The two moves that are not captures

Two rungs below, the Kōnane essay lists what its figures cannot show and puts this second:

The second thing not shown is the opening. Kōnane starts from a full checkerboard, and the first two moves are special: Black removes one of their own stones and White removes an adjacent one, which is the only non-capturing move either player ever makes. Nothing here models that, because a full board of any size is far past the evaluation, and every position drawn above is a middlegame or an ending arrived at by assumption rather than by play.

Two things have changed. The rung below evaluates rectangles rather than rows, so a full board is a position this machinery can hold. And a board small enough to solve completely is small enough to solve from the start, which means the opening stops being an assumption and becomes a table.

What the opening is #

Kōnane’s board begins entirely full, black and white stones alternating like a checkerboard. There is nothing to hop into, so no capture exists, so under the ordinary rules nobody could move at all.

The rules therefore have a preamble:

Black removes one of its own stones. White then removes one of its own, orthogonally adjacent to the empty square Black made. From the third move onward every move is a capture, and a player with no capture available loses.

That preamble is not a formality and it is not a variant. It is a pair of moves played under a different rule from every other move in the game, and it is the only time in Kōnane that a stone leaves the board without being jumped over.

Everything this site is built on applies from the third move. Nothing applies to the first two, because the first two are not moves of the game whose values the recursion computes — they are moves of a two-move game whose result is a position of it.

What the table says #

On a 4 × 4 board Black has eight legal openings, one for each of its stones. Each leaves a board with one hole in it; White then has some number of legal replies, and White chooses among them.

Solving the whole thing exactly — every opening, every reply, every position that follows — gives two distinct answers.

Four of Black’s openings lead to a position worth nought. Four lead to a position worth minus one, which is a whole free move for White.

The four good ones are the stones on the main diagonal. The four bad ones are the rest.

That is a fact about a 4 × 4 board and it is the sort of fact nobody could have guessed from the rules. Nothing in the rule set distinguishes a diagonal stone from any other; every black stone is a black stone, and the board is symmetric under a great many motions. The diagonal falls out of a complete evaluation of forty-seven thousand positions, and the evaluation was not told what a diagonal is.

What “worth nought” means here, and it is not encouraging for Black #

A value of nought means whoever moves loses. After the two removals it is Black’s move again, so a position worth nought after the opening is a position Black loses.

A value of minus one is a free move for White, which Black also loses.

So on a 4 × 4 board Black loses whatever it opens with, and the choice decides the margin rather than the winner. The same is true on 3 × 3 and 3 × 4, where every opening leads to nought or to minus one.

That is worth stating plainly rather than tucked into a caption, because it would be easy to present the diagonal as the winning opening and it is not. What the table establishes is that the openings are not interchangeable — that a preamble everybody treats as setup is a move with a computed cost attached to it — and separately that on boards this small the second player wins regardless.

The second half is almost certainly an artefact of size. Kōnane is played on 8 × 8, 10 × 10 and larger, and small partizan games routinely favour the second player for the reason Domineering’s small boards do: there is not enough room for the first player’s advantage to become a move. Who moves last is the whole of what a value of nought asserts, and on a board with four moves in it that is a very short argument. Nothing here can check that, because 5 × 5 is 3²⁵ positions before any play at all.

The middle row, and why a colouring is the wrong explanation #

The two bad openings on a 3 × 5 board are the two black stones in the middle row, and the first explanation anybody reaches for is the checkerboard: on a board coloured in the usual way the middle row’s stones sit on squares of a particular parity, so perhaps parity is what the value is tracking.

That explanation cannot be right as stated, because every black stone sits on a square of the same parity. The board is a checkerboard and the stones alternate with it, so all eight of Black’s stones are on squares of one colour and all eight of White’s are on the other. A parity argument distinguishing the middle row from the rest of Black’s stones would have to distinguish squares of the same colour, and colour is exactly what it has to work with.

What does distinguish them is available and is geometric rather than chromatic. On a board three rows deep, a stone in the middle row has neighbours above and below it and a stone in an outer row has one of the two. So a hole made in the middle row is a landing square reachable from four directions and a hole in an outer row is reachable from three, and the two are simply different objects.

