Heuristic — where it appears
Named by 71 essays across 8 fields — each of them below, with the objects they name alongside it.
The game in every exercise book
Dots and Boxes is played by more people than every game in this collection put together, and everybody is taught the same rule — take every box available. The rule is wrong. Establishing that takes a solver rather than an opinion, and the solver says how wrong, on which boards, and by how many boxes.
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.
When a real board falls apart
Amazons is played competitively, and late in a game the arrows have cut the board into regions no piece can cross. From that moment the position is a disjunctive sum — arrived at by the play rather than assumed — and the territory count every program uses can be measured against what the sum is actually worth.
The period is small and the proof does not say so
Every subtraction game repeats eventually — that is a theorem, and its proof gives a bound of sixteen thousand for a three-move set. Over 112 sets the longest period measured is twenty-two. The proof and the fact are four orders of magnitude apart, and the rule of thumb that closes the gap is broken by one set in the sweep.
A rule that is never right and cannot be far wrong
Playing the hottest component is not optimal — over 440 measured lines it costs something on 17 of them. What makes it worth having is that the cost is bounded by the temperature, provably, and that the same test run with the ordering reversed breaks the bound on 54.
Finding the parts
Decomposition turns a product into a sum and is the largest saving in the subject. Nobody labels the regions. The pass that finds them costs the same on every board of a size — including the boards where there is nothing to find — and what it buys ranges from four orders of magnitude to nothing at all.
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.
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 reading that survives too much
Counting the empty squares in front of each coin gets a Push position right half the time, and the rung below said the failures were exactly the positions with two coins of opposite colour side by side. Sixty-six of the 1,072 failures have no such pair, the smallest is five squares long, and the condition that does decide it is not about the board at all — it is about every position the board can reach.
A compound of two different games
Every rule the compound theory has survives mixing exactly — the minimum-remoteness rule is right on all 5,184 mixed pairs and all 7,560 triples — and the reason is not that the rules are strong. It is that each of them reads one number per component, and a number does not remember which ruleset produced it. The thing mixing damages is the shortcut a reader carries instead.
The moves a player can be talked out of
The difference of the two players' largest domino packings is the value of a Domineering region on 141 of the 315 worth numbers. The count is optimistic for its owner and pessimistic for the other, and one number cannot be both — so it becomes an interval, from what a player can be reduced to against what the opponent can achieve. The interval contains the value on 209, is a single point on 505 of the 1,042 regions, and never exceeds two moves wide.
The criterion that cannot exist
The rung below asked for a quantitative version of its condition — turn 'the reading survives mixing three quarters of the time' into a statement about the strip. Three strips of four squares settle it. `.LLR`, `.LRL` and `.RLL` have the same length, the same reading, the same coins and the same single run, and their readings are wrong by 1¼, ¼ and ½. The error is a fact about the order of the colours, and 207 of 805 statistical classes carry more than one of them.
How many moves are worth making
A value answers who wins and by how much, and the anchor below names the quantities it discards. This is the first of them counted. Over 1,034 Domineering regions and 125 values, 63 values have two regions disagreeing about how many placements are worth making and 52 disagree over whether there is any choice at all — and the count of good moves stays near one and a half however large the region gets.
When the catalogue starts paying
The rung below priced two questions — who wins one board, and what it is worth — and named the third: a program pays for a family of regions once and answers every board over them by addition. The crossover is between five boards and two hundred, depending on how far the catalogue reaches, and it falls as the board grows. The whole catalogue of every region to eight squares costs one part in seventy-six of one undecomposed five-by-five board.
What a game actually produces
Fifty-three per cent of the Domineering regions of at most eight squares are hot. Of the components eleven hundred random games actually produce, sixteen per cent are — and ten per cent once single squares are counted. The figure is the same on three sizes of board, so it is a property of play rather than of the board, and it says that every temperature census this site has taken over a catalogue overstates how hot the game is by a factor of three.
The margin a count needs
Leaving the opponent fewest replies names only surviving options nine times in ten, which leaves the question of what a bound stated in that count would have to be weakened to. It is a margin. Over 57,879 pairs of Domineering options, the one leaving the opponent fewer replies is the worse of the two 1,052 times at a margin of one and 72 times at a margin of two — and at a margin of three, never.
The weight that blunts the count
The rung below found that counting the opponent's replies gets the direction of a comparison right once the gap reaches three, and proposed a repair: weigh each reply by whether it leaves the opponent anything. Weighing it makes the count worse. The threshold goes from three to four, the failures from 1,124 to 1,320, and all seventy-two of the pairs the repair was written for come through it unchanged.
