Further generalizations of Wythoff’s game and minimum excludant function

Given non-negative integer a and b, let us consider the following game WY T (a, b). Two piles contain x and y matches. Two players take turns. By one move, it is allowed to take x′ and y′ matches from these piles such that 0 ≤ x′ ≤ x, 0 ≤ y′ ≤ y, 0 < x′ + y′, and [min(x′, y′) < b or |x′ − y′| < a]. The player who takes the last match is the winner (respectively, loser) in the normal (respectively, misere) version of the game. It is easy to verify that cases (a = 0, b = 1), (a = b = 1), and (b = 1, ∀ a) correspond to the two-pile NIM, Wythoff, and Fraenkel games, respectively. The concept of the minimum excludant function mex is known to be instrumental in solving the last two games. We generalize this concept by introducing a function mexb such that mex = mex1 and solve the normal and misere versions of game WY T (a, b).