In mathematics, specifically combinatorial game theory, on, written as , is the value of the loopy game where Left may only loop back the same position, while Right has no moves. may be represented by surreal formBecause of Right's inability to move, is positive, as Left automatically wins.
Its negative counterpart, off, written as , may be represented by and are not additive inverses. Their sum, dud (deathless universal draw), written as , always loops back to itself, that is, always results in a draw, hence its name.
is greater than any ender , as in the difference game , Left may play solely in , while Right has no option but to play in . Because is an ender, play in will eventually end, and Left will win. Similarly, is less than any ender.
Value- games
Hackenbush In Hackenbush, a position with infinitely many blue edges on the ground has value , as chopping an edge still leaves infinitely many edges remaining. This example illustrates the fact that ; adding a blue branch won't change the position.
Col The Col position to the left has value . Right cannot move, while Left may color any of the empty areas, while not changing the position.
Snort A similar Snort position has value , shown to the right. Again, Right cannot move, while Left may color any of the empty areas, which doesn't change the position.
See also Loopy game Surreal number
References