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.
The wider move is the easier game
An earlier essay ruled out every rule that reduces the heaps and reads the residues, and asked for a two-part statistic: the residues plus one more count. Four second parts are tested here and none of them decides. What turns up instead contradicts the premise the request was made under — a move that may reach three heaps is more predictable than one that may reach two, on every cap, every candidate rule, and after the change in the base rate is taken out.
What a strategy has to remember
A value answers who wins and by how much, and it settles neither how many moves achieve it nor whether the best one is unique. Counted over every position reachable inside the catalogue of regions, the gap has a size: 4,269 positions carry 128 values between them, and a player who wants to win rather than to predict has to store 3,308 choices — twenty-six entries for every number the theory supplies.
One fight makes a board a fight
The rung below found 16 per cent of the components a played game produces to be hot, against 53 per cent of the catalogue they are drawn from, and predicted that the share of hot boards would be much larger. Taking the same play-outs and tallying at the board gives 32 per cent — twice the piece figure and not ten times it, because a Domineering board carries only 1.68 pieces and the hot ones cluster on the same boards.
Two errors that cancel
Replacing the packing count with an interval left a doubt that the pessimistic half would add across a board. It adds, for a one-line reason. What is worth measuring is what the reading is then worth: over boards of one to four regions the count decays from exact on 45 per cent to exact on 11, and the interval's containment does not decay at all — it rises from 67 per cent to 74, because the interval's width adds and its error does not.
Where to stop building
The rung below priced a catalogue of small regions against the search it replaces and found the crossover. What it could not say is how far to build, and the coverage answers that: going from four squares of reach to ten multiplies the catalogue by 860 and lifts the share of regions it answers from 54 per cent to 74. The price of a point of coverage runs from five shapes to five thousand.
The size of a cake
Ω gives the sign of a Maundy Cake and says nothing about the size, and the rung below left four values — 7, 10, 13 and 16 — unaccounted for. For a one-row cake they are a formula: write the prime factors largest first and add up their running products. The rule behind it is greedy — cut by the largest prime — and it is exact on every one-row cake to two hundred and wrong on a fifth of the two-sided ones.
A check in front of a search
The rung below found a pairing one move away on 288 of the 767 even first-player shapes, and asked what a solver that tested for one before recursing would save on a real game. On an even Cram board it saves nearly the whole search — a 4 × 5 board takes 17,348 node expansions without the check and one with it — and the depth profile shows why that number flatters: the check settles every winning position at the opening and at the last two moves, and about one in ten in between.
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 numeral in the empty squares
The rung below ruled out a quantitative criterion for Push and asked for a numeral over the coins combined with a count over the gaps. The two ingredients are the right way round: the colours pick a fraction — −1, −1/3, −1/7, −1/15 — and the empty squares give the binary precision, so a run of k coins before one of the other colour with g gaps is worth exactly (1 − 2^(−kg)) ÷ (2^k − 1). And it does not compose: a strip of two runs is not the sum of them, on any pair tried.
The threshold was a fact about the census
Two rungs failed to account for the seventy-two pairs where a mobility count gets the direction of a comparison wrong, and the third looks at them one at a time. They are not a class of shapes. All seventy-two are on the largest board in the census, at two depths, and sixteen positions up to symmetry — and one board larger the count fails at a margin of three, which the ladder has been quoting as the point at which it never does.
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.
Three rules and a tie-break
An exhaustive table of what a Domineering strategy has to remember is 3,308 lines. Three rules applied in order answer 94.5 per cent of it — leave the opponent fewest replies, then keep the region whole, then take whichever placement comes first — and the fourth and fifth rules answer not one more. The residue is 181 decisions in which every rule scores the candidates the same and one of them is worse.
Half the difference in odd runs
The rung below asked what the regions the packing reading fails on have in common, and whether it is something a player could see. It is: the reading itself. The count has a closed form — half the difference between the region's odd horizontal runs and its odd vertical runs — and it is exact seven times in ten when it claims one move of advantage, on none of the largest regions where it claims two, and it exaggerates four times in five when it is wrong at all.
Cut small unless you are behind
The rung below found the greedy rule — cut at the largest prime — wrong on 104 of 552 Maundy Cakes and asked for a description of them. On all 104 the best cut is at the smallest prime, the exact opposite. A middle divisor is never needed on any cake in a sixty by sixty grid, and which of the two extremes wins is decided by Ω alone: cut small when Ω(m) + 1 ≥ Ω(n), large otherwise, and that is exact on all 3,540.
