Combinatorial Game Theory
who wins, and by how much
Generator

Mock Turtles on 8 coins: every lost position

Mock Turtles on 8 coins: every lost position
Mock Turtles on 8 coins: every lost position. The rows a player to move has already lost, drawn in full. A filled disc is a coin showing heads. The set is closed under turning over every coin two of its members disagree about, which is what makes it a linear code, and the count of heads in the sparsest of them is the fewest coin turns that separate two lost positions.

The rows a player to move has already lost, drawn in full. A filled disc is a coin showing heads. The set is closed under turning over every coin two of its members disagree about, which is what makes it a linear code, and the count of heads in the sparsest of them is the fewest coin turns that separate two lost positions.

3 essays call code-words. 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.

PositionWorth OutcomeDrawn in
heads at 1, 2, 3, 4 0 P The code names the move · The losing positions are a code
heads at 1, 2, 5, 6 0 P The code names the move · The losing positions are a code
no heads 0 P The code names the move · The losing positions are a code
heads at 1, 2, 3, 4 0 P The losing positions are a code
heads at 1, 2, 5, 6 0 P The losing positions are a code
no heads 0 P The losing positions are a code
heads at 1, 3 0 P The losing positions are a code
heads at 1, 5 0 P The losing positions are a code
no heads 0 P The losing positions are a code
heads at 1, 2, 3 0 P The losing positions are a code
heads at 1, 4, 5 0 P The losing positions are a code
no heads 0 P The losing positions are a code

Where it is called

Changing this generator changes every one of these figures.

Mock Turtles on 8 coins: every lost position. The rows a player to move has already lost, drawn in full. A filled disc is a coin showing heads. The set is closed under turning over every coin two of its members disagree about, which is what makes it a linear code, and the count of heads in the sparsest of them is the fewest coin turns that separate two lost positions. Impartial games

The losing positions are a code

Turn over one, two or three coins, and the rows a player has already lost turn out to be closed under adding two of them together. That makes them a linear code — and on eight coins it is the extended Hamming code exactly, sixteen words with a weight enumerator of 1 + 14x⁴ + x⁸, produced by a move rule that knows nothing about codes.

 Mock Turtles on 8 coins: finding the move is decoding. Every row of the game, sorted by what it takes to win from it. The lost rows are the codewords; a won row is a codeword with errors, and the winning move is the error pattern that turns them off. The distance column is a fact about the code and the coins column is a fact about the rules, and the two do not quite agree. Impartial games

The code names the move

If the lost rows of a coin-turning game are a linear code, then a won row is a codeword with errors in it and the winning move is whatever turns the errors off. Over all 256 rows of Mock Turtles on eight coins: 16 codewords, 240 won rows, none more than two coins from a lost one — and 64 of them whose cheapest winning move has to turn three coins anyway.

 The dual is the value table's span. The dual code against the span of the bit-planes of the one-coin Grundy values, on every game measured. Impartial games

The dual was the value table

A coin-turning game's losing rows form a linear code, and a code has a dual that nothing in the game appeared to read. It reads it constantly: the dual is spanned by the bit-planes of the Grundy values — the parity checks are the value table stood on end — and on Mock Turtles over eight coins the losing rows are exactly the span of the table that decides them.

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