Combinatorial Game Theory
who wins, and by how much
Generator

Every domino Left can topple in LRRL

Every domino Left can topple in LRRL
Every domino Left can topple in LRRL. A row of dominoes, blue for Left and red for Right, and each of the mover's options below it. Toppling a domino leftward removes it and everything to its left; rightward removes it and everything to its right. The value under each option is what the game recursion returns for the row that survives.

A row of dominoes, blue for Left and red for Right, and each of the mover's options below it. Toppling a domino leftward removes it and everything to its left; rightward removes it and everything to its right. The value under each option is what the game recursion returns for the row that survives.

4 essays call topple-row. 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

26 distinct positions, harvested by running this generator again at the options each essay passed it.

PositionWorth OutcomeDrawn in
empty 0 P A recipe instead of a census · How long a row a value needs · Topple it from either end
G ∗ N How long a row a value needs
GG ∗2 N How long a row a value needs
GGG ∗3 N How long a row a value needs
GRG {0 | 0, {0 | −1}} N How long a row a value needs
L 1 L How long a row a value needs · Topple it from either end
LGL 1∗ L How long a row a value needs
LGR 1 | −1 N How long a row a value needs
LGRG {1 | 0, {0 | −1}} N How long a row a value needs
LLLLLLL 7 L Topple it from either end
LLLLLLLR 6 | 0 N Topple it from either end
LLR 1 | 0 N How long a row a value needs · The other way to move a row
LLRLRLL 1↑ L How long a row a value needs
LLRR 1 | −1 N The other way to move a row · Topple it from either end
LR ∗ N The other way to move a row · Topple it from either end
LRL 1/2 L How long a row a value needs · Topple it from either end
LRR 0 | −1 N The other way to move a row
LRRL ↑ L A recipe instead of a census · Topple it from either end
LRRLL {1 | ∗} L Topple it from either end
LRRR 0 | −2 N Topple it from either end
R −1 R How long a row a value needs
RG 0 | −1 N How long a row a value needs
RLRR −1∗ R Topple it from either end
RRL 0 | −1 N A recipe instead of a census · Topple it from either end
RRLR −1∗ R Topple it from either end
RRRL 0 | −2 N Topple it from either end

Where it is called

Changing this generator changes every one of these figures.

The same row, cut and toppled. Rows of blue and red drawn once and evaluated twice: as a Hackenbush string, where a player cuts an edge of their own colour and everything above it falls, and as Toppling Dominoes, where a player knocks one over and everything on the chosen side falls. Both values are computed by the same recursion from the two rulesets. Particular games

Topple it from either end

A row of blue and red is the picture this site opens with, and under Hackenbush's rules it is always a number. Knock the pieces over instead of cutting them — everything on the chosen side falls — and 480 of the 510 rows up to eight pieces stop being numbers. The two games agree on sixteen rows, every one of a single colour, and the temperature of the hottest row climbs by exactly a half for each domino added.

 Push and Shove over every strip up to 6 squares. The same strips under both rules. A cliff lets coins fall off and a wall does not, and the census says what that one clause is worth: both games are entirely made of numbers, they never agree on a value, and the obvious board-reading is right far more often under the wall than under the cliff. Particular games

The other way to move a row

Shove has a cliff and Push has a wall, and that is the whole of the difference. Both games make every one of the 728 strips up to six squares a number, so neither ever has anything worth fighting over — and the two rules do not agree on the value of a single position. The obvious board-reading is exact on 446 strips under the wall and on 140 under the cliff, and 486 strips contain a coin its owner cannot move at all.

 What the third colour reaches. Every row of Toppling Dominoes up to 7 long, over two colours and over three, with the number of distinct values each set of rows carries. Each value was computed by the recursion; the last column is the count of values three colours reach that two do not, cumulatively. Particular games

How long a row a value needs

Add a third colour that either player may topple and a row of seven dominoes reaches 1,047 distinct values where two colours reach 149. That makes the length of the shortest row worth a value into a measure of the value's complexity — one a reader can hold in their hand — and it is not the birthday: 1↑ is born on day three and needs seven dominoes.

 Every domino Left can topple in LRRL. A row of dominoes, blue for Left and red for Right, and each of the mover's options below it. Toppling a domino leftward removes it and everything to its left; rightward removes it and everything to its right. The value under each option is what the game recursion returns for the row that survives. Particular games

A recipe instead of a census

Counting a thousand values in seven dominoes suggests Toppling Dominoes reaches every short game, and a count is not a construction. The obvious construction — lay the two options either side of a Left domino and a Right one — is exact on day one, right on a third of day three, and cannot be applied to nine in ten values at all.

The whole library · The position index · The figures that play back