A game with nothing at stake

End-Nim is Nim with one clause changed: Left may take only from the leftmost non-empty heap and Right only from the rightmost. Taking from the ends is where the game arrives, and the two rungs since have measured how little of the impartial theory reaches it — 410 of 5,460 rows are worth nimbers, and a row is worth a nimber exactly when it is its own negative, with no exception.

That last coincidence was explained with a claim the page could not make good on:

The argument that End-Nim contains no hot self-negative values is the whole explanation of this page’s coincidence, and it is stated and not computed. A census of End-Nim temperatures would settle it in one pass at this size.

The census is one pass. It says something considerably stronger, and the stronger statement has a one-line proof that was sitting in the rules the whole time.

Nothing is hot #

Not the self-negative rows. Not any of them. Over all 5,460 rows of up to six heaps of up to four counters, the temperature is nought on 5,099 and undefined-because-a-number on the other 361, and it is above nought on none.

That is unusual enough to be worth stating plainly. End-Nim is a thoroughly partizan game with 2,693 distinct values in which nothing is ever worth fighting over. Every position is a second-player-ish squabble about who runs out of moves, and no position offers either player anything worth taking now rather than later.

The corollary is what the rung below wanted. A hot self-negative value would have broken the nimber coincidence; there are no hot values at all, so it could not have.

Why, in one line #

The reason is not about values. It is about the move rule.

In End-Nim a player takes from an end of the row. Left has a move exactly when there is a non-empty heap, and so does Right. So either player can move precisely when the row is not empty — and that, in this subject, is the definition of an all-small game.

An all-small position has both its stops at nought, because whichever player is trying to bank a lead runs out of moves at the same moment the other does. Both stops at nought means both walls of the thermograph rise from the same point, which means the mast is vertical from the ground up and the temperature is nought.

So every End-Nim value is an infinitesimal: smaller than every positive number and larger than every negative one. The census confirms it on all 5,460 rows, and asserts it rather than reporting it — a row that were not all-small would mean the argument above is wrong even if the temperature came out right.

And the 361 numbers are one number. The rung below counted 361 rows worth numbers; an all-small game that is a number is the zero game, so those 361 are 361 rows worth nought. That is a fact the earlier census had in hand and had no reason to look at.

What that looks like on a diagram #

A thermograph is the picture of what is at stake, and an all-small position makes a particularly dull one.

Both stops are at nought, so the two marks on the base line coincide. The walls therefore meet at height nought, and the mast — the vertical line above the meeting point, which is where the value settles once moving is taxed — starts on the ground.

Compare that with a switch, where the two walls start apart at the bottom and lean in until they meet at the temperature, and the height of the meeting is the number a player is deciding whether to fight for. An End-Nim row has no such height. There is no tax at which the position becomes worth moving in and none at which it stops, because it was never worth moving in for the sake of the value.

That does not make the game trivial to play. Somebody still wins, and which of them wins is decided by an infinitesimal that took a full canonical-form computation to find. What it makes trivial is the first question the theory usually asks — how much is this worth — and the answer everywhere is nought.

Three shapes of board #

An argument from the rules is a claim about every position, and a sweep is a claim about a sweep. The two are checked against each other on boards of three different shapes.

Heaps up to four in rows up to six, heaps up to five in rows up to five, heaps up to six in rows up to four — 10,919 rows in all, and the answer does not move. It should not move: the deduction says the answer depends on the rule and not on the position, and a sweep where one shape behaved differently would mean the deduction is wrong.

That is what the shape sweep is for. It cannot confirm the argument, because no finite sweep can; it can refute it, and it does not.

What this costs the theory #

A game where nothing is hot is a game the whole apparatus of temperature has nothing to say about. Thermographs, cooling, the mean value, playing in the hottest component — the machinery that occupies a third of this site simply does not engage.

What is left is the harder half. The values are infinitesimals, and comparing infinitesimals is the part of the theory that this site brackets rather than computes. Atomic weight measures an infinitesimal in multiples of ↑ and it measures it as an interval, so a question like which of these two End-Nim rows is better for Left is answered by comparison when the comparison happens to settle it and by a bracket when it does not.

