Scientists finally figured out the absolute limit on how many different ways there are to juggle.
March 19, 2026
Original Paper
Enumerating Prime Patterns in Juggling Variations
arXiv · 2603.17284
The Takeaway
While juggling looks like a physical feat, it is bound by strict combinatorial rules. This research identifies new 'lower bounds' for the number of unique juggling variations possible, mapping the formal mathematical landscape of every catch and throw a human could ever perform.
From the abstract
Juggling patterns can be mathematically modeled as closed walks within directed state graphs. In this paper, we present a unified framework of unbounded juggling patterns and its variations (including multiplex, colored, and passing) primarily through the formalism of the juggling state. By extending this state-based approach and utilizing combinatorial tools such as set partitions and filled Ferrers diagrams, we find and prove a new lower bound on the number of $b$-ball prime patterns with peri