Abstract. Splittles is a nim variant in which a move consists of removing tokens from one heap, and (optionally) splitting the remaining heap into two. The possible numbers of tokens that can legally be removed are fixed, but the two players might have different subtraction sets. The nature of the game, and the analysis techniques employed, vary dramatically depending on the subtraction sets.

Introduction

Partizan Splittles is a game played by two players, conventionally called Left and Right. A position in the game consists of a number of heaps of tokens and a move requires a player to choose a heap, remove some positive number, s, of tokens from the heap and optionally to split the remaining heap (if there are two or more tokens remaining) into two heaps. Two sets of positive integers SL and SR are fixed in advance, and there is an additional restriction that when Left moves she must choose s ∈ SL, while Right must choose s ∈ SR at his turn. The sets SL and SR are called the subtraction sets of Left and Right respectively.

It is sometimes convenient to represent a position pictorially by one-dimensional blocks of boxes rather than heaps of tokens. A move is to remove a contiguous block of boxes; moves in the middle of a block are tantamount to splitting a heap.