.46, 4.56, 4.66 across the three distances — a spread of two tenths of a square — while the mean temperature runs 0.098, 1.428, 1.912. Twenty times the movement in one column and four per cent in the other.

The reason it comes out that way is the restriction rather than luck: capping the free squares at five compresses the size range to almost nothing, so distance is nearly the only thing left varying. The restriction that made the board affordable is the same restriction that controls the confound, which is a convenience worth naming rather than relying on silently. The sweep asserts the control: it refuses to draw if the mean free count ever spreads by more than half the cap.

Room pulls two ways is where this question was posed properly — distance against reachable squares, pulling opposite ways — and this rung does not settle that. It settles the narrower thing: on a population where room is held nearly fixed, distance alone still produces the whole curve.

Where the population came from #

The 315,068 regions deserve an account, because a population that large from a board that small is not obvious.

A 4 × 4 board has sixteen squares. Place a Left amazon on any of them and a Right amazon on any other: 240 ordered placements. Of the fourteen squares left, choose up to five to be free and burn the rest — 1,471 ways. That is 353,040 boards, and the ones whose free squares plus amazons do not form a single connected region are discarded, which is where the 38,000 difference goes. A region in two pieces is a sum of two smaller games and belongs to a different question entirely — a wall an amazon can walk through is the essay about how that connectivity has to be decided, and it is decided the same way here: across corners as well as edges, because that is how an amazon moves.

Twelve thousand of those 315,068 are evaluated, taken at a fixed stride of 26 rather than at random, so the same regions are read on every run and nothing depends on a seed. The stride is coprime with nothing in particular and does not need to be: the boards are generated in a systematic order that has no period near 26, and the distance distribution of the sample — 5,277, 4,871, 1,852 — is close to what the full population’s would be.

Nine seconds for the whole thing, against an eight-gigabyte failure for one unrestricted board on the same size of board. That ratio is the rung.

What levelling off means #

A rise that decelerates has a natural reading and it is worth stating so it can be argued with.

The temperature of a shared region is how much the first move in it is worth — how much a player loses by being made to play elsewhere. Two amazons far apart are each free to develop without immediately running into the other, so the region holds more that is worth doing and the move in it is worth more. That is the mechanism the rungs below identified and it predicts a rise.

It does not predict an unbounded one. Past some separation the two amazons are effectively not interacting at all, and a region where they do not interact is not one fight but two — at which point the temperature is set by whichever half is hotter and stops caring about the distance. So the curve should rise and then flatten, and flattening is what the third point shows.

What the data cannot say is whether it flattens to a ceiling or keeps creeping. Three points and a decelerating step are consistent with a limit and with a very slow continued rise, and separating those would need a fourth distance — which needs a 5 × 5 board, which needs an evaluation this site cannot do at any number of free squares that leaves room for the amazons to be four apart.

So the ladder ends here for a measurable reason. The first two points came from a board anybody can evaluate; the third came from a board that has to be cut down to five free squares to be evaluated at all; the fourth is not available.

What this does not settle #

Three questions the rungs below asked that a third distance does not answer, listed so the ladder’s state is clear.

Whether the arrangement effect turns. An effect that changes sign found that clumping the shared squares is hotter at one distance and colder at the next, on two distances. The same question at three distances would need the shared-square census repeated on this population, and the shared-square count is not a quantity a five-free-square region has much of — most of these regions have one or two shared squares and no room to arrange them. So that curve is still two points, and the board that would give it a third is the board that cannot be evaluated.

Whether room and distance still pull opposite ways. Room pulls two ways found that distance and reachable-square count together beat distance alone by half as much again, precisely because they pull in opposite directions. Here room is held nearly fixed by construction, which is what makes the distance reading clean and also what makes the two-variable question unaskable: nothing can measure how two variables trade off on a population where one of them does not vary.

And whether any of it survives a real board. Every rung on this ladder measures regions that a game of Amazons might produce, on populations chosen for what an evaluator can reach. When a real board falls apart is the one that looks at positions from actual play, and the regions there are larger than anything measured here. The distance curve is a fact about small regions and it is offered as one.

What a third point is worth #

Three things, and the first two are about this curve and the third is not.

The direction survived. Four rungs asserted further apart is hotter from two points, and a third point could have refuted it and did not. That is worth having explicitly: a two-point trend on this site is a hypothesis, and this one has now been tested once.

The rate is new information and it changes the reading. A linear rise and a decelerating one suggest different mechanisms — the first that distance keeps buying something, the second that it buys a great deal at first and then stops mattering — and only the second is consistent with the picture of two amazons ceasing to interact.

And the general lesson is about where a measurement stops. An effect that changes sign closes by noting that its arrangement effect predicts well and describes nothing, and the reason it could say no more is the same reason as here: two values of a variable are not enough to see a shape, and the third value cost a board size the evaluator cannot afford. Every quantity on this site that is measured at two settings is in that position, and the ones worth extending are the ones where the two settings already disagree about something.

The cost of extending is worth carrying too, because it is not linear in anything. Going from two points to three meant going from a 3 × 3 board to a 4 × 4, which meant an evaluation that fails outright until the regions are cut to a third of the board — and the cut is what made the third point affordable and what limits what it says. A fourth point would need a 5 × 5 board and a cut so severe that two amazons four queen moves apart would have almost no free squares between them, which is not a region so much as a pair of amazons in separate corners. So the ladder does not stop because nobody has run it. It stops because the next rung’s population would be empty.