A fight with no midpoint

A switch is the first position a reader meets that is not a number, and it comes with two formulas that are almost too tidy to be true. The value {a | b} with a above b has mean value the midpoint (a + b)/2 and temperature half the gap (a − b)/2, and both can be read off the two options without any machinery at all.

When a switch is not a switch established that the regime has an edge — at a = b the position is a number plus a star, and below it the simplicity rule takes over — and closed by naming what happens on the other axis:

{1 | ∗} has a temperature of ½ and no midpoint to speak of; {2 | {1 | −1}} has a bent wall and stops at 2 and 1 rather than at its options. What the mean and temperature mean once the hypothesis is dropped is the next thing to compute.

The count is the first thing worth reporting, because it is not the count a reader would guess.

Twenty-one of one hundred and sixty-seven #

One hundred and sixty-seven values of day three have exactly one Left option and one Right option. That is what a switch looks like — a single fight, one move each way, nothing to choose between.

Twenty-one of them are switches in the sense the formulas are stated for. The other 146 are positions like {∗2 | 0}, {↑∗ | −1} and { {0, ∗ | −1} | −2}, whose written form is identical in shape and whose options are not numbers.

So the hypothesis fails on seven of every eight positions that satisfy the description. That is worth sitting with, because the description is how the class is usually introduced: a switch is a position of the form {a | b}, and the form is not the hypothesis.

The gap between the two is not pedantry about wording. A reader who has learned the shape has learned to recognise a written expression, and the recognition fires on all 167; the arithmetic that goes with it is licensed on twenty-one. Nothing about the notation flags the difference, which is the standing complaint against brace notation arriving in the one place the notation is usually held up as helpful.

The obvious repair #

The formulas take two numbers and return two numbers. The options of {∗2 | 0} are not numbers, so the formulas cannot be applied — but a position has two other numbers attached to it that always exist, and where the fight stops is the essay about them.

The left stop of a position is the number play settles at with Left moving first and both playing well; the right stop is the same with Right moving first. Both are numbers for every short game, whatever its options are. And on a textbook switch they are the options: {2 | 0} stops at 2 and at 0.

So the repair writes itself. Replace the two options by the two stops:

$$\text{mean} = \frac{\text{left stop} + \text{right stop}}{2}, \qquad \text{temperature} = \frac{\text{left stop} - \text{right stop}}{2}.$$

That is the right shape for a generalisation, and it passes the first test any generalisation has to pass: it agrees with the old reading everywhere the old reading applies. On all twenty-one textbook switches the stops are the options and the two formulas coincide, which the census checks rather than assumes.

Three quarters, and no more #

Off the hypothesis the stop reading gets the mean right on 114 of the 146 and the temperature right on 106. Every position it gets the temperature right on it also gets the mean right on, so the two are not independent failures: there are 106 positions where the stop reading is wholly correct, eight more where it happens to get the mean and not the temperature, and thirty-two where it gets neither.

Three quarters is a good hit rate for a rule with no theorem behind it and it is not a theorem. What matters is which quarter it misses, and the forty misses are two unrelated kinds rather than one. Eight are numbers written as fights, where the reading gets the position exactly right and the temperature convention disagrees with it; thirty-two are genuine, where the reading names a mean and a temperature the position does not have. Only the second kind is about switches at all, and the rest of this page is about those thirty-two.

What the stops throw away #

Every failure has the same cause, and the cause is visible in the diagram rather than in the numbers.

The two stops are the two ends of the thermograph — where the left wall meets the ground and where the right wall does. The mean is where the two walls meet each other, at the top. If both walls are straight lines of slope one, the top is determined by the two ends: the walls close at the midpoint at height half the gap, and the stop reading is exact.

A wall bends when the option it is built from is itself a fight. { {0, ∗ | −1} | −2} has a Left option that is a switch, so its left wall runs at slope one only until that switch cools, and then it turns. The two ends of the diagram are unchanged and the top has moved, so the stop reading predicts the wrong meeting point.

