Mathematical viewpoints combine algebra, combinatorics, number theory, and complexity theory with other perspectives from economics, statistics, and computer science. A SUM of such games is naturally defined as the game in which each player, at his turn, may choose to make any of his legal moves on any single summand. The first small step toward the mathematical theory of combinatorial games was the solution of NIM in 1902 by Bouton at Harvard. In the 1930s, Sprague and Grundy extended this theory to cover all impartial games. John Conway axiomatized this important branch of the subject.
Checkers satisfies the normal termination rule, but it is loopy.
For example, if each player has only one king remaining, and each gets into a double corner, then neither player can bring the game to a victorious termination. Chinese Go IS loopfree, but only because of a somewhat artificial extra rule which forbids moving into any position which occurred earlier in the same game. He then opined that this point of view superseded the classical works of Dedekind, Cantor, and many others. Indeed, almost all examples of mathematical problems considered harder than NP are combinatorial games. Some consider this intimate connection with complexity theory to be the primary mathematical justification for studying the subject.