One rule makes it cold, the other hot

Two games, one board, and one word of difference in the rule. In Col a player may not paint a vertex next to one of their own colour. In Snort a player may not paint next to one of the opponent’s. Everything else is identical: the same graphs, the same alternation, and the same losing condition.

The first essay on the pair put them side by side and reported that the values come out different in kind. This one measures how different, over every position both games have on this site’s graph library.

What a temperature above zero means #

The number in that table is not a proxy for anything. A position’s temperature is how much a player should be willing to pay for the privilege of moving there rather than somewhere else, and a temperature of zero or below means the answer is nothing.

So the census says something stronger than “Col’s values are simpler”. It says that in Col, across every position on the library, there is never anything worth fighting over. A Col game is a sequence of moves nobody is in a hurry to make; the outcome is decided by how many moves each player will have, and by who runs out first.

Snort is the opposite everywhere it can be. Sixty-one of its positions are fights, and on the largest graph in the library moving first is worth four points against an opponent playing perfectly.

Why the mover is punished, and why that is cold #

The mechanism is in the rule and it is short.

A Col move damages the mover. Painting a vertex blue forbids Left from painting any of its neighbours. So each Left move shrinks Left’s own future supply, and moving is a cost as well as a gain — there is no advantage to grabbing a vertex first, because grabbing it is what takes the neighbours away.

That is precisely the shape of a position that is worth a number. A number is a position where neither player wants to move: every move gives something away, and the incentives are negative. Col arranges for that to be true everywhere.

A Snort move damages the opponent. Painting a vertex blue forbids Right from painting the neighbours, and leaves Left free to continue. So each move is a land grab, both players want the same vertices, and moving first in a contested region is worth something.

That is the shape of a switch: two options, both good for whoever takes them, and a gap between where the position lands under each.

One move, counted both ways #

The mechanism is easiest to believe with the move counts in front of it, so here is the path of four, empty and after Left paints the second vertex.

Empty, both games give both players four legal moves. Left paints vertex two.

Under Col Left is now down to one legal move and Right still has three. Painting has cost the mover three of their own options and none of the opponent’s — the neighbours of the painted vertex are closed to blue and open to red.

Under Snort Left has three legal moves left and Right has one. The same painting has cost the opponent three options and the mover one.

The values follow immediately. Col’s empty path is worth 0 and after that move it is worth −3/2: Left has given away a point and a half by moving. Snort’s empty path is worth { {2 | 1} | {−1 | −2} }, a fight of temperature 3/2, and after Left’s move it is worth 2 | 1 — Left has taken a position that guarantees at least a point.

That is the whole of the difference, and it is arithmetic rather than intuition: one rule subtracts from the mover’s supply and the other from the opponent’s, so one produces positions nobody wants to move in and the other produces positions both players are racing for.

The values themselves #

The census records what values turn up, and the two lists have nothing in common but zero.

Col produces numbers, and numbers with a star. Across the 540 positions the Col values are 0, ±1/2, ±1, ±3/2, ±2, ±3 and ∗ — halves, integers, and seven positions worth ∗ itself. Nothing else, and no $x\ast$ for any $x$ other than nought.

Every one of those is born by day three of the construction that builds the numbers, which is another way of saying that Col on a board of six dots is a game about counting to three. Nothing on the library needs a quarter, and nothing needs anything hotter than a star.

Snort produces switches. 2 | −2 on a path of three, 3 | −3 on a star, 4 | −4 on the bowtie, and on the path of four something with a follow-up on each side: { {2 | 1} | {−1 | −2} }, whose temperature is 3/2.

The word has a third setting #

Own colour and opponent’s colour are two of the ways that clause can be written, and there is a third that completes the picture: may not paint next to any painted vertex, whichever colour it is.

That game is not a third partizan variant. It is impartial — and the reason needs no computation. The legality test no longer mentions the painter, so at any position the set of vertices Left may paint and the set Right may paint are the same set. Both players have the same moves, from every position, all the way down. That is the definition, and by Sprague–Grundy every position of it is worth a nimber.

So the three settings of one clause produce three different kinds of game, and the classification is complete:

That table used to be an argument with nothing computed under it, and the third row is now drawn beside the other two.

