Every Domineering shape up to four squares
The pieces a partly played board falls into, each with the value the recursion gives it. Left plays vertically and Right horizontally, so a tall shape is worth something positive and a wide one something negative, and the quarter turn is not a symmetry of the game.
16 essays call
region-catalogue. 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
12 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
A board that is a sum of its regions
A table of rectangles is a table about the openings. A partly played Domineering board is not a rectangle, and evaluating one means splitting it into pieces no domino can straddle, looking each piece up and adding. The catalogue of 104 shapes does it correctly on every one of the 3,227 positions of a 3×4 board it covers — and among the shapes are two worth an up and a down, which no rectangle ever is.
Particular games
Which shapes are worth fighting over
Forty-four of the 104 Domineering regions of at most six squares are worth numbers and the rest are not, and the rung below said no visible property of a shape predicts which. Half of that is wrong: a region only one orientation fits in is a whole number, on all eleven of them, for a reason a reader can supply in a sentence. The other half stands, and thirty-three shapes are what makes it stand.
Particular games
Counting the moves each side has
How many dominoes could each player still place? Subtract, and there is a whole number computable from the drawing with no game theory in it. Over 1,042 regions it is the value on 141 of the 315 worth numbers, lands between the stops on 619 of the other 727, and its failures are two different kinds — one of which was inevitable and one of which is a fact about the game.
Particular games
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.
Values
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.
Temperature
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.
Values
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.
Particular games
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.
Values
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.
Particular games
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.
Values
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.
Particular games
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.
Values
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.
Particular games
Where the runs meet
A Domineering region's value interval is a function of its run lengths and its value is not — twenty-one groups of shapes share a multiset and disagree. The crossing count separates none of them, and neither do ten other local statistics, fifteen sets of which agree on everything and differ in value. What separates nineteen of the twenty-one is where along its runs each crossing sits.
Values
Close calls nothing resolves
The same value panel that settles 118 of the 202 silent decisions settles four of the seventy-two the rules get wrong. Every rule that helps at all must replace connectivity rather than follow it, and the cheapest one breaks twenty-six decisions for every one it saves.
Particular games
Three distances too many
The junction descriptor records how far a crossing sits from four ends, and the rung below asked what the value does when one crossing slides along its run. It reads one bit — the offset's parity — and only when the run has odd length. The other three distances reach the value not at all.
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