Nimbers

Nimbers are a class of numbers used in combinatorial game theory to assign values to impartial games under the Sprague-Grundy framework. For any impartial game G, its nimber, denoted g(G), is defined as the mex (minimum excluded value) of the set { g(H) : H is an option of G }. Nimber values generalize the Grundy numbers; they capture the outcome class of sums of independent game components. The disjunctive sum of two games satisfies g(G+H) = g(G) ⊕ g(H), where ⊕ is nimber addition. For finite nimbers, ⊕ coincides with bitwise XOR of binary representations; this is why Nim heaps add by xor of their sizes. The nimber 0 represents a losing position; 1, 2, 3, ... represent progressively richer positions. The class of nimbers is closed under nimber addition and multiplication, forming a commutative semiring; it also includes transfinite nimbers such as ω, ω+1, and so on. These infinite nimbers arise from certain impartial games and extend the finite ordinal values; they are not simply ordinals under standard addition. The theory was developed by Sprague and Grundy in the 1930s and provides a unifying method to analyze impartial games by reducing them to nimbers and using nimber arithmetic. An individual Nim heap of size n has nimber n; the nim-sum of several heaps is obtained by repeatedly applying ⊕ to their sizes. Example: two heaps of sizes 3 and 5 have nimber 3 ⊕ 5 = 6, predicting a winning strategy for the next player in the disjunctive sum unless the xor is zero. Further study often involves constructing the nimber values of more complex games or using the theory to classify game positions. Nimbers thereby connect combinatorial game theory with number systems and abstract algebra.