A Tractability Gap Beyond Nim-Sums: It's Hard to Tell Whether a Bunch of Superstars Are Losers

Abstract

In this paper, we address a natural question at the intersection of combinatorial game theory and computational complexity: "Can a sum of simple tepid games in canonical form be intractable?" To resolve this fundamental question, we consider superstars, positions first introduced in Winning Ways where all options are nimbers. Extending Morris' classic result with hot games to tepid games, we prove that disjunctive sums of superstars are intractable to solve. This is striking as sums of nimbers can be computed in linear time. Our analyses also lead to a family of elegant board games with intriguing complexity, for which we present web-playable versions of the rulesets described within.