Nim Games: Is it possible to reduce following using Grundy Number?
Your game is an impartial game, and so it is covered by the Sprague–Grundy theorem, which say that it is equivalent to a "nimber". Note that the state of the game consists of the current value of the sum as well as the current set of allowed addends. You can compute the Grundy function, which gives the equivalence to Nim, using dynamic programming.