$\ast2$ is worth noticing rather than passing over: it is not $\ast$, so the third setting is not merely a first-player win dressed up. It is a Nim heap of two, and the board it is drawn on is Node Kayles — painting a dot takes that dot and all its neighbours out of play, which is what selecting a vertex does there.

The clause’s reference to colour is what makes the game partizan at all. Drop the reference and the two players become interchangeable; keep it and which colour is named decides whether moving costs or gains.

Which turns the census into a prediction #

That reframes what the 540 positions establish. The essay’s mechanism — a move consumes the mover’s supply under Col and the opponent’s under Snort — is a fact about the rulebook, available before a single board is drawn, and it determines the sign of every incentive in the game.

A position is worth a number exactly when every incentive is negative. So a rule that guarantees every move costs its maker more than it costs the opponent guarantees coldness, everywhere, on every graph, without any board being evaluated. Col has that guarantee written into its legality test.

The census is therefore not the discovery. It is the confirmation, and it is confirming something that could have been asserted from the rule — which is worth saying, because it means the result extends past the library. No Col position on any graph has anything at stake, not merely none of the 540, and the reason is the clause rather than the sample.

Snort has no matching guarantee in the other direction, which is why the essay’s own claim there is 61 hot positions rather than all of them. A Snort move takes options from the opponent, so incentives are typically positive — but a board can run out of contested vertices, and a Snort position whose remaining vertices are unreachable by one player is a cold position in a hot game. The asymmetry between the two counts, 0 and 61, is exactly the asymmetry between a guarantee and a tendency.

And the third setting is the control the pair was missing. A reader shown two games differing in one word, one cold and one hot, might reasonably suspect that any change to the clause produces one or the other. It does not: the third setting produces neither, because it stops distinguishing the players and the whole partizan question dissolves. One clause, three settings, and the space of answers is larger than the pair suggests.

The seven starry positions #

Col is not quite a game of pure counting, and the exception is worth naming because it is what “cold” means precisely.

Seven of the 540 Col positions carry a star, every one of them is worth exactly $\ast$ rather than some $x\ast$, and all seven are on the star. A star is not a number: it is confused with zero, and its presence means the position contains a move that matters even though no points do.

That is the general shape of a cold game, and it is a stronger statement than “the values are numbers”. A cold game’s values are numbers or numbers plus infinitesimals: the points are settled, and there may still be a question about who is left holding the move.

So the honest headline for Col is not that it is a counting game but that it is a counting game with a parity question attached, and the parity question is exactly what the star records. It is also the one place in the library where the two games return the same value for a reason other than the board being finished.

What the census is evidence for #

The zero in the Col column is measured on 540 positions and it is also a theorem, and the two are worth keeping apart.

The theorem: every Col position is worth a number or a number plus a star. It is the standard result about the game, and its proof is an induction on the position rather than a search — the base case is a fully painted graph, and the inductive step turns on the fact that a Col move restricts only the mover.

The census: 540 positions computed independently of that argument, by the same recursion that evaluates every other partizan position on this site, none of them hot and seven of them starry.

The relationship between the two is the one this site keeps insisting on. The census cannot prove the theorem: 540 positions is not all positions, and a counterexample on a graph nobody drew would be invisible from here. What the census does is make the theorem checkable — it is a test the claim could have failed, run on a body of positions the claim was not derived from.

That is also why the seven starry positions matter more than their number suggests. A census that returned nothing but numbers would be evidence for a stronger and false claim, namely that Col values are numbers full stop. The stars are the part of the theorem a careless statement drops, and they are in the data.

Where the fights are, and what to do about them #

The practical difference between the two games is what a player has to think about.

In Col there is no move-ordering problem worth the name. Every component is worth a number or a number plus a star, the values add, and the whole board is settled by arithmetic and a parity check.

In Snort the ordering problem is the game.

That is why playing the hottest is a rule about Snort and not a rule about Col: it answers a question Col never asks.

Comparison settles what inspection cannot #

One caution about reading values off a table, which the census makes easy to forget.