The fractions that were not there
The rung below counted 1,452 fractions among the shared Amazons regions and asked which fractions they are. Fifty-six of them are fractions. The other 1,396 are hot positions with a fraction somewhere inside their options, counted by a regular expression looking for a slash — and the quantity the separation of the two amazons actually sets is not a denominator but a temperature.
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 obstacle was the catalogue
The rung below could not measure the early game because its regions were too large for the catalogue, and asked for a bracket rather than a value. No bracket is needed: a twelve-square region evaluates in five milliseconds and an eighteen-square one in under a second. What was expensive was cataloguing every shape rather than sweeping the positions a board actually reaches — and the sweep says a board is hot four times in five three moves in, and cools when it breaks up.
The option nothing names
The rung below found the arithmetic on reduced forms to be add and reduce again, needing the second pass on 431 of its sums, and asked whether the option that pass deletes can be named from the parts. Eight rules were scored and the best reaches four in five — and on a pool closed under negation it falls to under half, which says the near-miss is a property of the population. What the second pass does have is a shape and a cheap test that rules it out.
The count of odd heaps
The rung below refused a family of two-part rules for bounded Moore's Nim and asked what the 364 losing positions have in common as a set. They have an invariant, and it is a statistic of the whole position rather than of a heap: how many heaps hold an odd number. Every all-even position is lost, at every width of move, by a restoring strategy — and the count settles every position at one heap a move and at four, and a little over half at two.
The only way to split into three
Nineteen three-digit hexadecimal codes climb by three, and the rung below asked whether they share a form and what digits they have in common. The digits are exact: on eighteen of them the only way to split a heap into three is by taking exactly three counters, and taking three counters can do nothing else. The form is not shared — the eighteen carry four distinct sequences, and exactly one of the four counts in base three.
The check that was not a check
The rung below asked for a depth-conditioned solver and named the board to measure it on. Building it found two things first. The pairing check is unsound at interior positions — on a four by five Cram board it fires on 8,613 positions and 1,026 of them are losses — and the board it named has twenty-five squares, so the check can never fire there at all. Repaired, the check is right everywhere, and the policy that pays is the root alone.
A catalogue that knows what it will meet
The rung below priced a catalogue of regions by its reach and found the coverage saturating, and asked what a catalogue ordered by frequency would cost instead. Eight shapes answer half the components a played Domineering board produces; a catalogue by size needs fifteen for the same, and 1,042 for what 119 chosen by frequency reach. Three quarters of a size-ordered catalogue never turns up in play at all.
A threshold is a detection limit
The rung below had two points — a margin of three at fifteen squares, four at eighteen — and asked whether the mobility rule's threshold grows with the board. Eleven more sweeps say no property of a board orders the thresholds, that the same board at two depths gives two of them, and that a tenth of the sweep which produced the four reports three instead. What does move, on every board measured twice, is the depth.
Seventy-two of them were not silence
The rung below said its 181 unanswered decisions were all the rules falling silent and asked whether the position's value picks the placement once the geometry cannot. Seventy-two of the 181 are the rules speaking and being wrong, which is a different failure. On the 109 that really are silence, a rule chosen per value answers more than half — and the star class the rung below singled out is settled outright by leaving the younger position.
One domino every three cells
The rung below gave the optimistic packing count as a formula in odd runs and asked for the other end of the interval, expecting a formula in the even ones. Parity is the wrong arithmetic: the smallest maximal packing is a sum of ⌈(len−1)/3⌉ over the runs, exact on all 1,042 shapes. That makes the whole interval readable off a drawing — and shows it can never reach the value, because regions with the same runs have different values.
Room pulls two ways
The rung below found the distance between two amazons setting a shared region's temperature and asked for something finer — the squares each can reach, or the squares both can. Neither beats the distance on its own. Together they beat it by half as much again, and the reason is that they pull opposite ways: further apart is hotter, and sharing more reachable squares is colder.
The worst value in its own interval
The rung below scored a component by its temperature less its hottest answer's and asked what rate the answer should really be charged at. Every weight strictly between nought and one scores the same and beats the rung below's choice of one at every board size — because a ranking rule's score is a step function of its own coefficient, and one is exactly where two components tie.
