A thousand positions and no exception

The parity law is short. The first player wins exactly when the number of short chains is odd — unless there is no long component at all, in which case it is when the number is even and not nought. It was fitted to every bag of a few chains and loops inside a budget of fifteen strings and it has never been contradicted there.

The bags were constructed. A board’s positions are not.

Why a board is a different population #

A sweep over bags of components chooses the components. A board offers what its geometry allows, and that is a narrower and stranger list.

The sizes are bounded by the board. The law was fitted inside a fifteen-string budget and a board has its own. A six-box board has no chain of seven and no loop of eight in it, so half the sweep’s vocabulary is unavailable. What it does have is 32 distinct shapes, of which the commonest is a two-chain beside a three-chain, arriving 112 times.

The components come correlated. A bag of components is any multiset inside a budget; a board’s components have to fit together on a grid, using the strings that are there. A six-chain and a one-chain cannot coexist on six boxes, and the sweep would happily have tested them together.

And the rare shapes are very rare. Exactly one position of the six-box board is a loop of six, and exactly one is four one-box chains and a two-chain. A sample would have missed both; a census cannot.

Two computations that share nothing #

The agreement is worth something only because the two sides are independent, and they are.

The law’s side reads the position as components: walk the surviving strings, find the pieces, count the one-box chains, the two-box chains and everything longer, and apply the rule. Nothing in that touches the board.

The truth’s side is the walk an earlier essay here already used: a recursion over subsets of the board’s strings, asking whether the player to cut wins, with the capture rule handled by the recursion itself. It has no notion of a component, never groups anything, and would give the same answer on a board that had never decomposed.

So an agreement is two programs with no shared assumption agreeing, one of which was written to a law fitted elsewhere. All 1,192 positions across the two boards agree, and the figure refuses to draw if one of them does not.

The shapes a board actually produces #

The distribution is the useful part for anybody playing, and it is lopsided.

Of the 1,032 decomposed positions on a six-box board, 112 are a two-chain and a three-chain, 104 a two-chain and a four-chain, 80 a one-chain and two two-chains, and 80 a one-chain with a two and a three. Between them the top four shapes are more than a third of everything.

Loops are almost absent: seventeen positions of 1,032 hold one, and exactly one of those is a loop of six. On a board this small there is barely room for a cycle of boxes, and the loop theory — the fee of four rather than two, the extra cost of declining — is being exercised seventeen times.

That is a caution about where the endgame theory’s weight sits. The loop rules are a genuine part of it and they are not what a small board is about. A player learning the analysis on six boxes would meet them once.

What a failure would have looked like #

A sweep that comes back clean is only informative if a dirty one was available, so it is worth naming what the two ways of failing would have been.

A shape the sweep never met. The constructed sweep is built from chains of one to five boxes and loops of four and six, because those are what fit its budget. A six-box board produces a chain of all six boxes, twenty-four times, and that component is outside the sweep’s vocabulary entirely — a size it never valued and never tested. The law reads it as one long component and predicts a second-player win, and the walk agrees on all twenty-four.

A correlation the law cannot see. This is the subtler failure and it is the one worth having looked for. The law reads three counts and discards everything else — which components are adjacent, how the board was cut, which strings remain. If any of that mattered, two positions of the same board with identical counts would sometimes disagree, and the sweep would find them because it tests every position rather than one per shape. The commonest shape arrives 112 times and all 112 agree; the same is true down to the shapes arriving once.

That second check is the one a constructed sweep cannot make at all. It tests one position per multiset of components, so a correlation between components is invisible to it by construction. Here the same multiset arrives a hundred times over, assembled differently each time, and the answer never moves.

Where the law’s second clause earns its keep #

The rule has an exception in it — unless there is no long component at all — and the board is where it can be checked against positions that arise rather than positions somebody chose.

Thirteen of the six-box board’s 32 shapes have no long component: nothing but one-box and two-box chains. They run from a single two-chain, seventeen positions, to a string of four one-chains and a two-chain, one position. On every one of them the clause fires and the parity is the other way round: an even number of short chains is a first-player win, an odd number a second-player win.

That is the half of the law with the least intuition behind it, and the board supplies 351 tests of it. The reason is the one the fitted law gives: a short chain is tempo, opened and handed straight back, so an even number of them cancels — and with no long component there is nothing left to be forced to open, so cancelling is winning rather than losing.

The commonest shape, worked through #

The shape a six-box board produces most often is a two-chain beside a three-chain, and it is worth following once because it exercises both halves of the law at the same time.

One short chain, one long component. The law says the count of short chains is odd and a long component is present, so the first player wins. Played out: the first player opens the two-chain hard-heartedly — cutting its middle string, so that every remaining cut there completes a box and the opponent cannot decline. The opponent takes both boxes and is obliged to cut again, into the three-chain. The first player then takes all three and has made the last cut.

Every step of that is the law’s reasoning in play. The short chain is tempo, handed over and handed straight back; the long chain is the thing nobody wants to open; and the odd count of short chains is what decides who is standing there when it has to be opened.

The second commonest is the same shape with a four-chain instead of a three, arriving 104 times, and the law gives the same answer for the same reason — the length of the long component does not enter. That two shapes differing by a box are the two commonest, and are decided identically, is the interchangeability claim showing up as a fact about a real board rather than as a sweep result.

The shape that appears once #

The single position of the six-box board that is a loop of six is worth a paragraph, because it is the whole board.