So End-Nim is at once the easiest game here to say something general about and the hardest to say anything specific about. The general statement is a line of the rules. The specific ones need the calculus this site does not have.

There is a practical consequence worth drawing out, because it changes what a player should do. In a hot game the advice is move in the hottest component, and the whole of that advice is a comparison of temperatures. On a board made of End-Nim rows every temperature is nought, the rule says nothing, and the decision about which row to play in is decided entirely by the infinitesimals — which is to say by the part of the theory that has no rule of thumb attached to it at all.

That is the reverse of the usual situation. Most games on this site are easy to play approximately and hard to play exactly; this one is impossible to play approximately, because there is no approximation to make. Either the value is known or nothing is.

The nimbers arrive at odd lengths #

The census turned up something nobody asked it for, and it costs nothing to report.

Break the 410 nimber-valued rows out by how many heaps the row has. At two, four and six heaps every one of them is worth nought — 4 of 4, 24 of 24, 266 of 266, with no star anywhere. At one, three and five heaps the stars appear: all four rows of one heap, 10 of 16 at three, 35 of 96 at five.

The shape of an explanation is visible and this page has not written it. A row of an even number of heaps is one in which a Left move and a Right move can be paired off from the outside in, which is the sort of argument that produces a second-player win — and a second-player win among the nimbers is nought. Whether the pairing survives heaps of different sizes is what a proof would have to establish, and 5,460 rows are data for it rather than a substitute.

It is reported and not asserted for that reason. Six lengths are not enough to make a claim of a parity, and the shape sweep does not check it.

The coincidence, closed #

The rung below’s finding was that a row is worth a nimber exactly when it is its own negative, on all 5,460 rows. Half of that is a theorem for every game: a nimber is its own negative, so nimber implies self-negative everywhere and needs nothing about End-Nim.

The other half is the coincidence. In general the self-negative values are a strictly larger class than the nimbers — thirty values on day three are their own negatives and four of the thirty are nimbers — so the equality here is a fact about this game.

The temperature census is what closes it, and the argument runs the way the rung below hoped. The self-negative values that are not nimbers are, in the small cases, hot ones — a switch and its reverse cancel without either being impartial — and End-Nim has no hot values at all. What is left of the self-negative class once the hot part is removed is very nearly the nimbers, and on 5,460 rows it is exactly them.

That is an explanation and not a proof, and the gap in it is named by the same day-three census. Of its thirty self-negative values, four are nimbers and seven are all-small — so three of them are infinitesimal, self-negative and not nimbers, $\{\uparrow\!\ast, \uparrow \mid \downarrow\!\ast, \downarrow\}$ among them. Removing the hot values from the self-negative class does not leave the nimbers in general; it leaves the nimbers plus a residue.

So the coincidence goes from two unexplained halves to one. The temperature census rules out the hot part of the class everywhere in End-Nim, which is most of it; why the game also produces none of the three infinitesimal non-nimbers is a question this census does not answer, and the honest reading is that this game’s values are a small and well-behaved corner of the infinitesimals rather than an arbitrary sample of them.

What the census does not say #

Four limits.

Cold is not simple. A temperature of nought means nothing is at stake; it does not mean the position is easy or that the value is small. The rows here carry 2,693 distinct values on 5,460 positions, and the rung below measured that count running away as the rows grow — 641 values at five heaps and 2,053 at six. Coldness is a statement about the scale of the values, not about how many there are.

The all-small argument is about End-Nim and not about end-taking games generally. Change the rule so that a player may pass, or so that one end may be exhausted while the other is not, and the equivalence Left can move iff Right can fails immediately. It is a fragile property and it holds here exactly.

No exhaustive sweep proves a rule. The argument in the second section is the proof and it is three sentences of prose rather than an induction written out. What the census establishes is that the evaluator agrees with it on 10,919 positions, which is a check on the evaluator as much as on the game.

