Trainyard is NP-Hard

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

Trains, Games, and Complexity: 0/1/2-Player Motion Planning through Input/Output Gadgets

We analyze the computational complexity of motion planning through local "input/output" gadgets with separate entrances and exits, and a subset of allowed traversals from entrances to exits, each of which changes the state of the gadget and thereby the allowed traversals. We study such gadgets in the 0-, 1-, and 2-player settings, in particular extending past motion-planning-through-gadgets work to 0-player games for the first time, by considering "branchless" connections between gadgets that route every gadget's exit to a unique gadget's entrance. Our complexity results include containment in L, NL, P, NP, and PSPACE; as well as hardness for NL, P, NP, and PSPACE. We apply these results to show PSPACE-completeness for certain mechanics in Factorio, [the Sequence], and a restricted version of Trainyard, improving prior results. This work strengthens prior results on switching graphs and reachability switching games.Comment: 37 pages, 36 figure

Tracks from hell - when finding a proof may be easier than checking it

We consider the popular smartphone game Trainyard: a puzzle game that requires the player to lay down tracks in order to route colored trains from departure stations to suitable arrival stations. While it is already known [Almanza et al., FUN 2016] that the problem of finding a solution to a given Trainyard instance (i.e., game level) is NP-hard, determining the computational complexity of checking whether a candidate solution (i.e., a track layout) solves the level was left as an open problem. In this paper we prove that this verification problem is PSPACE-complete, thus implying that Trainyard players might not only have a hard time finding solutions to a given level, but they might even be unable to efficiently recognize them

On the Complexity of Two Dots for Narrow Boards and Few Colors

Two Dots is a popular single-player puzzle video game for iOS and Android. A level of this game consists of a grid of colored dots. The player connects two or more adjacent dots, removing them from the grid and causing the remaining dots to fall, as if influenced by gravity. One special move, which is frequently a game-changer, consists of connecting a cycle of dots: this removes all the dots of the given color from the grid. The goal is to remove a certain number of dots of each color using a limited number of moves. The computational complexity of Two Dots has already been addressed in [Misra, FUN 2016], where it has been shown that the general version of the problem is NP-complete. Unfortunately, the known reductions produce Two Dots levels having both a large number of colors and many columns. This does not completely match the spirit of the game, where, on the one hand, only few colors are allowed, and on the other hand, the grid of the game has only a constant number of columns. In this paper, we partially fill this gap by assessing the computational complexity of Two Dots instances having a small number of colors or columns. More precisely, we show that Two Dots is hard even for instances involving only 3 colors or 2 columns. As a contrast, we also prove that the problem can be solved in polynomial-time on single-column instances with a constant number of goals

On the PSPACE-completeness of Peg Duotaire and other Peg-Jumping Games

Peg Duotaire is a two-player version of the classical puzzle called Peg Solitaire. Players take turns making peg-jumping moves, and the first player which is left without available moves loses the game. Peg Duotaire has been studied from a combinatorial point of view and two versions of the game have been considered, namely the single- and the multi-hop variant. On the other hand, understanding the computational complexity of the game is explicitly mentioned as an open problem in the literature. We close this problem and prove that both versions of the game are PSPACE-complete. We also prove the PSPACE-completeness of other peg-jumping games where two players control pegs of different colors

Northern Pacific Railroad Shops of Brainerd, Minnesota

An adaptive reuse of the old Northern Pacific Railroad yard in Brainerd, Minnesota. This project will aim to restore the once bustling character of a site at the heart of the city. Comprising of 47 acres, the site includes several abandoned buildings on the National Register of Historic Places. Conformance to the Secretary of Interior's guidelines on buildings of the National Register will be followed for design consideration. A better utilization will include commercial, residential, and other tourist related functions, as well a rail link with other regional communities (Fargo, St. Cloud, Duluth, and Twin Cities). Linking people to other cities and old world charm atmosphere will be driving force behind the site. A phase in the design will be a consideration to allow for the project get underway

A Moral Influenza: An Historical Archaeological Investigation of the Prohibition Era in the United States 1920-1933

The following exploration of National Alcohol Prohibition in the United States employs an interdisciplinary approach to understand the impetus of the large-scale defiance of liquor laws and to identify the physical manifestations of the social process of mass resistance in the archaeological record. Historical documentation, newspaper publications, oral histories, landscape analysis of road development in Montana, and an archaeological survey of a mountain homestead site where manufacturer of illicit alcohol took place are used to contextualize the Prohibition Era from the perspective of the offender. The research goals of this work are varied and include documentation of the archaeological footprint of prolonged alcohol production and transport as well as highlighting the social pressures driving the lawlessness that characterized the era, including a case study on female offenders. The results of this investigation determined that liquor law infractions were primarily driven by socio-economic and environmental factors rather than a wave of immorality

Trainyard is NP-Hard

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 these features are somehow related to their computational complexity [6]. Indeed, many puzzle games – such as Mah-Jongg, Sokoban, Candy Crush, and 2048, to name a few – are known to be NP-hard [3,4,8,12]. 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 provide an implementation of our hardness reduction

Trainyard is NP-hard

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 [5]. Indeed, many puzzle games - such as Mah-Jongg, Sokoban, Candy Crush, and 2048, to name a few - are known to be NP-hard [3, 4, 7, 10]. 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 provide an implementation of our hardness reduction1