The catalogue a strong player needs
A Domineering catalogue built from random play faces an objection that could overturn it: random play is not play. A player that reads the board produces the same head — eight of the ten commonest shapes — and concentrates far harder: 114 entries answer nine tenths of what it meets, against 2,018. And a catalogue measured on random play over-serves it, while the reverse fails.
A heuristic that becomes a theorem
The mobility rule's failure rate had been measured at two depths on each of five boards and found to fall. Swept at every depth it does not merely fall — it accelerates, and it reaches exactly nought before the endgame. From four to eight empty squares onwards the rule has no exceptions at all, which turns a rule of thumb into a guarantee for the last few moves.
The price of taking the maximum
The seventy-two decisions where a Domineering strategy's rules name the wrong placement are never wrong by more than one reply, and a third of them are the second rule's fault rather than the mobility count's. The repair that follows — keep every placement within one reply of the best — retains a best placement every time and costs thirty decisions for every one it saves.
An effect that changes sign
Which squares two Amazons share turns out to matter about as much as how many — three shared squares in a line run at 0.63 where three scattered run at 2.51. But the effect of clumping is hotter at one distance and colder at the next, so the arrangement predicts well and describes nothing, which is not what the four rungs below it produced.
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.
The clause that turns the class off
Three rungs failed to find the dead-ending class doing measurable work, and each time the population was blamed. Toads and Frogs with and without the jump is the matched pair the anchor wanted — the same board with the class switched on and off — and on it the test the class licenses gains less from the class than a control that has never heard of it.
A catalogue that builds itself
A solver that stores every region it has to evaluate builds a catalogue out of its own games. After 650 games it holds 232 of the 1,042 shapes and is still growing — and the order things arrive in is nearly arbitrary while the order they are consulted in reproduces a census of a strong player's games almost exactly.
A second pool, designed differently
One designed pool put the rule's best coefficient between a quarter and a third, and all three board sizes agreed. A second pool, built by the identical greedy criterion from different material, has no cell that is best at every size — so the coefficient is a property of the pool and there is no number to find.
An environment instead of a stack
The guarantee behind playing the hottest component survives one step down a board's sorted temperatures and fails at two. Against a coupon environment as hot as the board it survives all of them — because the rule stops being approximate and starts being optimal, on every sum in the pool built to punish it.
The third point on the curve
Four rungs below this one measure a shared Amazons region's temperature against how far apart its two amazons are, and every one does it on a 3 × 3 board, where the distance can only be 1 or 2. Two points make a direction, not a curve. A 4 × 4 board reaches distance 3 — and cannot be evaluated at all until the regions are cut to five free squares. Restricted that far, the answer is neither a sign that flips nor an oscillation: the rise continues and it is running out, the second step being 36 per cent of the first.
A count that forgets
A Domineering solver with room for ten component values does better evicting whatever it used least recently than evicting whatever it used least often, and the explanation offered was that a use count never forgets. Halve every count at a fixed interval and the count overtakes recency at every table size — by less than half a point, and only with the right interval. The right interval grows with the table: a quarter of a game's worth of lookups at ten entries, five games' worth at forty.
One board, and recency still wins
A Domineering solver's table of component values did best evicting whatever it used least recently, and the explanation was that the run changed board size three times. Take the change away — play all 650 games on one board — and counting wins back its lead only on the smallest board. On 5 × 5, 6 × 6 and 7 × 7 recency still beats both counting and the best fixed table, by the most on the largest. The locality recency exploits is not between boards or between opening and endgame. It is inside a single move.
The order a solver tries the moves in
A memoised search asking who wins 4 × 5 Domineering expands 1,125 positions when it tries first the move that leaves the opponent fewest replies, and 30,202 when it tries losing moves first — the same answer at twenty-seven times the price. The ordering that already knows which moves win is not the cheapest. A win needs one move and a loss needs all of them, so the price of an order is paid one level down, in the replies it leaves.
A verdict that changes with the depth
A who-wins search of 4 × 5 Domineering cut at a fixed depth, guessing that the player with more placements wins where it stops, is right about 72.5 per cent of positions at depth 0 and about every one of them by depth 7. On the way, 4,697 positions are right at one depth and wrong at a deeper one. With a guess that knows nothing, going one move deeper makes the search worse — and its errors alternate in kind with the parity of the depth, so that half its verdicts are proofs.