The parity is an observation on six lengths. Three even lengths with no star and three odd ones with stars is a pattern over six cases, and this site has just spent an essay on a pattern over five cases that turned out to be a coincidence. The right weight to give it is that of a question.

And the sweeps are bounded in both directions. Six heaps of four counters, five of five, four of six. A row of twenty heaps is not evaluated here and there is no reason to expect it to differ — the argument does not mention the size — but there is a difference between no reason to expect and checked.

A game with no fights is a game with no theory to apply #

A ruleset every position of which is cold sits oddly in a collection about values, and it is worth saying what such a game does and does not need from the apparatus.

It needs the definition and nothing above it. A cold position is worth a number, the numbers add, the sign settles the board, and the answer is arithmetic. Temperature, thermographs, means, the hottest-first rule and the whole endgame accounting have nothing to do — not because they fail, but because every quantity they compute is degenerate.

And it needs the definition badly. Every position is a number is a strong claim about a whole game tree — no position anywhere has a move worth making — and establishing it requires exactly the machinery it then makes unnecessary. The apparatus is spent proving that it will not be needed.

That is a real service and a strange-sounding one, so it is worth stating plainly. A theory earns its place partly by identifying the cases where it is not required, and a game certified cold is a game a player can count rather than analyse, with the certification standing behind the counting.

It also makes the game a control. Every claim on this site about what heat does can be checked against a ruleset that has none: a rule that behaves the same on hot and cold games is a rule not reading the heat, and a measurement that produces an interesting number here is a measurement of something other than temperature. That is worth more than another hot game, and it is why a family with no fights in it is in a collection about fighting.

What it changes about the earlier rungs #

Two of this anchor’s earlier findings read differently once the coldness is in hand, and it is worth saying how.

Where the nimbers run out counted 361 rows worth numbers and 410 worth nimbers, and treated those as two overlapping classes of a general population. They are not: the 361 are one value repeated, so the whole of the number-valued part of End-Nim is the single value nought, and the interesting classification was always the nimbers against everything else.

And the two-heap rule — the larger end wins — reads as a statement about outcomes because that is all it can be. There is no value for it to be a statement about beyond the outcome class, since the values it ranges over are all infinitesimal and their comparison is exactly the outcome question. A rule of thumb about how much the larger end is worth could not exist here.

The convention, named #

Normal play: the player who cannot move loses. A row is a sequence of heap sizes, and it is a sequence rather than a multiset — the order is the whole game, since Left reaches one end and Right the other.

All-small means that in every position reachable from this one, Left has a move exactly when Right does. It is decided by walking the position tree rather than by reading the rules, and it is checked here on every row rather than deduced — so the census tests the argument of the second section rather than assuming it.

The temperature is the height at which neither player wants to move, read off the thermograph. A number is given temperature −1 by convention here, so that a sum’s stack of temperatures contains only its genuinely hot components; the 361 rows reported at −1 are the rows worth nought.

Where the ladder goes next #

The endnim anchor has four rungs: the game with its two-heap rule, how little of the impartial theory reaches it, the condition that says which rows are impartial, and now that the whole game is an infinitesimal.

The rung above is the atomic weight, and this page is the argument for why it is the rung that matters here rather than one option among several. A game with no temperature is a game whose only interesting quantity is the one measured in multiples of ↑, and End-Nim supplies 2,693 distinct infinitesimals to measure — a larger and better-organised supply than any other game on this site. Whether the atomic weights of End-Nim rows have a pattern in the heap sizes is a question nobody can ask until there is a calculus to ask it with, and it is the fifth essay on this site now waiting on the same piece of machinery.

Two neighbours are worth the trip. The class where nobody runs out first is where all-small games are taken on their own terms, and End-Nim turns out to be one of the largest examples on the site — hiding in a field about particular positions rather than in the field about values. And big is not the same as hot is where the independence of size and temperature is established, and this game is the extreme case of it: values born deep into the third day and beyond, every one of them with nothing at stake.