Combinatorial Game Theory
who wins, and by how much
Generator

Toads and Frogs on T.FT.F — and who wins

Toads and Frogs on T.FT.F — and who wins
Toads and Frogs on T.FT.F — and who winsA Toads and Frogs strip with the outcome stated before anybody moves. Toads move right, frogs left, and either may jump one opposing piece into an empty square. The position is worth zero, so whoever moves first loses — and every reply the machine makes was worked out in advance.worth 0whoever moves first losesyou move the toads · it moves the frogswith the script running, the toads become clickable and this claim can be tested

A Toads and Frogs strip with the outcome stated before anybody moves. Toads move right, frogs left, and either may jump one opposing piece into an empty square. The position is worth zero, so whoever moves first loses — and every reply the machine makes was worked out in advance.

4 essays call play-toads. 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

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

PositionWorth OutcomeDrawn in
T.FT.F 0 P The same strip without the jump
TT..FF 0 P The same strip without the jump · The strip where every number is a whole one
T.FT.F 0 P The strip nobody has a formula for · Toads and Frogs

It plays back

The winner is named before a move is made, and every reply was worked out in advance.

Where it is called

Changing this generator changes every one of these figures.

Toads and frogs. Toads move right and frogs move left, one square into a gap or hopping over exactly one opponent. A player unable to move loses. It can be played on squared paper by anybody, and its values are immediately stranger than the game looks. Particular games

Toads and Frogs

Toads shuffle right, frogs shuffle left, and either may jump over one of the other. A strip six cells long is worth exactly up. Another six-cell strip is worth exactly down. Nobody has a formula for which.

 Toads and frogs. Toads move right and frogs move left, one square into a gap or hopping over exactly one opponent. A player unable to move loses. It can be played on squared paper by anybody, and its values are immediately stranger than the game looks. Particular games

The strip nobody has a formula for

Some toads, a gap, some frogs. Two counts and a spacing is the whole description, and the values that come out of it are integers, stars, switches with eighth-point options and a down — four classes inside one two-parameter family, which is why nobody has written the formula.

 The same strips, with no jumping. Elephants and Rhinos: toads move right and frogs move left, one square into an empty one, and nothing may hop over anything. The pieces keep their order for ever, and the values are computed by the same recursion as the game with the jump in it. Particular games

The same strip without the jump

Delete one clause from Toads and Frogs — the hop over an opponent — and the game is Elephants and Rhinos. Over the same 3,279 strips the values do not become simpler in the way a reader would guess: every value that is a number becomes an integer, against 172 fractions with the jump in, and the count of positions worth fighting over nearly doubles. Removing a move made the game hotter.

 Every strip, without the hop. Toads and Frogs with the jump deleted, over every strip up to eight squares. The fourth column is the argument: whenever the value is a number it is a whole number, without exception, so the fractions the ordinary game produces are made by the hop and by nothing else. Particular games

The strip where every number is a whole one

Delete the hop from Toads and Frogs and the halves, quarters and ups vanish completely: over 9,801 strips, every value that is a number is an integer, without a single exception. The guess that the hopless game therefore has a formula reading the gaps is half right and exactly wrong — 1,460 strips of eight squares are switches, and three strips with the same counts of toads, frogs and empty squares are worth 1, {2 | 1} and 2.

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