Trainyard is NP-Hard (1603.00928v1)

Abstract: Recently, due to the widespread diffusion of smart-phones, mobile puzzle games have experienced a huge increase in their popularity. A successful puzzle has to be both captivating and challenging, and it has been suggested that this features are somehow related to their computational complexity \cite{Eppstein}. Indeed, many puzzle games --such as Mah-Jongg, Sokoban, Candy Crush, and 2048, to name a few-- are known to be NP-hard \cite{CondonFLS97, culberson1999sokoban, GualaLN14, Mehta14a}. In this paper we consider Trainyard: a popular mobile puzzle game whose goal is to get colored trains from their initial stations to suitable destination stations. We prove that the problem of determining whether there exists a solution to a given Trainyard level is NP-hard. We also \href{http://trainyard.isnphard.com}{provide} an implementation of our hardness reduction.