Two-pile and three-pile games of a new variant of Nim known as Halve Nim

We investigate a variant of Nim called Halve Nim, which in addition to the standard moves of Nim, we allow replacing each pile of coins with half its amount. We determine the P-positions of all two-pile games of Halve Nim. Also, we determine the P-positions of all three-pile games of Halve Nim in which one pile has at most ten coins.