A coin with three strings is worth something

Every chain and every loop is worth nought — a one-chain, an eight-chain, a four-loop, a six-loop, all of them second-player wins on their own. That is the fact the whole endgame analysis is built on, because it is what makes the nim-sum say nothing and forces the parity law to replace it.

It is also a fact about two shapes out of the shapes a board holds. For most of a game the board has a coin with three or four strings on it somewhere, and nobody here has asked what one of those is worth.

What a branching component is #

A fork is a coin with three strings on it, each running off into a chain of its own. An arm of $k$ boxes is $k$ coins in a row with a string out to the ground at the far end; an arm of nought is a bare string from the centre straight to the ground.

That is the smallest departure from the endgame theory’s vocabulary there is: one coin holds three strings and everything else holds two. Written as the boxes on each arm, 1/1/1 is a centre with three single boxes hanging off it — four boxes, six strings — and 0/0/0 is a single box with all three of its remaining strings running to the ground.

Worth something, which nothing else is #

Of the 35 forks measured, 31 are worth something. Ten are worth one, fifteen two, six three, and only four are worth nought.

That is the finding, and it is worth saying why it is a finding rather than an observation. A component worth nought is a component whose presence changes nothing when it is added to a position that already has an even count of them — the nim-sum’s nought. Every component the endgame theory names has that property, which is precisely why the theory had to be rebuilt around a parity rule: with everything at nought there is nothing for the arithmetic to do.

A fork at one or two or three is a component with something to contribute. Whether the contribution composes is a separate question and the answer is still no, for the reason the earlier essay gives — a capture keeps the turn, so the components are not a disjunctive sum however nice their values are. But the values themselves are no longer uniformly blank.

The values that appear #

The four values are not spread evenly and the distribution is worth reading.

Ten forks come out at one, fifteen at two, six at three and four at nought. Sorting the arms and reading down the list, the value is not a function of the total boxes, nor of the number of coins, nor of the longest arm alone. Three things are visible and only two of them are safe.

A fork with a bare arm and a single-box arm is worth one, whatever its third arm. Every 0/0/k and every 0/1/k in range, ten of the thirty-five, and the third arm runs from nought to four without moving the value. So the long part of a fork can be arbitrarily long and contribute nothing, which is the same insensitivity to length the parity law has.

A fork worth three always has an arm of four, and four is the longest arm measured. Six forks reach three and every one of them sits on the edge of the sweep, so three may be a real value or may be where the range stops. That is the one number on this page to distrust.

And the value is neither a sum nor a maximum of anything. 1/1/1 is one, 1/1/2 is two and 2/2/2 is two; adding a box to one arm moves the value and adding it to another does not. Four distinct values arise from thirty-five components, which is four more than the chains and loops manage between them at any size.

The four that are nought #

The exceptions have a description and it is short.

The four forks worth nought are exactly the four whose arms are all three boxes or longer. 3/3/3, 3/3/4, 3/4/4 and 4/4/4, and nothing else in range.

That is the same three-box threshold the parity law runs on. A chain of one or two boxes is short in the law’s sense — it can be opened and handed straight back, because the opponent cannot decline it profitably — and a chain of three or more is long. A fork with a short arm has a cheap way in; a fork with three long arms does not, and behaves like the components the theory knows.

So the threshold is the same threshold, arriving in a different measurement. What makes a chain long is what makes a fork nought.

And four strings is nothing again #

The pattern does not continue upward, which is the part nobody would guess.

Every one of the fifteen crosses — a coin with four strings — comes out at nought, whatever its arms. A cross with four single-box arms is nought; the fork with three of them is one.

So it is not branching that gives a component a value. It is branching to an odd degree, or at least it is on everything in range: degree two is nought, degree three is usually not, degree four is nought again. That is a parity in the degree of the coin, and nothing in the endgame theory anticipates one.

The reason is worth guessing at even though nothing here establishes it. Cutting a string at a branching coin reduces its degree by one, so a player at a degree-four coin can hand over a degree-three coin and a player at a degree-three coin can hand over a degree-two one — a chain or loop, worth nought. The parities of those two exchanges differ, and the values differ with them. That is a sketch rather than an argument.

What a board’s branching coins actually look like #

The census is over constructed forks and it is worth checking it against the shapes a board holds, since the two need not overlap.

A box on a Dots and Boxes grid starts with four strings: two along it and two across, with a border line counting as a string to the ground. So the first branching coin a board meets is a cross rather than a fork, and a fork appears only after one of a box’s four strings has been cut.

That makes the crosses — all fifteen of them worth nought — the shape the opening of a game is made of, and the forks the shape the middle is made of. A board in its first few moves is a lattice of degree-four coins; by the middle it is a tangle of degree-three ones; and only at the end is it chains and loops.

Read that way the three regimes line up with the three value patterns: nought at the start, something in the middle, nought again at the end. Whether that is worth anything is not established — the values do not add, so a board is not the sum of its coins — but it is a much more legible picture of a game than the theory applies to under four per cent of it.

Can a fork be replaced #

If a fork behaved like some chain or loop in every company, the parity law could be extended to boards that have not fully decomposed: replace each fork by its equivalent and apply the rule. That would be the cheapest extension available and it is the one to test first.

It fails. Seven of the eight forks tested match no chain and no loop across all six companies, and each of them matches two of the four candidates in some company and not in others. A component that is a one-chain beside a two-chain and a three-chain beside a loop is not a component with an equivalent.

The eighth is 3/3/3, which appears to stand in for a one-box chain — and it was tested in three companies rather than six, because at twelve strings the rest of the tests run past the budget. Three agreements is not a stand-in; it is three agreements, and the figure prints the company count for exactly that reason.

