Transverse Wave: an impartial color-propagation game inspired by Social Influence and Quantum Nim

Abstract: In this paper, we study a colorful, impartial combinatorial game played on a two-dimensional grid, Transverse Wave. We are drawn to this game because of its apparent simplicity, contrasting intractability, and intrinsic connection to two other combinatorial games, one inspired by social influence and another inspired by quantum superpositions. More precisely, we show that Transverse Wave is at the intersection of social-influence-inspired Friend Circle and superposition-based Demi-Quantum Nim. Transverse Wave is also connected with Schaefer's logic game Avoid True. In addition to analyzing the mathematical structures and computational complexity of Transverse Wave, we provide a web-based version of the game, playable at https://turing.plymouth.edu/~kgb1013/DB/combGames/transverseWave.html. Furthermore, we formulate a basic network-influence inspired game, called Demographic Influence, which simultaneously generalizes Node-Kyles and Demi-Quantum Nim (which in turn contains as special cases Nim, Avoid True, and Transverse Wave). These connections illuminate the lattice order, induced by special-case/generalization relationships over mathematical games, fundamental to both the design and comparative analyses of combinatorial games.