The second dimension is not the deep end

The rung below evaluates every row of Kōnane up to eight squares and finds the theory’s whole vocabulary in it: integers, halves, quarters, stars, switches, and infinitesimals of several kinds, out of a game played on carved lava long before any of those had names. It then qualifies the finding carefully. Every corner of the vocabulary is there and the range in each corner is narrow — no denominator finer than a quarter, no nimber above star-two, no temperature above one — and it says why:

The narrowness is a fact about the board, not about Kōnane, and a longer row would supply eighths and hotter switches by exactly the mechanism that supplied quarters and cool ones.

There are two ways to buy a longer game. Make the row longer, or use the second dimension the game actually has. The second is the interesting one, because the rung below plays on a row and admits it: a real Kōnane board is a rectangle, hops run in four directions rather than two, and the two-dimensional game is where the difficulty lives.

The expectation is obvious. More directions, more options, richer values.

The measurement #

A row of nine squares has 3⁹ arrangements — 19,683 of them, since every square is empty, Black or White — and evaluating all of them gives fifty-eight distinct values.

A 3 × 3 board has exactly the same 19,683 arrangements, because it has exactly the same nine squares. Evaluating all of them gives twenty-five.

The same code does both. The recursion is lib/cgt.js’s, the canonical form is the site’s, and the only difference between the two runs is that a hop on a board may go up and down as well as left and right. More moves, fewer values.

That is not a small gap and it is not a boundary case. The line holds more than twice as many values as the square, and it holds them across a wider range: the row reaches quarters, double-ups and a handful of forms with no short name, and the square reaches halves and stops.

Why, and the reason is in the shape rather than the game #

A hop needs three squares in a line — the stone, the enemy stone it goes over, and an empty square to land on — and may continue in the same direction so long as each landing square is empty.

So the number of places a hop can start, and the longest chain one can run, are properties of the shape before any stone is on it. Both are counted rather than argued.

A line of nine has fourteen places a hop can start, and a hop can continue up to four times. The middle of the row is a long straight in both directions.

A square of three has twelve places, and a hop can never continue at all. Three squares is exactly enough for one hop and one square short of two, in every direction there is.

That second number is the whole explanation. A chain of hops is the mechanism that gives a Kōnane position deep options — a single move that can stop after one enemy stone, or two, or three, offering the mover a genuine choice about how far to go. A board three squares wide has no chains, so every move is a single hop, so the option lists are short and the game trees are shallow, and shallow game trees produce shallow values.

The rung below explains the row’s own ceiling the same way and it is the same explanation. A row of eight goes two levels of the simplicity rule deep and reaches quarters; a row of nine goes further; a 3 × 3 board goes one level and reaches halves.

What the twenty-five values are #

The counts are the argument and the lists are what make them checkable, so it is worth reading both censuses rather than their sizes.

A 3 × 3 board reaches integers to ±4, halves, a single nimber, and a handful of switches and infinitesimals — 9,159 of the 19,683 boards are worth nothing at all, 3,236 are worth one to Black and as many to White, and the tail beyond that is short.

A row of nine reaches integers, halves, quarters, star and star-with-a-number, ups and double-ups with and without a star, several switches, and a scattering of forms with no short name — {∗ | −2}, {0 | 0, ↓∗}, and their relatives — which the square never produces.

The kinds are what the rung below is about, and the square supplies most of them: a number, a finer number, a nimber, an infinitesimal, a switch. Where it falls short is not in kind but in reach, which is exactly the axis the rung below identified as a property of the board — and the square is worse on that axis than a line of the same area, which is the part nobody would have guessed.

There is one more difference worth the sentence. The square produces no quarters at all, and quarters are the clearest evidence of the simplicity rule doing two levels of work. A quarter is the simplest number strictly between nought and a half, so producing one requires a position whose two options are already a nought and a half — which requires a game tree with a half in it two levels down. Three squares in a direction cannot build that, and a line of nine can.

The direction the expectation ran #

It is worth saying plainly that this was measured rather than predicted, because the prediction was wrong.

The rung below names the two-dimensional game as the obvious next step and describes it as a real piece of work rather than a variation on this one, on the grounds that the search grows by a factor of three per square and the positions stop being drawable. Both halves of that are true. What neither half says is that the values get simpler, and they do.