That is a description rather than a proof, and it is offered as one. Whether the neighbour count is what the value is tracking, or whether the two facts are both consequences of something else, is a question a 3 × 5 board cannot settle — two bad openings out of eight is a very small sample to fit an explanation to, and the 3 × 4 board’s single bad opening is not in the middle row’s interior at all.

Why the choice matters at all #

A reader might reasonably expect all openings to be equivalent, and the reason they are not is worth following because it is the same mechanism the rung below is about.

Removing a stone creates a hole, and a hole is a landing square. Every capture in Kōnane needs one, so the position of the first hole decides which hops exist in the first few moves, and the hops that exist decide everything after that.

A hole in the middle of a board has four neighbours and sits on more lines than a hole in a corner. A hole on the main diagonal of a 4 × 4 board sits differently again with respect to the checkerboard colouring, because the diagonal squares of a checkerboard all carry the same colour.

So the opening is a choice about where the game’s first landing square is, and the reason it is worth something is the reason a Kōnane row’s value depends on where its gaps are: a gap is not a wall, it is a place to land, and putting one somewhere is putting a resource somewhere. That is why the census of rows two rungs below finds five rows of seven squares worth exactly a half with no arrangement in common — what a Kōnane position is worth depends on where its emptiness is, and the opening is the game’s first decision about that.

Both halves of the opening are choices, and they are not symmetric #

The preamble has two moves in it and they are usually described together, which hides that they are different kinds of decision.

Black chooses first, from every one of its own stones. On a 4 × 4 board that is eight options, and Black picks the one whose forced sequel is best for Black.

White chooses second, from its own stones adjacent to the hole. That is at most four options and often fewer — a corner opening leaves White two — and White picks the one best for White.

So the two players are not choosing from comparable sets. Black picks from the whole board and White picks from a neighbourhood Black has just decided. That is an asymmetry built into the rules and it is the reverse of the asymmetry a reader might expect, since the second player is the one the small boards favour.

The table reports White’s best reply to each opening, which is the right thing to report: an opening is worth what it is worth against correct play, not what it is worth against a convenient reply. On several of the openings White’s replies differ from each other, so the second removal is a real choice as well, and reporting the best of them is what makes each row a value rather than a range.

The preamble is a game with its own rules #

There is a general point here about rule sets, and Kōnane states it more cleanly than most games because the departure is so short.

Almost every game has a preamble of some kind. Chess has a starting array; Go has an empty board and a handicap convention; Sylver Coinage has an opening move that is unlike the others because there is nothing on the board to be a sum of yet. What Kōnane has is a preamble that is a move, played by a player, under a rule that appears nowhere else in the game.

That is precisely the shape that makes a game hard to write down as one object. The value recursion needs a single move rule; the two-move preamble is a different one; so the game as played is the composition of a two-move game and an ordinary combinatorial game, and only the second half is what this site computes.

The composition is not a difficulty of principle — the table above computes it by solving the preamble by hand, in the sense of enumerating both choices and looking up the values underneath. It is a difficulty of notation. There is no expression for “this position, arrived at by a rule that no longer applies”, and a reader handed the value of the position after the opening has been handed a fact about a different game from the one Black was choosing in. That is the same complaint the brace notation’s two hard edges are about, one level up: the notation writes games, and a rule that changes between moves is not a thing a game expression has a slot for.

What a player would take from this #

Two things, and one of them is more useful than the other.

Stay out of the middle row, on a 3 × 5 board. That is real and it is exact and it is worth a move, which in a game this small is the difference between losing narrowly and losing to a free tempo. It is also almost certainly not a rule about Kōnane; it is a rule about that board, in the way a threshold fitted to a census is a fact about the census until somebody checks it somewhere else.

And expect the opening to matter more as the board grows, not less. That is a guess rather than a measurement and it is offered as one, but the mechanism is visible: the opening chooses where the first landing square is, the rest of the game is hops that need landing squares, and a bigger board has more places for the choice to be wrong in.

The honest version of the second is that nobody knows, and the reason nobody knows is arithmetic rather than indifference. A 5 × 5 board’s opening table needs twenty-five full solves of a game with 3²⁵ possible positions, and the machinery here manages a 4 × 4 board in about a minute.

