Topple it from either end

The first picture on this site is a row of coloured segments standing on the ground. Left may cut a blue one, Right a red one, and everything that loses its footing falls. The value of the row is a number, always, and reading the colours from the ground upward writes that number in binary. The picture is not a diagram of the value; it is the value.

Toppling Dominoes is the same picture with the pieces standing on edge in a line rather than stacked on the ground, and one rule changed. Left knocks over a blue domino in either direction, and everything on that side goes down with it. Right does the same to a red one. Nobody who has not been told which game they are looking at can tell the two apart.

They could hardly behave less alike.

The rule, and why it is cheap #

A move topples one domino and clears one side. So every position reachable from a row is a contiguous substring of it, and a row of $n$ has at most $n(n+1)/2 + 1$ positions beneath it however long the game runs. That is a small enough state space to solve exhaustively for rows far longer than any figure can draw.

It also means the move is two slices. Toppling the domino at position $i$ leftward leaves everything after it; rightward, everything before it. Each domino therefore offers its owner exactly two options, and a row of $k$ blue dominoes gives Left $2k$ moves, several of which coincide.

That row is worth $\uparrow$, which is worth pausing on. A blue-red-red-blue Hackenbush string is worth $\tfrac38$ — a number, positive, and Left is comfortably ahead. The same four pieces under toppling are worth up: positive, but smaller than every positive number, and a value from the infinitesimal end of the theory entirely. Left is ahead by an amount no number can measure.

The census #

Every row of up to eight dominoes is 510 positions — two of length one, four of length two, and so on. Evaluating all of them gives three counts.

Thirty are numbers. Under six in a hundred. As a Hackenbush string every one of the 510 is a number, so the change of rule takes a game made entirely of numbers to one that is almost entirely not.

The thirty are not a mystery. Sixteen of them are the rows of a single colour, worth $\pm n$ for a row of $n$; the rest are short rows and a handful of accidents. A row with both colours in it has a fight in it, because whoever moves gets to decide which end of the row survives.

Two are star. $\mathsf{LR}$ and $\mathsf{RL}$ — one domino of each colour — and they are worth $\ast$ for the reason $\ast$ always turns up: each player has exactly one useful move and it clears the board.

The temperature climbs, and that settles the matter #

The column worth staring at is the last one. The hottest row of length two has temperature $0$; of length three, $\tfrac12$; of length four, $1$; and so on to $3$ at length eight. Half a unit per domino, with no exceptions in the range swept.

That is a growth no game whose values are numbers can have. Temperature is what is at stake — the difference between moving first in a position and moving second — and a number has nothing at stake, which is why the site’s convention puts every number at $-1$. A family of positions whose temperature grows without bound is a family in which the advantage of moving first grows without bound.

The row that achieves it is not subtle: seven blue dominoes and one red, $\mathsf{LLLLLLLR}$, worth $\{6 \mid 0\}$. Left topples the rightmost blue domino rightward, which takes the red down with it and leaves six blue standing — six free moves. Right topples the red leftward, which takes the whole row down and leaves Left nothing. One piece of the minority colour, at the end of a long run of the majority, is a fight over the whole run.

The climb has a formula #

“Half a unit per domino, with no exceptions in the range swept” is a measurement over eight lengths, and the family achieving it is regular enough to write down — which turns the observation into a statement about every length rather than about the sweep.

The hottest row of length $n$ is $\mathsf{L}^{n-1}\mathsf{R}$, and

$$\mathsf{L}^{n-1}\mathsf{R} \;=\; \{\, n-2 \;\mid\; 0 \,\}, \qquad \text{temperature } \tfrac{n-2}{2}.$$

Both halves come straight from the two moves already described. Left’s best is to topple the last blue rightward, which takes the red with it and leaves $n-2$ blue standing; Right’s best is to topple the red leftward, which clears everything. Two options, both numbers, so the value is a plain switch and its temperature is half the gap.