The reason the expectation is so natural is that in almost every other game on this site, more room is more depth. A bigger Domineering board holds hotter values; a longer Toads and Frogs strip holds more awkward ones; more counters in a heap reach further up the Grundy sequence. Kōnane inverts it at a fixed area because Kōnane’s move is a hop, and a hop needs a straight line of three.

A game whose move needs a straight line rewards being straight. That is a sentence about geometry rather than about combinatorial game theory, and it decides a value census. It is the same kind of statement the decomposition field makes about how often a board falls apart: a property everybody reads as being about the game turns out to be about the shape it is played on.

A cut through a board is not a wall either #

The rung below asks whether a row can be cut at a run of empty squares and the pieces added up, and finds it cannot — at any width, because a stone hops into the gap on one move and out of it on a later one. Cutting a row of seven at a single empty square is right 842 times out of 1,266, and widening the gap helps a little and never fixes it.

Two dimensions do not repair that. They make it worse, and the controlled comparison is the second row of the sweep. Splitting a position is a claim about the position rather than about the drawing, and here the claim gets less true with more room rather than more.

A 3 × 4 board cut at its second column — an empty column, with a 3 × 1 piece on one side and a 3 × 2 on the other — gives the right answer on 12,119 of 19,683 boards, which is 61.6%. The row’s single-square cut managed 66.5%.

Then widen the band. A 3 × 5 board with two empty columns has the same pieces, 3 × 1 and 3 × 2, and a wider gap between them. It gives the right answer on 52.1%.

Widening made it worse, by nearly ten points, on a comparison where nothing else changed. That is the opposite of what happens in one dimension, where widening the gap raises the agreement a little, and the reason is worth stating because it is the mechanism rather than a curiosity.

An empty band is not distance; it is landing squares. A stone hops into the band and out of it, and a wider band offers more places to land in and more places to hop back out from. In one dimension a wider gap adds a little of both and roughly cancels. In two dimensions it adds a whole column of them at once — three landing squares rather than one — and the extra room helps the crossing more than it hinders it.

Widen it again and nothing further happens. A 3 × 6 board with three empty columns, still with the same pieces, agrees on 52.3% — within two tenths of the two-column figure. So the damage is done by the first extra column and the rest is free, which is what a mechanism about landing squares predicts and what a mechanism about distance would not.

There is a second mechanism available in two dimensions and not in one, and it is worth separating from the first even though the sweep cannot tell them apart. A stone in a row that wants to reach the other side of a gap has one route. A stone on a board has several — it may cross the band anywhere along its length, and it may go round rather than through if the board’s edges permit. The counts above pool both, and what they establish is that the pooled failure rate is roughly double the row’s; which of the two mechanisms dominates is a question this sweep does not ask.

The check that says both sweeps are about the same object #

There is a claim buried in the last section that would be easy to get wrong and is checked rather than assumed.

A Kōnane board and its transpose are the same game. Rotating the picture ninety degrees relabels the directions and nothing else, so a board cut at a column and the transposed board cut at the corresponding row must have identical answers.

The sweep does both — a 3 × 4 board cut at a column, and a 4 × 3 board cut at the corresponding row — and the counts come back identical: 19,683 boards, 12,119 agreements, 7,564 failures, in both. That is two independent enumerations of what ought to be one set, and the figure refuses to draw if they differ.

What the census does and does not settle #

The rung below is careful about this and it needs saying again in two dimensions, because a census of 19,683 boards sounds like a lot.

Kōnane is not solved. The full game on an 8 × 8 board starts from a filled checkerboard and its position count is far past anything here; Hearn proved the two-dimensional game PSPACE-complete with respect to board size, so there is no prospect of a general answer of the kind Nim has; and a 2008 result of dos Santos and Silva shows the game is universal in a strong sense.

What has been computed is: every arrangement of a 3 × 3 board, every arrangement of a 3 × 4 board with an empty second column, and every arrangement of a 3 × 5 and a 3 × 6 board with two and three. Those are complete answers about three stated families and about nothing larger.

The gap is worth measuring rather than gesturing at. A 3 × 3 board is 3⁹; a 4 × 4 board is 3¹⁶, which is forty-three million; and the real game is 8 × 8. Every figure on this page sits at the bottom of an exponential, which is the ordinary situation here and the reason what “solved” means has to be restated every time.

What this says about buying depth #

Put the two halves of the page together and there is a general statement in them about how a game’s value range is bought.

