The game positions of Nim with a pass (heaps of sizes at most four)

Nim, since it came to the attention of Charles Bouton in 1890s, it has been the most classic and famous combinatorial game. Hundreds of Nim variants appeared and have been investigated extensively since a century. Nim with a pass (Nim*) is a Nim variants which challenges the power of the traditional approaches in combinatorial game theory. Because of the high complexity in 3-heap Nim*, it is believed that a complete analytical solution is difficult to give. The impact of a single pass introduced into the standard game remains an important open question and none of the methods provide solution for Nim*. Nevertheless, a partial analysis of Nim* was published a half year ago and it provides solutions for Nim* played on heaps of sizes at most three. Based on this previous research, Nim* played on heaps of sizes at most four will be analyzed in this paper and the N- and P-positions will be located.