Sente — where it appears
Named by 27 essays across one field — each of them below, with the objects they name alongside it.
What is at stake
Some positions both players are desperate to move in, and some neither player wants to touch. The difference is a number — how much the move is worth — and it turns out to be the most useful single quantity for deciding where to play.
Playing the hottest
Given several independent fights, play in the one with most at stake. The rule is simple, it is what strong Go players do without being told, it is provably close to optimal — and it is provably not optimal, which is the interesting part.
The endgame, accounted for
Add up what each region is worth, then add the biggest thing at stake, subtract the next, and so on down. On a board of simple fights the result is exact — and the moment one region has a fight inside it, the account is out by a point.
Two hot fights that add to a cold number
The mean of a sum is the sum of the means, every time. The temperature is not — it is bounded by the hottest part and is often far below it. Two positions each worth fighting over can add to a plain number that neither player wants to touch.
Big is not the same as hot
A player sizes a move by how much changes hands when it is played, which is the number in every endgame book. The theory sizes it by temperature. On a plain switch the two agree exactly, so nothing shows; on a move with a follow-up they come apart, and the count gives up more than the guarantee the theory's rule carries.
Sente is a fact about the rest of the board
A move that must be answered is called sente, and the word is used as though it described the local position. It does not. The same fight is answered while the rest of the board is quiet and ignored once it is busy, and the crossover — measured by solving the whole board at every temperature — sits at the follow-up's own temperature.
Double sente is not a property of the position
A fight with a threat on each side is called double sente, as though the phrase named a shape. Swept against a rising ambient temperature, one such fight — { {5 | 3} | {2 | −4}} — is double sente up to an ambient of 1, sente for one player only from 3/2 to 3, and gote for both from 7/2, and the two band edges are the temperatures of the two follow-ups.
An environment made of coupons
Beside the game sits a stack of coupons worth 4, 3, 2, 1, 0, and a player may always take the top one instead of moving. Play the whole thing out and two quantities the theory computes are measured instead: {4 | 0} comes out worth exactly 2, its mean value, and the coupons stop at 2, its temperature. For {10 | {9 | 1}}, whose temperature is 1, they stop at 7/2 — because what the stopping coupon measures is the hottest temperature anywhere in the tree.
A thermograph with two bends
A wall bends where the option holding it up stops holding it up, and most drawn thermographs bend once. { {5 | {3 | 1}} | 0} bends twice, at 1 and 3/2, and neither height is its temperature of 7/4 — every bend lies strictly below where the walls meet. Over a census of 17,255 hot positions two levels deep, a second bend in one wall never happened once.
Two games in one environment
A coupon stack measures a position: play the whole board out and the coupon the players stop at is the temperature, the score is the mean. Put a second position beside the first and one of the two measurements stops working. Over 36 ordered pairs the contributions still add to the means every time, and the coupon a fight is entered at moves on 13 of them — without either position changing.
A rule with a guarantee
Evaluating a sum of a dozen fights is impossible; following a rule is not. Move where the stake is largest, and over 220 sums of three hot components the rule scores exactly what perfect play scores in 196 of them, is never more than one point behind, and never ends more than the largest single stake below the mean. The rule that is supposed to be different — answer the threat — chose differently in none of the 220.
A rule with no promise at all
Playing in a hottest component comes with a bound: never more than the largest single temperature below the mean of the board. Over 220 sums the bound holds 220 times — and so does the bound for a rule with nothing behind it, which scores exactly what perfect play scores on 205 sums against the hottest rule's 196. The control that shows the bound is doing work is the rule that plays the coldest component, which breaks it 74 times and loses up to eleven points.
What a move nobody makes is worth
If Right's move in a fight has to be answered, then Left's move in the same fight prevents an exchange Right was going to get for nothing. That is a reverse sente, and pricing it is the awkward case: over sixty measurements the gain matches the local temperature once and the follow-up's swing twice, and the band over which the move is worth taking is not the band over which the move it reverses is sente.
A pool built to punish greed
The rung below found a rule with no theorem behind it beating the rule with one, and predicted that a pool of deliberate traps would reverse the result. It does not. The traps miss, because the rule called greedy scores a move by the stop it leaves and a stop already contains the follow-up — and the rule the traps do catch, losing sixteen points where the guaranteed rule loses five, had to be written to make the point.