Three of the census values compared against zero settle it. A half is genuinely greater than zero. A star is confused with it — neither above nor below. And the symmetric Snort switch $\{2 \mid -2\}$ is confused with it as well, despite being worth nothing on average: two of those three are positions the mover wins, and no arithmetic on their means distinguishes either from a draw.

A value of 2 | −2 has mean zero, and a reader who summarised it as “worth nothing” would have the outcome exactly wrong: it is a first-player win, worth two points to whoever moves. The census counts positions with a positive temperature precisely because temperature is the number that catches this and the mean is not.

The pair as a test case #

Two games this close together are useful for more than their own sake, and the census is a reminder of what they are useful for.

Any claim of the form “games of this shape behave like that” can be run against both. They share a board, a move alphabet and a losing condition, so anything that distinguishes them has to be about the one word that differs — which makes them an unusually clean control.

Three examples from elsewhere on this site. The claim that temperature is the right way to size a move is vacuous on Col, where every temperature is zero or below, and is the whole game on Snort. The claim that outcomes do not determine the sum is demonstrable in both, and the Col demonstration needs the starry positions to make it. And the disjunctive sum behaves identically in the two, because addition of values knows nothing about where the values came from.

So the pair separates the parts of the theory that are about fights from the parts that are about sums. The first half applies to one of them, the second half to both, and a reader who wants to know which half a technique belongs to can ask what it says about Col.

Where the model stops #

Six graphs is a library, not a theorem. Every claim above is measured over the graphs this site keeps — paths of three and four, a star, a triangle, a square and a bowtie — and their 540 colourings. That Col has no hot position on those is a fact about those. It is also a known theorem about Col in general, and the census is evidence for it rather than a proof of it.

Snort is not solved. Its values on larger graphs are not known in general, and the hottest value on this library is a fact about the library. A bigger graph with more contested structure would be hotter, and how much hotter is exactly the question nobody has a formula for.

And the temperature is computed on a single component. Both games are played on one graph here. A real Snort game on a disconnected board is a sum, and the sum’s temperature is not the sum of its parts’ — the census says how hot each component is, and the board’s answer needs the disjunctive sum machinery on top.

What a player does with a cold game #

If Col contains no fights, the natural question is what is left to think about, and the answer is a different skill entirely.

The whole of Col is counting spare moves. A position worth +2 is one where Left has two moves in hand; a position worth 0 is balanced; a position worth ∗ is balanced with an odd move left over. The board is a sum of regions, the values add, and the player with the larger total wins by arithmetic rather than by tactics.

That is the same discipline Cutcake demands — every Cutcake value is an integer, and the game is a count from beginning to end — and it is the discipline a chess player uses in a blocked pawn ending, where the material is level and the question is who runs out of waiting moves.

So the two games are not a hard one and an easy one. They are two different games with two different skills, and the census is the measurement that says which is which: a game where every temperature is zero is a game of accountancy, and a game with a value of 4 | −4 in it is a game of grabbing.

What the picture cannot show #

The census figure is a table of counts, which is the honest way to state a claim about 540 positions and is also the least illuminating way to look at any one of them.

What no figure here shows is which Snort positions are hot. The count of 61 is a fact about a set, and the set has structure — the hot positions are the ones with an uncontested region large enough to be worth claiming — that a table of totals cannot display. Drawing all 61 would be a wall of small graphs; drawing three of them, as this essay does, shows the mechanism and not the distribution.

The other invisible thing is the negative result. “No Col position is hot” is a claim about an absence, and an absence has no picture: the zero in that column is the entire content of the strongest statement on this page, and it is one character wide.

The convention, named #

Normal play, and one rule convention that decides everything above.

A player who cannot paint loses. That is what makes Col a game about running out of moves and it is the reason Col’s values are counts. Under misère play — the player who cannot move wins — a Col position that is worth a comfortable +2 becomes something else entirely, and the value theory does not survive the change.

The second convention is that a vertex, once painted, stays painted, and that both games are played to exhaustion rather than to a score. A scoring version of Snort, where the winner is whoever has painted more, would be a different game whose answer is a margin rather than a value, and counting at the end changes everything is the essay about how much that changes.