Integers, Game Trees and some Unknowns.
Samee U. Khan · 2004
Over the number of years, Conway [1, 2] has presented a fruitful theory to link numbers (number theory) and combinatorial games. This theory is useful in analyzing and to some extent predicting the outcome of games. Combinatorial games require the basic assumption that all moves are visible to both of the players and chance (dice) does not play any role. Thus, a game can also be represented as a tree with left child of the tree node capturing the outcome or the effect of one player on the entire (or the subgame) game and the right child of the tree node as the play of the second player. It is to be noted that for this reason we often seen players referred to as Left and Right. In this paper, we will introduce the link between number theory and combinatorial games, and we will also show that the current theory although very powerful, is in some cases not complete. In the next few passages, we will introduce some of the basic concepts of number theory related to combinatorial games. We will show the effectiveness of this theory by giving some examples from real board games. These examples will also lead to some open problems in Combinatorial Game Theory (CGT). Conway [2, pp.4-6] defines (surreal) numbers as: “If L and R are any two sets of numbers, and no member of L is ≥ any member of R, then there is a number {L | R}. All numbers are constructed in this way.” From this definition a number x can be written as: {x L |x R}, and each of the individual elements as: x L (left element), x R (right element). This breakup of number(s) allow us to apply simple set operations that will mimic addition, subtraction, multiplication, etc. For example, the addition of two numbers x and y