A switch, its mean and its temperature
Positions of the form {a | b} with a above b: both players want to move there, so neither is settled. The bar spans the two options, the marked point is the mean the position is worth once the fighting is over, and the temperature is half the gap — which is exactly what moving first is worth.
16 essays call
switch-diagram. The drawing above is what it returns with no arguments at all; every
call below passes it something, because a placement that passes nothing draws whichever
member of the family the generator happens to default to rather than the one its essay argues
about.
The positions it draws
61 distinct positions, harvested by running this generator again at the options each essay passed it.
Where it is called
Changing this generator changes every one of these figures.
Particular games
One board, two rules
Col forbids painting next to your own colour. Snort forbids painting next to your opponent's. One word differs, the boards are identical, and the values that come out are not the same kind of object.
Values
Worth nothing, and worth fighting for
A switch is a position both players want to move in. Its average value can be zero while the difference between getting there first and second is enormous, and that gap is a second number every position carries.
Temperature
A number and a fight
Charge a position exactly what it is worth fighting over and the fight disappears, leaving the mean value — with something still attached to it. Over all 1,122 hot values born by day three the residue is smaller than every positive number, it is a star in 942 of them, and it is never nothing. So a hot game is its mean plus a fight plus a remainder that no number reports, and the remainder is what decides close games.
Values
When a switch is not a switch
A position {a | b} with numbers on both sides is a fight only while a is above b. Sweeping the boundary with a fixed at 2 and b climbing from −2 to 3 turns up three regimes rather than the two the definition suggests: eight fights whose temperature is exactly half the gap, one position at a = b that is 2∗ and is not a number, and beyond that numbers chosen by the simplicity rule — which on 8 of 12 sampled cases is not the midpoint.
Temperature
Cooling adds and heating does not
The two operators are presented as a pair, and they are not one. Cooling a sum is the same as cooling the parts and adding, on every one of the 1,768 pairs tried, at two taxes and on two pools. Heating fails on 263 — and not for the obvious reason: in every failure neither part and not the sum is a number, so the clause exempting numbers never fires at the top. It fires two levels down, where an option of a sum is one part's option plus the whole of the other.
Temperature
How hot a day gets
A day of construction buys exactly one degree of temperature — nought, then one, then two — and the value attaining the maximum is unique on every day: ∗, then {1 | −1}, then {2 | −2}. The distribution underneath is not tidy at all: it peaks at a half, leans to the right of the peak, and has a hole in it at one and three quarters where nothing is born.
Sums and comparison
How hot a background has to be
The reduced canonical form throws away infinitesimals, and the rung below asked for a bound: how hot must the rest of the board be for the discarded part not to matter? There is no such bound. Safety is not monotone in the background's temperature — an eighth is safe, a quarter is not, two is safe again — and the quantity that does decide it is not a temperature but a stop.
Values
A fight with no midpoint
The mean of {a | b} is the midpoint and the temperature is half the gap — on the twenty-one values of day three where a and b are numbers. One hundred and forty-six others have the same shape and not the hypothesis, and the repair that suggests itself, reading the two stops instead of the two options, holds on about three quarters of them and no more.
Sums and comparison
The thirty that cancel themselves
Thirty values born by day three are equal to their own negatives, and every one of them has a mean of exactly nought and two stops that are exact opposites. Neither property comes close to picking them out — 496 values of the day have a mean of nought — and half of the thirty are hot, one of them the hottest value the day produces.
Values
The bend is the condition
The rung below offered a description of the class its stop reading is exact on — neither wall bends below the meeting point — and a route to proving it: that a bend happens precisely when some option is neither a number nor an infinitesimal. The first is exact on all 138 values, both directions, no exception. The second is half right: every bent value has such an option and 49 unbent ones do too. And the eight apparent exceptions to the first turn out to be a bookkeeping convention.
Values
Half a follow-up out
The rung below settled which values the stop reading is wrong about — the ones whose walls bend — and left the size of the error unmeasured. It is not bounded by anything readable off the diagram; it equals something readable off the diagram. On all thirty-two, the mean and the temperature are each out by exactly half the follow-up's temperature, and the temperature is always read too low.
Values
A second level of stops
The rung below found the stop reading's error to be exactly half the follow-up's temperature and asked whether the correction survives a wider pool, survives two bends, and can be stated without a thermograph. It survives eleven times the pool, missing two values in 1,459. It needs no thermograph — the follow-up's temperature is half its own stop gap. And it does not survive two bends, because day three contains no value with two of them.
Values
The bend is in the stops
The rung below reduced the whole stop reading to one question — does this wall bend? — and asked whether that could be answered from the options' stops instead of from a diagram. It can, in four lines, and it gives more than the bend: on all 1,459 non-number values born by day three the options' stops determine the entire thermograph. One day deeper it breaks, and every failure is a value with a bent-walled option.
Values
A bend that never reaches the surface
How many levels of the recursion a thermograph needs before its stops suffice is a number attached to a position, and the rung below conjectured it was the depth of the deepest bend in the tree. It is not: on 124 values a bend one level down costs nothing at all. What the number counts is the longest unbroken chain of bends running down from the top, exact on 2,400 of 2,403.
Values
The bend above the top
The chain reading gets three values in 2,403 wrong because it counts bends that the diagram never reaches. Counting only the bends below the position's own temperature fixes all three and breaks none — the first exact reading on this ladder, and it needs one comparison rather than the envelope the rung below expected.
Out in the world
What has to break before a pawn is worth a number
Every value the blocked-file model can hold is an infinitesimal, and the reason is one sentence about the move rule rather than anything about pawns. Break that sentence — a pawn stuck behind a friend, a square only one side can hold — and integers, switches and positions worth fighting over arrive at once.
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