Tiny, miny, and the sizes below every size

An empty Domineering board, two squares by four. Left places dominoes upright, Right places them flat, and a player with nowhere to put one loses. It is as ordinary a position as this subject contains — no cleverness in the setup, nothing removed, nothing prepared.

Right wins it, whoever moves first. That much a search settles in a moment. The interesting part is by how much, and the answer is a value that no amount of arithmetic will produce.

Being worth less than zero and more than every negative number is not a contradiction; it is the ordinary state of affairs once positions stop being numbers. Up is the same statement with the signs flipped: greater than zero and smaller than every positive number.

What is new here is that this position is not down, and not two downs, and not any multiple of down. Comparison says so directly. Subtract down from it and neither player wins moving second; the two are confused, which is the relation that exists in this subject and does not exist among numbers.

So the class of things smaller than every positive number is not exhausted by the multiples of up and down. It contains a second family, and the family has a scale.

Tiny, with a subscript #

The family is written with a subscript, and the subscript is a game rather than an index:

$$+_x \;=\; \{\,0 \;\|\; 0 \mid -x\,\}, \qquad -_x \;=\; -(+_x) \;=\; \{\,x \mid 0 \;\|\; 0\,\}.$$

Read the first as a position: Left may move to zero. Right may move to a position in which Left may move to zero and Right may move to $-x$. That is all. The whole of tiny is one asymmetry — Left’s escape is immediate, Right’s escape costs a move and then exposes a threat of size $x$.

Two things about that position are worth having before the comparisons start. Nothing in it is dominated and nothing is reversible, so $+_1$ is already its own canonical form — which is why the subscript cannot be simplified away, and why a tiny with a larger threat inside it is a different game rather than a different spelling of the same one. And the threat is genuinely a threat rather than a move: Right’s escape does not reach $-x$, it reaches a position from which Right could reach $-x$ if allowed a second move, and Left is never obliged to allow it. The whole family is built out of that one gap.

The two-by-four board above is $-_2$: miny-two. The threat inside it has size two, and everything else about the value follows from that.

The larger the subscript, the smaller the value #

Here is the part that is easy to get backwards, and the reason the figure at the head of this essay exists.

$+_1$ is larger than $+_2$, which is larger than $+_4$. The subscript measures the size of Right’s threat, and a bigger threat is worse for Left, so the position is worth less. Every one of them is still greater than zero, and every one is still smaller than every positive number.

The ordering is strict, and it goes on for ever downward. $+_x$ is defined for every game $x$, not merely for integers, and $+_{1/2}$ sits strictly above $+_1$ because half a threat is a smaller threat. There is no smallest tiny and no largest, which is a sentence that could be said about the positive rationals and means something quite different here: these are all below every positive rational at once.

The last column of that table deserves a paragraph of its own, because it is the property the family is named for in the literature and the one that makes tiny worth having.

Add twelve copies of $+_1$ together. The total is still less than $\uparrow$. Add sixteen; still less. There is no number of copies that reaches up, in the same way there is no number of copies of up that reaches $1/1024$.

That is a second layer of infinitesimality inside the first. Up is infinitesimal with respect to the numbers. Tiny is infinitesimal with respect to up. And atomic weight, which measures positions in multiples of up, therefore has nothing to say about a tiny except that it is somewhere between zero and one up — which is exactly what the bracket comes back as.

Every relation in the table, and how each one was got #

It is worth slowing down on one row, because the whole family is ordered by a single operation and the operation is not obvious.

To ask whether $+_1 > +_2$, the site builds $+_1 - +_2$ — the first position beside a mirror-image of the second — and asks who wins the resulting sum moving second. That is the definition of the order and there is no other. The answer comes back: the second player is Left, so $+_1 - +_2 > 0$, so $+_1 > +_2$.

One subtraction settles one relation, and a table is nothing but a stack of them — no theory anywhere in it. So the chain can be run out as far as anybody has patience for, and running it out is the way to see that it does not thin, taper or approach anything.

There is a relation those columns do not contain, and it is the one that keeps the family honest. $+_1$ and $\ast$ are not comparable at all: their difference is a first-player win, so neither is at least the other, and no amount of smallness forces a value onto the line. The order is partial among the infinitesimals exactly as it is everywhere else, and the tinies are a chain inside a much wider structure rather than being the structure.

The layers do not stop at two #

Up is infinitesimal with respect to the numbers; tiny is infinitesimal with respect to up. That is stated above as a second layer, and it is worth asking whether there is a third. There is, and it is inside the family already on the page.