There is a third thing a player might want and this page does not supply: whether the opening choice interacts with what the opponent does next. It does — White’s best reply differs between openings, and on some openings White has two replies of equal value and on others only one — so a player choosing an opening is choosing a position and a set of replies, and the table collapses that to a single number by taking White’s best. A reader who wanted the shape of the second choice rather than its outcome would want a different table.

What the picture cannot show #

No board here is a board anybody plays on. Kōnane is played on 8 × 8 at the smallest, and every table above is 3 × 3, 3 × 4 or 3 × 5. The opening on a real board offers thirty-two choices to Black rather than eight, and nothing here says whether they collapse onto two values or spread across twenty.

The reason the tables stop there is arithmetic and it is worth quoting, because it is the same wall every rung of this ladder meets. A 3 × 5 opening table is a couple of seconds and about three hundred megabytes of interned games. A 4 × 4 table — sixteen squares against fifteen — is seventy-five seconds and four and a half gigabytes, and its answer is that four of Black’s eight openings are worth a move less than the other four, the good ones being the main diagonal. One square, and a factor of fifteen in memory.

The tables report values and not lines. An opening worth nought and an opening worth minus one differ by a move, and what a player would want to know is which move — where the free tempo comes from, and what shape of play cashes it. The recursion produces a value and the value says nothing about when.

Nor does anything here model the handicap conventions a real game has. Kōnane as played has traditions about who removes what and where, and some accounts have the two removals restricted to the centre or to a corner. Every table here takes the general rule — any own stone, then any adjacent enemy stone — which is the widest reading and therefore the one that makes the choice largest.

And the second player’s dominance is a fact about size. Every board solved here is a second-player win after the opening, and reporting that as a property of Kōnane would be reading a small-board artefact as a theorem. It is stated as what it is: a fact about three boards, none of which anybody has ever played on for fun.

The convention, named #

Normal play, again and for the last time on this ladder: a player with no capture available loses. That is Kōnane’s own rule, and the rung two below makes the case that arriving at it independently, centuries early, is the reason this game belongs on this site at all.

The opening is where the convention has nothing to say, and it is worth being precise about why. The normal-play convention is a rule about what happens when a player cannot move. On a full board neither player can capture, so a strict reading has Black losing before the game starts — which is obviously not what anybody means, and is exactly the gap the preamble exists to fill.

So the preamble is not a decoration on the convention; it is the thing that makes the convention applicable. A game whose starting position has no legal moves needs a rule to get out of it, and Kōnane’s rule is two removals. Every value on this page is a value of the game after that rule has done its work.

The surprise: the choice is worth exactly one move, on every board solved #

The expectation, before the tables were computed, was one of two things: either the openings are equivalent — a preamble is a preamble — or they spread out, with some substantially better than others on a board with any room in it.

Neither happened. On all three boards the openings take at most two values, and where there are two they differ by exactly one move. Not a fraction, not a fight, not a range: nought or minus one, on 3 × 3, on 3 × 4 and on 3 × 5 — and on the 4 × 4 board that was solved once and is too expensive to draw, the same two values again.

That is a stronger regularity than the tables were built to find, and its cause is visible in the rung below. Small Kōnane rectangles produce integers and very little else, because three or four squares in a direction leave no room for a hop to continue and therefore no room for a game tree deep enough to be worth a fraction. The opening inherits that ceiling: it is choosing between positions of a game whose values are integers, so the choice is worth an integer, and on boards this size the only integers available are nought and one.

So the opening’s regularity is the board’s regularity, one level up. The same reading applies to every census on this ladder: canonical form collapses thousands of arrangements onto dozens of values, and what bounds the dozens is how deep the game tree goes rather than how many arrangements there are. A reader who wanted to know whether the first move of Kōnane is worth a fight rather than a tempo would need a board with chains on it — which is to say a board nobody can solve — and that is the same answer the rung below gives to the same question about the middlegame.

Where the ladder goes next #

konane has three rungs: the row and its vocabulary, the rectangle and what the second dimension costs, and now the two moves that are not moves of the game.

The rung the first of those most wanted is still open, and it is comparative rather than computational. Kōnane is one old game that turns out to hold the whole vocabulary, and one is an anecdote. Two would be a pattern — another game somebody played for centuries without anybody meaning it to display halves and switches and infinitesimals, evaluated the same way, with its census set beside this one. What that rung would establish is whether the vocabulary is something the theory found in games or something games turn out to have.