The formula holds where the sweep looked and it does not stop there: at nine dominoes the row is $\{7 \mid 0\}$ and the temperature is $\tfrac72$. So the temperature is not merely growing over the range measured — it is unbounded, by a family with one member per length and a closed form for each.

Which is a stronger statement than the census makes #

That matters because a census can only ever report what it saw, and the claim the essay wants is about the game rather than about eight lengths.

A family of positions whose temperature grows without bound is the sentence in the section above, and a table of eight rows does not establish it — a sequence can climb for eight steps and level off. A family with a formula does establish it, and the formula is two lines of play rather than a search.

And it says exactly what is being paid for. The hot row is one red domino at the end of a run of blue, and what the two players are fighting over is the run itself: whoever topples the red decides whether $n-2$ blue moves survive or none do. The stake is the length of the row, so the temperature is half the length, and a game in which a single move can decide the fate of the whole board is a game with no ceiling on what a move is worth.

Set that against the game the picture is borrowed from. A Hackenbush string of the same $n$ pieces is worth a number between $-n$ and $n$, and no move ever decides more than one edge’s worth of it, because a cut removes what is above and above is fixed. The unbounded temperature is the price of letting the mover choose a direction — the same clause the section below identifies as the whole difference, now with a number attached to it.

Sixteen agreements, and what they are #

The two games agree on exactly sixteen of the 510 rows, and every one of them is a row of a single colour.

A monochrome row is the one case where neither player has a decision to make. In Hackenbush a row of $n$ blue edges is $n$ free moves for Left; in Toppling Dominoes it is also $n$ free moves for Left, because Left can topple the outermost one and lose nothing, and Right cannot move at all. Both games are $n$, and they agree because there is nothing in the position for a rule to disagree about.

Everywhere else the rules differ in a way that matters, and it is worth naming precisely what the difference is.

In Hackenbush, a cut removes the piece cut and everything above it, where “above” is a fixed direction — away from the ground. The structure is a stack and every move is a truncation from one end.

In Toppling Dominoes, a topple removes the piece and everything on a side of the mover’s choosing. The structure is a line with two ends and every move is a truncation from an end the mover picks.

So the change is not really “cut versus topple”. It is the addition of a choice of direction, and it is that choice that makes the game hot. A player who can decide which half of the board survives is a player whose move is worth fighting over.

The values that come out, and what they are made of #

Two hundred and eighty-four distinct values from 510 rows is a rate no other row game here comes close to — Shove manages 286 from 728 and it is a game of nothing but numbers, so its values are simply a lot of fractions. Toppling’s are not.

The eight-domino sweep produces integers, halves, quarters and eighths; it produces $\ast$, $\uparrow$, $\downarrow$ and their relatives; and it produces switches of every temperature up to three. That is a spread across the whole of the theory’s vocabulary from a game whose entire rule fits in one sentence.

The infinitesimals are the surprise, and $\mathsf{LRRL} = \uparrow$ is the smallest instance. A position is worth $\uparrow$ when Left is ahead by less than every positive number, which sounds like a delicate condition and turns out to be what a symmetric row of four produces. The symmetry is doing the work: Left’s two moves reach $0$ and $\{0 \mid -1\}$, Right’s reach $0$ and $\{1 \mid 0\}$, and the position that results is the canonical $\{0 \mid \ast\}$ once the reduction has run.

What the picture cannot show #

Nothing in the drawing distinguishes the two games, which is the whole point of the essay and also its warning. A row of blue and red is not a position; it is a position together with a ruleset, and the ruleset is not in the picture.

This site’s figure standard states the general caution — show the position beside the notation, because notation is a compression that throws the position away. This is the mirror of it: a picture is a compression that throws the rules away. The reader has both halves only when the caption supplies the one the drawing cannot.

That is a caution the essay on independence makes in the other direction, about a drawing that hides a dependency. Here the drawing hides a whole ruleset.

There is a second thing the picture cannot show, and it is specific to this game. The value of a row is not a function of any local property of it. Two rows with the same number of blue and red dominoes, in different arrangements, are usually worth different things, and no reading of the picture from either end produces the answer — unlike Hackenbush, where reading from the ground upward is the answer. Toppling Dominoes has no numeral.