So the law cannot be extended by a lookup table, which is the answer to the obvious next question and the reason the endgame theory stops where it does.

What this adds to the failure of the nim-sum #

There is a temptation to read a non-zero fork as good news for the arithmetic, and it is worth blocking.

The nim-sum fails here because a capture keeps the turn, not because the values happen to be nought. The nought was what made the failure invisible — a prediction of nought everywhere is right two thirds of the time by accident — and it is not what caused it. Components with interesting values would fail to add for exactly the same reason.

What the branching values do add is a diagnostic. A position of chains and loops has every component at nought, so the nim-sum predicts nought and the parity law has to be consulted. A position with a fork in it has a component at one or two, so the nim-sum predicts something other than nought — and it will still be wrong, but it will be wrong visibly, which is a slightly better position to be in.

Why this is where the theory had to stop #

Putting the three measurements of this sequence of essays together says something the individual ones do not.

The chain-and-loop theory reaches under four per cent of a six-box board’s quiet positions. On those positions its law is exact, 1,192 times out of 1,192. And the positions it does not reach differ from the ones it does by a component whose value is not nought and which no component of the theory stands in for.

So the boundary is not arbitrary and it is not a matter of effort. The theory’s domain is exactly the positions in which every component is worth nought, which is exactly the positions in which the nim-sum’s failure can be replaced by a parity — and one step outside it, the components have values, the values do not add, and no substitution recovers the law.

A theory that is exact on its domain and has no continuous extension past its boundary is a different kind of object from one that is approximate everywhere. This one is the first kind, and the measurement above is what says so.

The one thing a non-zero value does buy #

Values that do not add are usually worth very little, and there is one exception here worth naming because it costs nothing to use.

A component worth nought is a second-player win on its own: whoever has to cut in it, alone, loses. A component worth one or two is a first-player win: whoever cuts in it, alone, wins. So a fork with a short arm is a component somebody wants to be handed, which is the reverse of every chain and every loop.

That inverts the whole shape of the endgame for a position containing one. The endgame theory’s central fact is that opening a component is a liability — the long chain nobody wants to open is what the entire manoeuvring is about. A fork with a short arm is a component it is safe to be given, and a player who can arrange to be handed one has been handed a free move rather than an obligation.

None of that survives addition, so it is not a strategy. It is a fact about single components, and it says that the sign of the endgame reverses at the theory’s boundary rather than merely its arithmetic breaking. Outside the domain, being made to move is not automatically bad.

Turns rather than moves, and an arm of nought #

Normal play and the Nimstring convention throughout: no score, and the player who makes the last cut wins. A turn is a run of cuts, each but the last taking a coin, and the Grundy value is computed over turns rather than over single moves — which is the apparatus the earlier essay built and checked against the solver.

Three conventions of the census.

An arm of nought boxes is a string from the centre to the ground. That is a real shape — a box at the edge of a board with a border line on it — and excluding it would leave out the smallest fork there is.

Arms are written sorted and counted once. 1/2/1 and 2/1/1 are the same component, so the census walks non-decreasing arm vectors.

And the budget is sixteen strings. A fork with three arms of four boxes uses fifteen and one with three of five uses eighteen, so the sweep stops where the walk over subsets stops being affordable. The four nought-valued forks all sit at the top of that range, which is a caution about the threshold rather than about the values.

Degree three and four, and nothing higher #

Degree three and four, and nothing higher. A coin on a real board holds at most four strings, so that is not a gap in practice — but the parity suggested above is a parity across two values and two values do not establish one.

The values do not compose, so they are not a theory. Knowing a fork is worth two does not let it be added to anything. What the numbers establish is that the components outside the endgame theory are different in kind from the ones inside it, which is a statement about the boundary rather than a step past it.

Nothing here is about the scoring game. Every value is a Nimstring value — who makes the last cut, with the boxes thrown away. What a fork is worth in boxes is a different quantity and it is not measured.

And the threshold is measured at the edge of the range. The claim that a fork with all arms long is worth nought rests on four components, three of which use arms of exactly three or four boxes. A fork of 5/5/5 would be the first real test of it and it has eighteen strings.

The shape of the answer, in one line #

Read as a single sentence, the census says: being a chain or a loop is what makes a component worth nothing, and it is not a property of size or of shape but of every coin holding exactly two strings.

That is worth saying because the obvious summary of the endgame theory — every component is worth nought — is true only of the components the theory admits, and it reads as though it were a fact about Nimstring. It is a fact about degree two. Raise one coin to three strings and the value moves; raise it to four and it comes back.

Which reframes the theory’s domain from an accident into a definition. The chain-and-loop positions are not a convenient subset somebody chose to analyse; they are exactly the positions on which every component is worth nothing, and therefore exactly the positions on which a parity rule is the only thing left to say.

Still open: whether a long fork is a long component #

The natural conjecture sits in plain view and this essay deliberately does not take it.

The four nought-valued forks behave, on their own, exactly like a chain of three or more. If they also behaved like one in company — counting as a single long component for the parity law, interchangeable with a three-chain and a loop of four — then the law would extend to boards holding a fork with long arms, and the domain measured in an earlier essay here would grow.

The stand-in test above cannot settle it, because the only long fork inside the budget is 3/3/3 and the companies it fits beside are three. What would settle it is a cheaper computation rather than a bigger budget: a fork’s value in company depends on the turns available at its centre, and there may be a way to reduce a long-armed fork to a chain before the walk runs rather than after.