1. Introduction

The games STRATEGO, OVID's GAME, THREE-, SIX-, and NINE-MEN's MORRIS [3] and also BUILDING NIM [7] are examples of combinatorial games that have two phases. Specifically, in Phase 1 the board is set up and in Phase 2 the game is played. There are many other “math games” (which mathematicians want to analyze regardless of whether people actually want to play them) in which Phase 1 is not even defined, but instances of Phase 2 are analysed. For example, BOXCARS [2], END-NIM [1], PUSH [2], TOPPLING DOMINOES [8] and their variants.

In this paper, we consider playing Phase 1 as a combinatorial game as well as Phase 2 and analyze two specific games. We were introduced to this concept by Kyle Burke and Urban Larsson (personal communication).

To avoid confusion with the multiple meanings of the word “game”, we refer to the ruleset, which describes the legal moves, and a position, which is an instance of the game after several (including zero) legal moves. By necessity, the position also describes the board upon which play takes place.

In an impartial game, both players have the same moves available.

Definition 1. Let F and H be two impartial rulesets. The conjoined ruleset (F ▸ H) is to play Phase 1 under the F ruleset and when play is no longer possible to start Phase 2 which is played under the ruleset of H.

Forming a conjoined game allows for an interesting Phase 1 battle before the “real” game begins. Since play in Phase 1 sets up the board, it is convenient to have the corresponding game be a placement game [5], i.e., pieces are placed but not moved or removed. The positions at the beginning of Phase 2 will have structure reflecting the Phase 1 rules, and this allows for some partial analysis.