The second column is the sharper observation. Toppling Dominoes cannot tell a row from its reverse, because the rule treats the two ends alike, so $\mathsf{LRRR}$ and $\mathsf{RRRL}$ are one position drawn twice. Hackenbush is built on a preferred direction — the ground — and distinguishes them by more than a unit and a half. Two of the four pictures above are the same game and two are the same game, and nothing about the drawing says which pairs.

The other row game on this site, and the contrast #

Shove is the other strip on this site made of blue and red pieces, and its census reads the opposite way: all 728 of its positions are numbers, so its temperature is $-1$ everywhere and there is never anything worth fighting for. Three row games, then, with three answers:

• Hackenbush strings — every position a number, and the number is written on the board.
• Shove — every position a number, and the number is not written on the board.
• Toppling Dominoes — hardly any position a number, and the temperature unbounded.

The three rulesets differ in what a move destroys and in nothing else, which makes the trio about as controlled an experiment as this subject offers. What destroying a choosable half buys is heat; what destroying a fixed half buys is a numeral; and what shoving buys is a number nobody can read.

Playing one, and what it feels like #

A row of eight with one red domino in it is small enough to solve completely and large enough that the right move is not obvious, which is the band this site’s playable figures live in.

The instructive part is how lopsided Right’s two options are. One is worth $0$ and the other is worth $7$, from the same domino, differing only in which way it falls — and there is no other move on the board. A player who topples the red domino without thinking has a one-in-two chance of handing over seven free moves.

Left’s side is the mirror of it and less dramatic: every blue domino gives Left the choice between keeping the run to its left and keeping the run to its right along with the red one, and the best of those is to topple the sixth blue outward, keeping six.

So what the position is really about is a single question — which side of the red domino survives — and both players’ whole move list is a way of answering it. That reading is available from the value and from nowhere else. The row does not look like a seven-point swing, and the two moves that differ by seven points look identical in the picture.

Who found it, and when #

Toppling Dominoes is from Lessons in Play, the introductory text by Albert, Nowakowski and Wolfe, where it does a specific job: it is the game used to show that every short value occurs. Allow green dominoes, which either player may topple, and there is a construction taking any option list to a row realising it. The game is universal, and it is the friendliest universal game there is, because the construction is a row rather than a graph.

The universality claim is worth setting beside the one made for green Hackenbush, which is a different kind of completeness — that every impartial game is a green Hackenbush position, via the colon principle. Toppling’s is the partizan version and needs all three colours.

Universality is a stronger statement than anything in this essay and a differently useful one. It says: for every value, some row. It says nothing about which values the short rows produce, which is the question a reader with a picture in front of them is in. The 510 rows here produce 284 distinct values, which is a great many for eight pieces and a vanishing fraction of what the construction promises.

The two-colour census above is deliberately kept to blue and red, without green, because the comparison is with Hackenbush strings and a Hackenbush string with a green edge in it is not a number either. Holding the alphabet fixed is what makes the sixteen agreements a measurement rather than a coincidence.

Where the ladder goes next #

The first rung out is green: the third colour, the universality construction, and the question of how long a row a given value needs. That is a measure of a value’s complexity in a currency a reader can hold, and it should agree with the birthday sometimes and not others.

The second is the arithmetic of rows. Most of the board games here fall apart during play and are analysed as sums thereafter. A row that has been cut in two by a topple is a single row, not a sum, so Toppling Dominoes never decomposes during play — which makes it unusual among the board games here, most of which fall apart. What it does instead is shrink, and a game that only shrinks is a game whose whole analysis is about substrings.

And the third is the temperature growth, which the census shows and does not explain. Half a unit per domino is a clean enough rate to want a proof, and the row achieving it is regular enough to suggest one: a run of $n$ of one colour with a single piece of the other at the end is worth $\{n-2 \mid 0\}$, and the whole question is whether anything beats it.