Where a search may stop
A search deepened until two consecutive depths agree carries a proof of its verdict, and on 4 × 5 Domineering it stops before the longest line on 17,589 of 48,670 positions. It also costs three times what the search that simply finishes costs. The rule that pays is the other one. Search on wherever the two players' counts of placements are within one, and at depth 2 the wrong verdicts fall from 2,140 to 86 for about a quarter more work per search.
A potential that names every move
Strategy stealing names no move, and the pairings that do name moves need a board with the right symmetry. The Erdős–Selfridge potential needs neither: Down, moving second in Hex, takes the empty cell through which Across's unfinished chains weigh most. Its guarantee reaches only boards two rows deep. It wins far past the guarantee — on every board of three rows that Down can win — and then, on a four-by-five board that a table of pairs wins for Down with certainty, it answers Across's first stone in a different cell and loses along the bottom edge.
The winning reply is the fourth choice
The repair proposed for the potential was to weigh an edge chain more heavily. Fifty-five weightings later, none holds the four-by-five board, and an edge bonus costs Down four boards it was already holding. The reason is not the numbers: over 393,660 turns of the pairing that does hold that board, the potential would take the same cell 26.1% of the time, and the winning cell is its 3.7th choice on average and as low as its seventeenth.
The bound names the hottest part and the cost does not
Moving in the hottest component costs at most the largest temperature on the board, and that bound is attained: 100 lines of 4,240 pay exactly it. It is still the wrong quantity. Across four pools and boards of two, three and four parts the cost is nothing on 90.8% of lines and otherwise takes one of two values — half a point or one — on boards whose largest temperature runs to three, and it exceeds the coolest component on 13 lines and twice it on none.
A shortlist with nothing at the top
The one-pass test leaves 9.58 moves of 12.27 and names none of them. Seven quantities a scan could compute about the survivors were turned into orders and scored on 583 positions: the best puts a winning move first on 24.9% against 22.3% by chance, and every one of the seven places the winner deeper in its order than chance does. Read as sieves instead, the gentlest keeps half the list and throws the only winner away on 264 positions.
The cheap fights make the rule cheaper
A conjecture stands that playing the hottest part costs at most the coolest temperature times the number of parts sharing it — proposed on a range where that number never exceeds two. Swept to five-part boards over 10,410 lines it is false, and false the other way round: every line costing more than the coolest part has one or two parts at that temperature, and over the 3,230 lines with three or more, not one does.
Search on in pairs of moves
Deepening until two depths agree gives a proved verdict, and searching on where the counts are close gives a better one; put together the obvious way, they stop on a wrong verdict at 3,231 positions of 4 × 5 Domineering. A guess one move past the cut has the other player to move and flatters the wrong side. Searching on two moves at a time keeps the proof, and the window that suits it is one-sided — but however it is widened, the certificate gets cheaper only by turning into the search that finishes, and on four boards it never gets below it.
A move whose every reply is struck
Read two moves at a time, the Sylver Coinage shortlist is no better ordered than read one at a time: preferring the survivor that leaves the opponent the fewest surviving replies puts a winner first on 24.7% of 583 positions against 22.3% by chance, and places it deeper than chance does. Read as a proof, the same count does what no order could. On 57 positions a survivor leaves no surviving reply at all, and wins by a certificate a few lines long; searched deeper, the survivors prove every position by thirteen moves — at a price that is never below the search that simply finishes.
The opponent stops choosing
Replace one player by a rule with no search in it and the question has one chooser left, which is a puzzle rather than a game. Nim recovers five of its six lost positions that way, and six of seven on three heaps of five. Domineering recovers six of a hundred and twenty-two while the fixed rule throws away a winning move eighty-eight times, and one Clobber board recovers none at all — because on that board no rule can misplay.
The best chance is the wrong move
Maximising a probability and denying an opponent a reply are different objectives, and on 189 of the 904 day-three positions where Left has a choice and a winning move, the option the coin prefers is one that loses the alternating game outright. The smallest case is two options and one line of arithmetic: five eighths beats a half, and a half is the move that wins.
Four boxes for every chain after the first
Nimstring answers who is forced to open and says nothing about the score. The margin has a formula: the controller nets the total less four boxes for every long chain after the first — two surrendered and two not taken, each time control is kept. Checked against the solver on 175 endgames it is exact on 172, never too generous, and exact wherever it promises the controller anything at all. The three it misses are the three where it promises nothing.
Named alongside it
The objects these essays reach for when they reach for this one.
EnumerationCounterexampleApproximationDomineeringExhaustive searchDecompositionTemperatureValueInvariantStrategyDisjunctive sumBound