The Mathematics of Tick-Based Timekeeping

Abstract

Timekeeping, via a discrete tick-based system, is a commonly utilized approach for determining the ordered progression of turns, within both the simulation and gaming contexts. Presented in the subject work are the mathematics that underlie such a timekeeping system. This presentation is made within the context of a very popular game, RAID: Shadow Legends. Within this context, the specific instantiations of speed, tick rate, turn meter, speed divisions and critical turn meter value are used to show the transformation of continuous variables into discrete variables with either stepwise continuous values or discrete integer values. When coupled with a simple series of rules that establish the criteria for winning turn meter and which dictate a lack of simultaneous turns, the resultant system is shown to be one in which conflicts at contested ticks are resolved in a pairwise and sequential manner, regardless of the number of combatants involved. Presented in the theory section is the mathematics, based upon modular arithmetic, which is used to determine the relationship between ticks and turns. The solutions are presented in multiple forms, including those based upon using integer sequences and generating functions. Numerical examples are provided to aid in the understanding of the application of the presented theory.