Game-theoretic Values
Most of the research around Toads-and-Frogs has been around determining the game-theoretic values of some particular Toads-and-Frogs positions, or determining whether some particular values can arise in the game.
Winning Ways for your Mathematical Plays showed first numerous possible values. For example :
In 1996, Jeff Erickson proved that for any dyadic rational number q (which are the only numbers that can arise in finite games), there exists a Toads-and-Frogs positions with value q. He also found an explicit formula for some remarkable positions, like, and formulated 6 conjectures on the values of other positions and the hardness of the game.
These conjctures fueled further research. Jesse Hull proved conjecture 6 in 2000, which states that determining the value of an arbitrary Toads-and-Frogs position is NP-hard. Doron Zeilberger and Thotsaporn Aek Thanatipanonda proved conjecture 1, 2 and 3 and found a counter-example to conjecture 4 in 2008. Conjecture 5, the last one still open, states that is an infinitesimal value, for all (a, b) except (3, 2).
Read more about this topic: Toads And Frogs (game)
Famous quotes containing the word values:
“Our culture is ill-equipped to assert the bourgeois values which would be the salvation of the under-class, because we have lost those values ourselves.”
—Norman Podhoretz (b. 1930)