## Submission + - Mathematicians find optimal video game double jump strategy. (jstor.org)

*Mathematicians Aaron Broussard, Martin Malandro, and Abagayle Serreyn have cracked the code for the optimal video game multi-jump, a normal jump followed by additional jumps initiated in midair without the aid of a platform, to determine the highest achievable jump, and have described strategies human players or AI can use in real time to select successful multi-jumps in real time. Their results (doi) are published in the December issue of*

*The American Mathematical Monthly*. From the paper's introduction:

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. ...We therefore assume that the character has a known finite sequence of jump arcs available to her and faces the problem of selecting when to jump in midair, i.e., to switch from the arc of one jump to the next, so as to land on the 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.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.