A multi-jump is a finite sequence of jumps where the first jump is initiated from the ground and the rest are initiated in midair. The number of jumps in a multi-jump is the length of the multi-jump, so a double jump is a multi-jump of length two. Several video games, such as Chair Entertainment Group(R)’s Shadow Complex(TM) and Nintendo(R)’s Super Smash Bros.(TM) Melee, feature triple jumps or multi-jumps of even longer length.
The basic problem we consider in this paper is the following. Suppose that a character in a two-dimensional side-scrolling video game wishes to use a multi-jump to jump to the right from a fixed starting point across a gap and land on a fixed platform.
Provided the platform is reachable by a multi-jump, we give strategies for solving this problem on the fly for both player-controlled and artificial intelligence (AI)-controlled characters. In the simplest situation all jumps available to the character are equal and fully concave (Definition 5). In this situation we give a simple strategy (the line method) that is usable by both players and AI. In our experience the majority of games featuring multi-jumps are covered by this situation. We give two further strategies for AI-controlled characters in more-complicated situations. Our first AI strategy is very general, in that it applies to any collection of standard jump functions (Definition 1). We also give a faster (less computationally intensive) AI strategy for collections of standard jump functions whose derivative inverses are known and computable exactly.