The number nobody needs

How long it lasts took the three compounds — disjunctive, conjunctive, selective — and found that the second needs a quantity the value does not carry and the third needs none at all. It computed a fourth quantity on the way and set it aside:

The suspense number. The shortened selective compound — move in any non-empty set, and stop when the first component stops — needs a third quantity, computed by the same recursion with both preferences reversed. This essay computes it and does not use it, because the compound it governs is not one of the three above.

The compound has now been solved, and the answer is that the quantity is correct and unnecessary.

The suspense rule holds. So does the remoteness version of it. So does the plain Grundy one. And the shortened compound has the same winner as the unshortened one on every position swept.

The four compounds, briefly #

A compound is a board of several components with a rule about how many of them a move touches, and each rule needs a different quantity.

The disjunctive compound — move in exactly one component — is the one the whole subject is built on, and it needs the value.

The conjunctive compound — move in every component — needs the remoteness: the length of the game under play in which the winner hurries and the loser stalls. The mover wins exactly when the minimum remoteness is odd.

The selective compound — move in any non-empty set of components — needs nothing at all beyond the outcome of each part: the mover wins exactly when some component is an N-position.

The shortened selective compound is the fourth. The move rule is the selective one, and the stopping rule changes: play ends as soon as any component has ended, rather than when all of them have.

That last change is a real one — it makes the game shorter, sometimes much shorter — and it is what the suspense number was defined for.

What the suspense number is #

It is the remoteness recursion with both of its preferences reversed, and the reversal is the interesting part.

Remoteness asks how long the game lasts when the player who is winning wants it over quickly and the player who is losing wants it to run on. So a position with a winning move takes the shortest even option plus one, and a position without one takes the longest.

Suspense reverses both. A position with an even option takes the longest such option plus one, and one without takes the shortest. That is what a player wants when the game ends as soon as any component ends: the winner is trying to stretch the shortest component out, not to finish it.

They are genuinely different numbers. On the set $\{1, 3, 4\}$ they differ on ten of the first seventeen heaps — a heap of three has remoteness $1$ and suspense $3$, and a heap of seven has remoteness $4$ and suspense $2$.

And they never differ in parity #

Here is the whole of the negative result, in one observation.

For a single component, the remoteness is odd exactly when the mover wins — that is what the recursion computes, since a position with a winning move takes an even option and lands on an odd count. And the suspense number is odd exactly when the mover wins, for the same reason: the reversal changes which option is taken and not whether an even one exists.

So both quantities have the same parity everywhere, on every subtraction set here, with no exception in sixty-eight heaps. Two different numbers, one parity.

And a compound rule built on parity cannot tell them apart. The shortened rule — the mover wins when some suspense number is odd — picks the same winner as the remoteness version of it, always, because the two sets of parities are identical. And both pick the same winner as some component is an N-position, which needs no compound theory at all.

The shortening changes nothing about the winner #

That is the sharper half, and it is the one the sweep is for.

On all 1,176 positions, the shortened selective compound and the ordinary selective compound have the same winner. Changing the stopping rule from when all components stop to when the first one stops alters the length of the game and does not alter who is winning it.

The reason is the move rule rather than the stopping rule. A selective move may touch any non-empty set of components, so a player faced with several offending components can repair all of them at once — and the winning condition is therefore some component is bad for the opponent in both games. Making the game stop sooner removes moves the winner did not need.

So the four compounds do not need four quantities. They need three, and the fourth was defined for a distinction that turns out not to exist at the level of the winner.

What the suspense number does measure #

Saying the quantity is unnecessary for the winner is not saying it is empty, and it is worth being precise about what it is for.

It is a length. Remoteness is how long the game runs when the winner hurries; suspense is how long it runs when the winner stalls. On $\{1, 3, 4\}$ a heap of seven takes four moves under the first reading and two under the second, and a player who cares how long they will be at the table has been told two different things.

What a value leaves out lists length first among the quantities the equivalence discards, and both of these are answers to it. They are answers to two different questions about length, and neither is an answer to a question about winning that the outcome does not already settle.

The pair of them is more informative than either alone. A position whose two numbers agree has a length that optimal play cannot vary — a heap of six in take one, two or three lasts three moves however the winner feels about it. A position whose numbers are far apart has a length the winner controls, and the gap is the size of that control. On $\{1,3,4\}$ the largest gap in the first seventeen heaps is four moves, on a game whose longest line is ten.

That is a real thing to know and it is not a compound theory. It is the observation that a game has a range of lengths under optimal play, and that the two ends of the range are computable.

One heap, worked through #

The set $\{1, 3, 4\}$ is the only one of the four where the two numbers come apart, and a heap of seven is the clearest case.

A player may take one, three or four counters. From seven, the options are heaps of six, four and three.

Remoteness. The heap of seven has remoteness four. Reading down: three has remoteness one, four has remoteness one, six has remoteness three. All three are odd — every option is a win for whoever moves there — so seven is a loss for the mover, and its remoteness is one more than the longest option’s, which is three. Four.

Suspense. The same heap has suspense two. Three has suspense three, four has suspense one, six has suspense three. Again all odd, so again a loss — and the suspense recursion takes one more than the shortest, which is one. Two.

Same position, same tree, same verdict, and two answers to how long it goes on: four moves if the winner is hurrying, two if the winner is stalling. Both numbers are odd or both are even, always, because both are one more than something odd whenever the mover is losing and one more than something even whenever the mover is winning.

That last sentence is the proof, and it is worth having in full because the sweep is only a check on it. A position’s parity in either recursion is decided by whether any option has even count. That test is the same in both recursions. So the parities are the same, at every position, on every game, with no computation needed — and the 1,176 positions are confirming an argument rather than establishing a fact.