Add eight copies of $+_2$ together. The total is still below $+_1$. The same is true of eight copies of $+_4$, and of any number of copies anybody cares to try.

So $+_2$ is not merely a smaller tiny than $+_1$ — it is infinitesimal with respect to it, in precisely the sense the essay has already used twice. The subscripts do not index points on one scale; each one names a scale of its own, and every scale sits entirely inside the gap between zero and the one above it.

One instance of that is a coincidence and two are a construction, so the same move is made again a floor lower.

$$\text{numbers} \;\gg\; \uparrow \;\gg\; +_1 \;\gg\; +_2 \;\gg\; +_4 \;\gg\; \cdots$$

with $\gg$ meaning no number of copies of the right-hand thing reaches the left-hand one. The chain has no end: the subscript takes any game, so a larger threat produces a scale below every scale already named.

Which is what “smaller than every positive number” was hiding #

That changes what the opening sentence of the whole subject is doing.

Smaller than every positive number sounds like a description of a narrow class sitting just above zero — a fringe, with the real content on the number line. It is not a fringe. It is an infinite tower of scales, each one invisible from the one above, and the numbers are only the top floor.

And it says why the measuring instruments come in a sequence. Temperature reads the top floor and reports nothing below it. Atomic weight reads the next floor down, counting in ups, and reports nothing below that — which is exactly why a tiny’s bracket comes back as “somewhere between zero and one up” and cannot be sharpened. A third instrument would be needed to separate $+_1$ from $+_2$, and this site does not have one, because the comparison is the instrument at that depth and comparison does not aggregate.

That is the honest form of the limitation the last section reports. The bracket is not crude; it is calibrated in ups, and the objects it is being asked about live a floor below its units. Every measurement in this field is a floor of the tower, and running out of floors is what happens when the questions get small enough.

A tiny and its miny cancel exactly #

$-_x$ was defined as $-(+_x)$, and that is not a definition to be taken on trust. It says the two positions annihilate: put a tiny beside its own miny and the result is worth precisely zero, a second-player win, whoever starts.

Three instances of that are checked directly: $+_1$ beside $-_1$, the two-by-four Domineering board beside the same board turned through a right angle, and $\uparrow$ beside $\downarrow$. Each sum is built by the recursion and each comes out a second-player win, which is what a value of zero means and the only thing it means.

The middle one is worth reading as a claim about Domineering rather than about notation. Right is ahead on a two-by-four board by an amount smaller than every number. Rotate a second board through a right angle — so that Left is ahead on it by the same unmeasurable amount — and the two boards together are a dead heat that the second player wins. Two positions neither of which is worth anything a number can express, cancelling exactly.

That is what being a group buys, and it is the reason subtraction is available as a tool at all. It is also the first thing misère play destroys.

Two conditions that are not the same condition #

The essay on atomic weight defines the all-small games: those in which either player having a move implies the other has one, all the way down. Every all-small game is infinitesimal, and it is tempting to run the implication backwards.

It does not run backwards. Tiny is infinitesimal and tiny is not all-small.

The reason is visible in the notation. $+_1$ has a Right option $\{0 \mid -1\}$, which has a Right option $-1$, which is a number — and in the number $-1$, Right has a move and Left has none. One asymmetric option, several levels down, is enough to fail a condition that quantifies over every subposition.

That has a consequence for the instrument the rung below this one built, and the consequence is a refusal rather than a bad answer. A bracket in multiples of up is drawn only for a position that has passed the all-small test, and a tiny does not pass it, so no bracket is drawn — not a wide one, not a vague one, none. The measurement declines the object rather than reporting a measurement of nothing, which is the correct behaviour and is also why nothing in this essay is drawn on that axis.

So the picture is two nested classes rather than one. The all-small games sit inside the infinitesimals, and tiny is in the outer ring: small enough to be invisible to every number, structured enough to be invisible to atomic weight, and still perfectly ordered.

Where it turns up #

$-_2$ came out of an empty Domineering board, which is worth saying plainly because most of the values in this family are met first as notation.

The contrast in that figure is the useful one. The two-by-two and three-by-three boards are worth $1 \mid -1$, a switch with a whole move at stake. The two-by-four is worth a miny, and a miny is what is left when a position has an advantage that no exchange can convert into a move. Right is ahead on the two-by-four, and cannot cash it.

Nothing else drawn on this site is worth a tiny. That is a searched claim rather than an impression: every Toads and Frogs strip up to seven cells, every Clobber row up to six stones, and every colouring of every graph in the site’s Col and Snort repertoire were evaluated and compared against $+_1$, $+_2$ and their negatives, and none of them matched. Domineering supplied the one. Games that produce tinies in quantity — Toppling Dominoes is the standard example — are not yet in this collection, and that is a gap rather than a discovery.