The rung below asks for depth and names two ways to get it: a longer row, or a second dimension. The census says the first works and the second does not, and the reason is that depth in the values comes from depth in the game tree, and depth in the game tree comes from long options rather than from many of them.

A 3 × 3 board has more moves available per position than a row of nine does, in the sense that a stone has four directions rather than two. It has fewer deep moves, because no move can continue. And a value is built by a recursion over options, so what makes a value complicated is how far down the recursion goes, not how wide it is at any level.

That distinction has a name elsewhere on this site. The birthday of a value is the depth of its tree, and the values a game can produce are bounded by the birthdays it can reach. A game whose moves are all short reaches small birthdays however many of them there are, and small birthdays are exactly the values a census like this reports. The same reading explains why a game’s realisable values are bounded by what its positions can be made to look like rather than by how many positions there are.

So the honest answer to where does a Kōnane row’s depth come from is: from the chain. And a rectangle of side three has no chains, which makes it a worse laboratory than the row it was supposed to improve on — and a perfectly good demonstration that a game’s value range is a property of its geometry.

What the picture cannot show #

A 3 × 3 board is not Kōnane and neither is a row. Both are legal positions of the real game with everything else burnt away, and both are chosen for being small enough to sweep. A board wide enough for chains — five squares in each direction, say — is where the two-dimensional game presumably gets its depth back, and it is 3²⁵ arrangements, which is past anything a page build can hold.

The geometry counts assume an empty board. The number of places a hop can start is counted over the shape, ignoring what is on it; in a real position most of those places have the wrong stones in them. That makes the count an upper bound on the game’s option width rather than a measurement of it, and the census beside it is what says the bound is informative.

And the split sweeps hold the band empty by construction. Only boards with the stated empty column or row are enumerated, which is the right population for the question and is not a random sample of boards. A board whose emptiness is somewhere else has not been asked about.

Nor are the pieces the same size as each other. Cutting a 3 × 4 board at its second column leaves a 3 × 1 piece and a 3 × 2 one, which is what makes the sweep affordable — a cut leaving two 3 × 2 pieces needs a 3 × 5 board, twelve free squares, and half a million boards rather than twenty thousand. That larger sweep was run once and agrees on 35.6%, which is worse still; it is not drawn here because a figure costing several gigabytes of interned games is a figure the site pays for on every build.

The convention, named #

Normal play, as everywhere in this game: a player with no capture available loses. That is Kōnane’s own rule, arrived at some centuries before anybody wrote it down as a convention, which is the rung below’s opening claim and is not weakened by anything here.

One thing does change in two dimensions and it is worth naming. In a row, a side with no capture is easy to see — the stones are in a line and the reader can check. On a board it is not, because a capture may be available in a direction the eye is not scanning, and the figures state the move counts for each side rather than leaving them to be read off. Every board drawn here carries a check that a side with no capture is not also drawn as winning, which is the same assertion the row figures make and is worth more in two dimensions than in one.

The surprise: more moves, fewer values #

The finding is the inversion, and it is worth stating in the form that generalises.

A reader who wanted to make a game’s values richer would add moves. It is the obvious lever, it is what every parameter of every game on this site does when it is turned up, and here it produces the opposite: a 3 × 3 board has hops in four directions where a row has two, and it holds less than half as many values.

What actually produces value range is the length of the deepest option, and adding directions to a small board shortens options rather than lengthening them — because a direction on a small board runs out immediately, and a hop that runs out is a hop that cannot continue.

That is a fact about the game and a warning about censuses. A count of distinct values over a family is a measurement of the family’s shape, and a reader comparing two such counts is comparing two shapes rather than two games. The row and the square here are the same game, the same code, the same nine squares, and the two numbers differ by a factor of more than two.

And it repairs the rung below rather than contradicting it. That essay’s claim is that Kōnane reaches every corner of the vocabulary and goes far into none of them at the size drawn, and that the narrowness is a property of the board. It is — and the property is not the number of squares. It is whether the squares are in a line.

Where the ladder goes next #

konane now has a row, a rectangle, and a measurement of what the second dimension costs rather than buys.

The rung above is the one every position on this ladder has assumed away. Kōnane starts from a full board, and the first two moves are the only non-capturing moves anybody ever makes: Black lifts one of its own stones and White lifts one of its own beside the gap. Everything on this site applies from the third move onward and nothing applies to the first two, which makes the opening a game with different rules sitting on top of the game — and a choice worth a computed amount.