Why a negative result is worth a page #

Three of the four predictions in the table are cheaper than the one the compound was given, and the cheapest of them — is any component an N-position? — is a single bit per component.

A reader meeting the compound theory finds four compounds and four quantities laid out in parallel, and the parallel is misleading. The remoteness genuinely does work that nothing cheaper does: the conjunctive compound needs it, and a sweep there finds the Grundy prediction wrong on a substantial fraction. The suspense number does no such work.

So the honest shape of the compound theory is three quantities and four compounds, with one compound’s quantity being an elaboration rather than a requirement. Recording that is worth as much as recording a rule, and it took solving 1,176 positions to find out.

The general form of the lesson is one this site keeps meeting from different directions. A theory laid out in parallel invites a reader to assume each row is load-bearing, and the only way to find out which are is to point the same instrument at all of them — which is what the four move rules did for strategies and what this page does for compounds.

What a compound theory is for, restated #

It is worth putting the four compounds back together with this in hand, because the corrected picture is tidier than the one with four quantities in it.

A compound is defined by two rules: how many components a move must touch, and when the game stops. The first is what decides which quantity is needed.

• Touch one component: the parts interact through their whole values, and the theory needs the value.
• Touch every component: a player cannot repair one part without spending a move in all of them, so the shortest part governs, and the theory needs a length.
• Touch any set: a player can repair everything at once, so nothing about lengths or values is needed — only whether any part is bad.

The stopping rule adjusts the length and, in the third case, adjusts nothing else. That is why the shortened selective compound needs no new quantity: its move rule is the one that makes the winning condition existential, and an existential condition does not care when the game stops.

Three ways to add the same games lays the compounds out and asks why the disjunctive one is singled out. The answer this page adds is that the three move rules are the whole taxonomy, and the stopping rules are a refinement of the length rather than of the theory.

Redundant is not the same as wrong #

The finding here is easily mis-stated in either direction, so it is worth pinning both.

The suspense number is not wrong. It is defined by a recursion, it computes what it claims, and it governs the compound it was built for — a reader who applies it gets right answers. Nothing on this page is a correction to anybody’s arithmetic.

It is also not merely unhelpful. An unhelpful quantity is one that fails to decide things; this one decides everything it is asked, and so do two cheaper quantities already in hand, on every position measured. That is a stronger statement than it adds little: it says the information it carries is contained in information the theory already has, on this pool, and that a solver holding the remoteness and the Grundy value has nothing left to compute.

The honest form is a conditional and it is worth stating as one. Either the suspense number is genuinely redundant, in which case the compound theory has three quantities and needs two — or the pool is not wide enough to contain the positions that separate them, in which case those positions are the interesting object and nobody has exhibited one. The measurement here cannot distinguish those, and saying which of the two it is would need a construction rather than a census.

That is why a negative result of this shape is worth publishing rather than filing. It converts here is a third quantity into here is a third quantity and 1,176 positions on which it earns nothing, which is a specific claim somebody can refute with a single example. A quantity nobody has tried to make redundant is in a much weaker position than one that has survived an attempt.

What the sweep does not settle #

The components are subtraction-game heaps, four sets of them, three at a time. That is a narrow family in one specific way: every component is the same game. A compound whose parts come from different rulesets is the case a real broken board presents, and the claim that any of these rules works across games is stated in the rung below and tested nowhere.

That matters more here than usual. The parity coincidence between remoteness and suspense is argued above for a single component of any kind, so it should survive a mixed compound — but should is not a measurement, and the argument turns on the recursion having a base case at a position with no moves, which is a property of the component and not of the compound.

The second limit is the size. Three heaps of at most a dozen counters is 294 positions per set, and a compound with six components has more room for a rule to go wrong. The shortening removes moves, and removing more moves from more components is where a difference would show up if there is one.

And the sweep is of winners rather than of lengths. The suspense number’s real content is a length, and nothing here checks that the shortened compound actually runs for as long as the suspense numbers say. That would be the natural next measurement and it is a different one.

The convention, named #

Normal play: the player unable to move loses. In the shortened selective compound that means the player whose turn arrives after some component has stopped, whether or not the other components still have moves in them — which is exactly what shortened means and is the whole difference from the ordinary selective compound.

Every position swept has a move available in every component. A compound with a stopped component in it has already ended, and including such positions would be counting the same trivial answer many times over — so the class the rule is stated for is the class of positions in which the game has not already finished.

Every winner above was obtained by solving the compound directly, not by any rule, and every quantity was computed by its own recursion over the same tree.

Where the ladder goes next #

remoteness has two rungs to here: the rule stated, checked and sabotaged, and now the fourth quantity the theory carries and what it decides.

The rung above runs the mixed-compound test this page names, and the result is a clean pass with an explanation better than the result. A compound of two different games finds the minimum-remoteness rule right on all 5,184 mixed pairs and all 7,560 triples — no exception, no degradation, nothing to report about the mixing at all.

The reason is the one this anchor keeps arriving at from different directions and is worth stating once more here. Each rule of the compound theory reads one number per component, and a number does not remember which ruleset produced it. A rule of that shape cannot distinguish a mixed compound from an unmixed one, so a sweep over mixed components is testing the same statement on a wider set of number combinations rather than testing a new statement. It was never in danger.

What mixing does damage is the thing a reader carries instead of the rule: the habit of reasoning about a component from its shape — a heap looks like this, a strip looks like that — which is exactly the information the number discards and exactly what stops being reliable when two rulesets are on the board.

Read beside this page that is a useful pairing. Here a quantity turns out to be redundant because two others already settle everything it governs; there a rule turns out to be unbreakable because it reads so little. Both are findings about how much of the compound theory is really one idea, which is that a component reduces to a number and the compound is a rule about numbers.