The stops are the boundary conditions and the walls are the function. Two boundary conditions determine a function only when the function is known to be a straight line, and what makes it a straight line is exactly the hypothesis the formulas were stated under.

That gives a description of the successful three quarters as well, and it is a better description than the count. The stop reading works on the positions whose walls do not bend below the meeting point — which includes plenty of positions whose options are not numbers, because an option that is an infinitesimal has a wall that is straight even though the option is not a number. {↑∗ | −1} has stops at 0 and −1, a mean of −½ and a temperature of ½, and the stop reading gets both: up-star is not a number and its thermograph is a vertical line at nought, which is as straight as a wall gets.

The star is worth a sentence of its own, because it is the commonest way an option fails to be a number and it is invisible to both readings. {1 | ∗} has stops at 1 and 0 and is neither above nor below the number 1: comparison comes back confused, exactly as it does for ↓ against nought and for ∗ against nought. A reading built out of two stops has already treated that star as though it were the number it is drawn on top of, which is the one thing a star never is.

The class the reading is exactly right on #

Putting the two halves together gives something more useful than a hit rate.

The formulas hold, in their stop form, exactly when the two walls are straight from the ground to the point where they meet. That happens when each option’s own thermograph has already frozen at the height in question — which is to say when each option is a number or an infinitesimal, since an infinitesimal freezes at nought.

That is a hypothesis a reader can check, it is strictly weaker than the textbook one, and it covers a class seven times larger. And it is not what the twenty-one are: the twenty-one are the positions where both options are numbers, and the wider class also takes in every position whose options are numbers plus something invisible to the temperature scale.

The eight positions where the mean survives and the temperature does not are a separate story, and it is not a story about walls at all. All eight are numbers. {−1/2 | 0} has one option a side and is worth $-\tfrac14$ by the simplicity rule; so are $\tfrac14$, $\tfrac12$, $\tfrac34$, $\pm\tfrac32$ and $-\tfrac34$, each written as a fight and each worth a point on the line. Their stops coincide at the number, so the stop midpoint is the number and the mean is right; their stop half-gap is nought and this site gives a number the temperature $-1$, so the two disagree by construction.

That is a disagreement about bookkeeping rather than about the position, and it is worth separating from the thirty-two, which are disagreements about the value. Counting the eight as failures makes the stop reading look worse than it is on numbers and better than it is on everything else; excluding them leaves 138 non-numbers, of which the reading is exactly right on 106 and wrong on 32, with nothing in between.

All eight are worth drawing, because seeing them together is what makes them look like a bookkeeping convention rather than a class of positions.

Nothing in the written form distinguishes those nine rows. The top one is a fight of temperature 2 and the other eight are points on the number line, and a reader recognising {a | b} recognises all nine equally.

What a player loses by using the wrong one #

The thirty-two positions the stop reading misses are not a rounding error for anybody who is using the answer, and it is worth pricing them.

The mean is what a position contributes to a board once the fighting is over, so an error in the mean is an error in the count. The temperature is what decides where to move, so an error in the temperature is an error in the move — the hottest-first rule compares temperatures and nothing else, and feeding it a temperature that is too large by a quarter makes it play in the wrong component whenever the true gap is smaller than that.

On the thirty-two the stop reading gets both wrong, the errors are not small. { {0, ∗ | −1} | −2} has stops at −1 and −2, so the stop reading calls its mean −3/2 and its temperature 1/2; the thermograph makes them −5/4 and 3/4. A quarter of a move in the mean and a half in the temperature, on a position whose whole stake is three quarters of a point.

And the direction of the error is consistent, which is the part a player could actually use. On every one of the thirty-two the stop reading under-states the temperature and mis-states the mean toward the losing side of it, because a bent wall is a wall that has been pulled in — the option it is built from gives something back — and pulling a wall in moves the meeting point away from the midpoint of the stops. So a reader working from stops alone will systematically think a bent position is colder than it is, which is exactly the mistake sente is made of.

Where the count comes from #

One hundred and forty-six against twenty-one is a lopsided ratio and it is worth knowing which way the lopsidedness runs.