What the solver computed, and how #

Every relation printed in the tables is a comparison, and comparison here is one thing: build the difference, and ask who wins it moving second.

$+_x$ is constructed by the same game(L, R) call that builds every other position on this site, with $x$ built first. The ordering claims — $+_1 > +_2 > +_4$ — are three subtractions and three outcome computations. The claim that a tiny is below up is another, and the claim that a tiny is below another tiny is the same operation with a different second argument, which is the whole of what makes the deeper tables above cost no more than the shallow ones. The claim about twelve copies is a sum of twelve identical games, canonicalised, and then one more comparison.

That last one is by far the most expensive thing on this page, and it is expensive for a reason worth reporting rather than hiding. The first version of this figure summed sixty-four copies, on the assumption that repeated addition would behave the way it does for a switch — where $n$ copies of $\{2 \mid 0\}$ have a canonical form two lines long however large $n$ gets. Tinies do not behave that way. The canonical form of $n$ copies of $+_1$ grows with $n$, and the intermediate sums grow faster again, because a sum is built before it is reduced. Twelve copies cost 53 MB. Sixteen cost 223. Twenty-four cost 915. Sixty-four exhausted a four-gigabyte heap and took thirty-eight seconds, inside a figure, inside a build — which is how the assumption was found to be wrong.

Two things were changed rather than one. The running sum is now reduced at every step instead of at the end, which keeps the intermediate tree the size of its value rather than the size of its history; and the generator refuses more than sixteen copies outright, with the reason in the error. The counts in the figures above are twelve, sixteen and eight, and each is the smallest that makes its own point: sixteen where the yardstick is a genuine number and the gap to be crossed is largest, eight where the yardstick is another tiny and one copy would nearly do.

The lesson generalises past this page. Adding games is cheap when the values are simple and is not cheap in general, and which of those a particular family falls into is not something to guess at.

None of that is special-cased. The generator is handed a list of subscripts and a value to compare against, and it prints whatever compare returns, including the confusion sign if that is what comes back. A tiny confused with up would print $\|$ and the figure would still build; the claim in the caption would then be visibly false, which is the correct behaviour for a figure whose job is to be checkable.

The site’s gate adds what a figure cannot see. It requires the ordering to be strict in the stated direction, which a bug that collapsed the family to one value would fail. It requires each tiny to be greater than zero and less than $1/1024$, which a sign error would fail. And it requires the all-small test to reject tiny, which is the assertion that would quietly disappear if the two conditions were ever conflated in the code — an assertion that has never rejected anything proves nothing, and this one rejects every time the figures are drawn.

Where the model stops #

The subscript is a game, and this essay only varies it over integers and one fraction. $+_x$ makes sense for any $x$, including values that are not numbers, and the resulting order is much richer than a single descending chain. Nothing here explores that.

Tiny is not a measurement. Atomic weight assigns a number to an all-small position; there is no comparable single number for a tiny, and the bracket that comparison returns — between zero and one up — is honest and nearly useless. What orders the tinies is the subscript, and the subscript is not a size, it is a threat.

The comparisons are exact and the range is small. Every value here has a birthday of four or five. The recursion is not troubled by them. It is troubled by the Domineering boards, which is why the largest drawn is three by four, and a two-by-eight board — where more interesting minies live — is out of reach.

Who found it, and when #

Tiny and miny are Conway’s, and they appear in On Numbers and Games (1976) as part of the apparatus for handling the games left over once numbers and nimbers have been accounted for. The notation $+_x$ and $-_x$, and the reading of the subscript as the size of a threat, are from Winning Ways (1982), where they are put to work on Toppling Dominoes and on Domineering endgames.

The observation that an empty two-by-four Domineering board is worth exactly $-_2$ is not new either; small Domineering values were tabulated by hand long before anybody computed them. What is worth noticing is how ordinary the position is. The theory did not go looking for a strange value in a constructed example — it found one on the fourth-smallest board of a game invented as a pastime.

Where the ladder goes next #

This is the third rung on the infinitesimals ladder, after the class itself and the count of ups that measures part of it. The fourth asks what happens when the structural condition holds everywhere: the all-small games, where a player has a move exactly when the opponent does, and where the whole apparatus of atomic weight applies without qualification.

The rung after that is the one this essay keeps brushing against and declining: the order these values sit in is partial, and the tinies are a chain inside it rather than the whole of it. How wide the class really is — how many mutually confused infinitesimals there are between zero and up — is a question the canonical form can answer and this essay has not asked.