Six boxes in two rows of three; cut every string that touches the ground and every string but the six that run around the outside of the block, and what is left is a cycle of six coins each holding two strings. No box can be taken, nothing is open, and whoever cuts first hands the entire board over.

The law reads it as no short chains and one long component, and predicts a second-player win. The walk agrees. It is one position of 131,072 and it is the only position of that board in which the endgame is a single object.

What the test does not establish #

It is worth being exact about what a clean sweep is worth here, because two boards is two boards.

It is not a proof. The law has now survived 207 constructed positions and 1,192 from real boards, and a rule that survives 1,399 chances to fail is a rule with 1,399 chances behind it. The argument for why it should hold — short chains are tempo, long components all end the same way, the last one settles everything — is in the earlier essay and it is an argument rather than a derivation.

The boards are small. Under four per cent of a six-box board’s quiet positions decompose at all, and six boxes is where an exhaustive walk over subsets stops being a moment’s work, and a real game is played on twenty-five or more. The components a large board produces are longer and there are more of them, and nothing here reaches that.

And the sweep tests the law where it applies. Under four per cent of a six-box board’s quiet positions fall into chains and loops, so 1,032 clean tests sit beside 27,000 positions the law cannot be asked about at all. A clean sweep over the domain says nothing about the size of the domain.

What the four-box board adds #

The smaller board is not a weaker version of the same test and it is worth saying what it contributes.

It has twelve distinct shapes against the larger board’s 32, and its rarest is a loop of four, appearing once. Eight of its twelve shapes have no long component at all — a much higher proportion than the larger board’s thirteen of 32 — because four boxes cannot hold many long chains and a great many of its decompositions are just short ones.

So the second clause of the law, the one with the inverted parity, is proportionally much better exercised on the small board, and the first clause on the large one. The two boards test different halves, which is the argument for running both rather than only the bigger.

They also agree about something the sweep could not have shown. Both boards produce their decompositions with the same commonest shapes in the same order — pairs and triples of short chains, then a short chain beside a long one — and neither produces the wide, many-component positions the constructed sweep spends most of its budget on. A bag of four long chains is a position the sweep tests and a six-box board cannot hold.

Nimstring, and what a component’s size means #

Normal play throughout: the player who makes the last cut wins, and the score is thrown away. That is Nimstring rather than Dots and Boxes, and the whole point of it is that it answers the question the scoring game keeps asking.

Three conventions of the test.

A position is a subset of the board’s strings, and every subset is tried. The ones that do not decompose are skipped rather than counted as failures — the law makes no claim about them, and counting them would be scoring a rule against positions outside its own statement.

A component’s size is its number of coins, so a chain of three is three boxes and four strings. The distinction matters because the law is stated in boxes and the walk is over strings.

And the empty board is excluded. It decomposes into no components at all, which makes the law’s count of short chains nought and its second clause require not nought — so the law declines to predict, correctly, and there is nothing to test.

What a clean sweep is and is not worth #

Nothing about the scoring game. Every verdict here is who makes the last cut. The margin in boxes is a different quantity and it is the subject of its own essay.

And nothing about frequency under play. The 1,032 positions are weighted equally and a game does not visit them equally — it visits the ones its own moves produce, and a good player steers towards particular shapes. The distribution above is a distribution over the space rather than over play.

Why this is the right test to have run #

There is a version of this essay that would have been easier and worth less, and the difference is worth stating.

The easy version extends the constructed sweep: raise the string budget from fifteen to eighteen, test a few thousand more bags, report no disagreement. That is more of the same evidence and it answers the same question — does the law hold on bags of components — with a bigger number attached.

The question that was actually open is different. A law fitted to constructed objects is a law about constructed objects until it is asked about the ones that arise. Nimstring’s whole point is that it is the skeleton of a game people play, and the positions that game reaches are not a uniform sample of anything.

So the value of a board test is not in the count. It is that the population was chosen by the geometry rather than by whoever wrote the sweep, and the law had no say in which positions it was shown. A clean result on a population somebody else chose is worth more than a larger clean result on one the tester chose, and that is the whole argument for this essay existing.

One more thing the board settles #

There is a claim in the earlier essay that a board is in a position to test and a bag of components is not, and it is worth extracting.

The law treats every long component as interchangeable: a three-chain, a five-chain and a loop of four all count as one long thing and nothing else about them enters. The constructed sweep checks that on 27 shapes realised more than one way, which is a good check and a small one.

A board checks it harder without being asked to. Among the six-box board’s 1,032 positions, a two-chain beside a long component arrives with the long component being a three-chain 112 times, a four-chain 104 times and a loop of four eight times — three different objects, 224 positions, and one answer. The interchangeability is not being tested on a handful of constructed pairs; it is being tested on a population the board handed over.

That is the same argument as the one above about correlations, applied to a different clause, and it is the reason a census beats a sample even when the sample is chosen carefully. A sample tests what somebody thought to include; a census tests what is there.

Still open: the third board #

The obvious extension is the nine-box board, and it is out of reach by this route for an exact reason worth recording.

A three-by-three board has twenty-four strings, so a walk over its subsets is sixteen million positions, each needing a recursion over whole turns rather than single cuts. That is hours rather than seconds, and the answer it would give is almost certainly no disagreement again.

The interesting board is the first one large enough to hold the shapes a real game produces — a chain of eight, two loops at once, a loop of eight — and none of those fits on nine boxes either. So the test that would actually extend this is not a bigger sweep but a targeted one: construct the positions a large board reaches in play, a few hundred of them, and check those. That is a different kind of measurement and it needs a player rather than an enumeration.