The answer that starts another fight
A local move is answered while the ambient temperature stays below the follow-up's — and that rule, which this site has carried since the anchor opened, is exact only when the answer ends the fight. When the answer starts another one the crossover is exactly half the follow-up's temperature, on every position tested, and a third level of fight does not halve it again.
What the halving is a function of
A move is answered while the board is cooler than the follow-up's temperature — or half of it, depending which of two classes the position is in, and the classes were stated in terms of forms. They are a feature of one wall: whether the follow-up's right wall rises straight up before it leans. Fifty-five positions, no exception, and the crossover becomes something a reader can see.
When to leave the environment
A Go player's question is not which fight to take but when to stop taking the small stuff. Put two fights beside a stack of coupons and the orthodox answer — leave when the coupon falls to the hottest temperature on the board — is exact on sixty of eighty-one pairs. All twenty-one departures have a fight with a follow-up in them, and every pair of plain switches leaves on time.
How big the answer is
The rung below found every early departure from a coupon stack caused by a position with a follow-up, and could not say more: its follow-ups were all of a similar size, so the class it measured was one bit. A pool graded by follow-up size answers it. With the position's own temperature held at one, the departure runs from coupon 1 to coupon 3.5 as the follow-up's temperature runs from 1 to 4 — and over the whole grid the players leave at the larger of the two temperatures.
A subtraction, not a factor
The crossover factor was a half on fights whose answer starts another fight, measured on a pool with two three-deep positions in it. A pool built to be deep gives twenty, and the factor does not survive them: the crossover is the follow-up's temperature less a half on eighteen of the twenty, and a factor of a half agrees with that only where the temperature is one — which nearly every position in the earlier pool had.
Half of the smaller temperature
The correction to the sente crossover has been priced twice — first as a factor of a half, then as a subtraction of a half — each time on a pool whose answers were all about the same size. Over 128 fights with answers from a number up to a temperature of six, the correction is half the answer's temperature, saturating at half the fight's own. Both earlier readings are regions of that one law.
The quantity that does not order a board
The rung below found the players leaving an environment at the larger of a position's two temperatures, and proposed that a board should therefore be played in the order of that quantity. Over 220 boards of three components it plays exactly on 124 against playing-in-the-hottest's 196, loses 85 of the 97 disagreements, breaks Hotstrat's guarantee on six boards, and costs nine points on its worst one.
A rule that beats the hottest
The rung below proposed the reverse of the rule that had just failed — discount a component by its answer's temperature rather than promoting it — and predicted, before the sweep, that it would not beat playing in the hottest component. It does. It plays exactly on 201 of 220 three-component boards against 196, wins two thirds of the boards where the two disagree, keeps inside a guarantee proved for the other rule, and the gap widens as the board grows.
The two numbers at the top
The rung below found the crossover of a sente fight to be its temperature less half its answer's, and said a proof would settle the depth question with it. The depth question is settled without the proof, by construction: group the positions by their two top temperatures and the crossover is single-valued on every group, however far apart the third temperature is — and the formula survives a fourth level of fight, which the rung below never reached.
The premises an induction would need
The rung below settled by a grouping test that a position's crossover depends on its own temperature and its answer's and on nothing below them, and asked for the induction. The four paragraphs are not written here; the checking they would rest on is. The law holds at five levels, survives translation, heating and cooling — and none of that is the step.
A pool built to have an answer
The coefficient in the rule score a component by t − λa scored identically for every λ in the unit interval, because the rule reads an ordering and that pool's orderings changed at three places. A pool designed to have twelve crossings turns the interval into thirteen different rules, and all three board sizes agree on one cell: between a quarter and a third.
Which top is the top
The crossover law's proof rests on the walls above the crossover being governed by the top two options, and the check was never run. Run on 23,586 heights it holds exactly — but only when the options are ranked by mean value. Ranked by the temperatures the law is stated in, it fails on a fifth of them.
The four paragraphs prove something else
Three rungs earned the right to write the crossover law's proof as two straight walls meeting where the law says. The walls are straight — one of them everywhere, for a trivial reason, and the other only above the answer's own temperature. The crossover sits below that height, so the geometry holds nowhere the law is about, and where it does hold it proves the temperature instead.
Named alongside it
The objects these essays reach for when they reach for this one.
TemperatureMean valueThermographFollow-upSwitchDisjunctive sumEnumerationHot gameAmbient temperatureCounterexampleStrategyExhaustive search