Small Domineering boards and what they are worth
Every value here was computed from the moves rather than looked up. Even on boards this small the values are switches and infinitesimals rather than numbers, which is the ordinary situation for a partizan game and the reason the theory needs more than arithmetic.
13 essays call
domineering-values. 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
27 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
Domineering
One player places dominoes vertically, the other horizontally, on a shared grid. The rules take one line, the values are a mess, and that mess is the point — this is what the theory looks like applied to a game nobody designed for it.
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.
Sums and comparison
Turn the board through a right angle
A two-by-four Domineering board is worth something no number can express, and Right is ahead on it. Turn a second board through a right angle, put the two side by side, and the total is exactly zero. Every position has an exact opposite, and that single fact is what makes subtraction — and therefore comparison — possible at all.
What it costs
What counts as the same position, and what that is worth
Folding a 4×4 Domineering board by its symmetries takes the table from 5,700 entries to 1,522 — a saving of 3.75, against a ceiling of exactly 4. An orbit cannot be larger than the group acting on it, so this is the one saving in the subject that can never change an exponent.
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.
Values
Tiny, miny, and the sizes below every size
An empty two-by-four Domineering board is worth less than nothing and more than every negative number. It is not up, not down and not a fraction — it is a miny, and the minies come in sizes, strictly ordered among themselves below a floor no number reaches.
Sums and comparison
Where the impartial theory stops
Sprague–Grundy gives every impartial position one number, and the number is complete. The moment the two players have different moves no number works at all — not a harder one to compute, none — and three positions here are compared against every nimber to show it.
Temperature
Cooling by exactly one
Cooling is usually met as a way of reading a thermograph: a tax, and two numbers at the height of the tax. Applied as an operator it returns a position instead — and then the obvious question is whether heating gives the position back. Over the 1,474 values born by day three, 27 survive the round trip and 15 of those are numbers that never moved. Cooling throws away almost everything it touches.
Particular games
The values of every small board
Thirty Domineering rectangles, every value computed from the moves rather than looked up. The 1×n row obeys a formula and the 2×n row does not: its outcomes run L N N R three times over and then 2×13 comes out worth exactly 0, and its temperatures climb to 19/16 and fall back without settling.
Sums and comparison
What can be struck out
From G + X = H + X it follows that G = H, in one line, by adding −X to both sides. It is the shortest theorem here and the most used: it is what makes comparing two boards region by region legitimate. Over 10,648 triples the hypothesis fires 484 times and the conclusion holds 484 times — and the licence expires in three separate directions, each of which loses the same axiom in a different way.
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.
Values
Which option the reduction keeps
Domination deletes an option when another is at least as good, so what survives is the top of an order. On a board that order is made of moves, and two descriptions of the surviving move suggest themselves. Over 1,586 Domineering option lists one of them is right 47% of the time and the other 90%, and the one that wins is not the one a player would guess.
The whole library · The position index · The figures that play back