A textbook switch on day three needs two numbers born by day two, with the left one larger. Day two has seven numbers in it — −2, −1, −1/2, 0, 1/2, 1, 2 — so there are twenty-one such pairs, and that is the twenty-one, exactly. Every one of them occurs and no two give the same value.

The six pairs of neighbours in that list are the tightest of the twenty-one, and they show where the whole count comes from.

Widen any of those rows by skipping a neighbour and the temperature grows without the row leaving the twenty-one, which is how the other fifteen pairs are reached. The largest is {2 | −2}, at temperature 2, and there is nothing above it because there is no number above 2 on day two to take.

The other 146 use one or two of the fifteen non-numbers of day two, of which there are far more combinations. So the ratio is not telling anything about games; it is telling how many of day two’s values are numbers, which is seven of twenty-two.

That has a consequence for the wider question and it is the honest caveat on the whole page. The population being measured is a day of the construction, and a day is heavy in exotic values by design. A population of positions from actual boards is a different population and a much colder one — two thirds of the positions this site has enumerated are worth numbers outright — so the hit rate for the stop reading on a real game is a different number and this page does not have it.

What the census does not say #

Three limits.

Nothing here defines a switch. The word is used for a class of positions and for a shape of written form, and the two have come apart on this page. Which of the two the literature means depends on the sentence, and this site’s usage from now on is the first: a switch is a position with two number options and a gap between them, and a position with one option a side and something else in it is not one, however it is written.

The stop reading is not the only candidate. The mean and the temperature can also be read off the mast — the vertical line the two walls become above their meeting point — and the mast reading is exact by definition rather than three quarters of the time. That is not a rival formula, it is the definition restated, and its cost is that it needs the whole diagram rather than two numbers.

The count of one option a side is a count about forms. A value with one option a side in canonical form may have had many before the reduction ran; dominated options are deleted and reversible ones are bypassed, so the 167 are the values whose reduced shape is a single fight. That is the right population for a page about a formula read off the written form, and it is not the population of positions a board produces.

And 146 is a sample of one day. The failures are all of one kind, which makes the description above look like a theorem, and this page has not proved it. It has measured 167 positions and found no exception to the description; a day further out has values whose options’ options are fights, and those may bend in ways day three has no room for.

The convention, named #

Normal play, and the temperature is the standard one: a tax of t on every move by both players, with the temperature the height at which neither wants to move. The mean is the value the two walls meet at, computed by the thermograph rather than by a formula, and the stops are computed by playing the position out under the two orders of play.

A number is given temperature −1 rather than nought throughout, which is the convention below zero sets out and which is why no number appears in the counts above as though it were a cold fight.

Where the ladder goes next #

The switches anchor has four rungs to here: the class and its two formulas, the switch a player imagines when the position is not one, the edge of the regime, and now what the formulas become once the hypothesis is dropped.

The rung above settled the description this page offered as a candidate, and it settled it in two directions at once. The bend is the condition checks the stop reading is exact exactly when neither wall bends below the meeting point against every one of the 138 non-numbers here, and finds it exact: 106 straight and correct, 32 bent and wrong, and not one position in either of the off-diagonal cells. So the class this page could only describe has a definition, and the definition is a property of the diagram rather than of the options.

The second half of the candidate does not survive. A wall bends precisely when some option is neither a number nor an infinitesimal is true in one direction — all 32 bent values have such an option — and false in the other, 49 times over: eighty-one of the 138 have an option outside the two classes and only thirty-two of those bend. The nearest better guess, some option is hot, agrees on 118 of 138 and is also not it. Which means the proof sketch this page gestured at cannot be completed as written, and the reason is instructive: an option can be neither a number nor an infinitesimal and still have its wall freeze below the meeting point, so what matters is where the option’s own temperature sits relative to this position’s, not what kind of object the option is.

Two neighbours are worth the trip. Where the fight stops is where the two stops are defined and first shown not to determine the position, and this page is the sharpest form of that finding: the stops are exactly the two numbers the formulas want and they are not enough. And a thermograph with two bends is what a bend means when there is more than one of them, which is the mechanism this page’s failures are made of.