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

Video Games as Time Machines: Video Game Nostalgia and the Success of Retro Gaming

This article conceptually integrates research on the experience of nostalgia—defined as a predominantly positive, social, and past-oriented emotion—into the fold of video game research. We emphasize the role of nostalgia as an explanation for contemporary retro gaming trends, and suggest that nostalgia towards gaming events is a necessary area of research. To those ends, we broadly review existing literature on nostalgia before specifically focusing on media-induced nostalgia, and demonstrate how theoretical and empirical observations from this work can be applied to understand video game nostalgia. In particular, we argue that engaging in older gaming experiences indirectly (via memories) and even directly (via replaying or recreating experiences) elicits nostalgia, which in turn contributes to players’ self-optimization and enhanced well-being. Moreover, as gamers and the medium mature together, nostalgic experiences with the medium are likely to become increasingly prevalent. The broad aim of this article is to offer future directions for research on video game nostalgia and provide a research agenda for research in this area

Super Mario Bros. is Harder/Easier Than We Thought

Mario is back! In this sequel, we prove that solving a generalized level of Super Mario Bros. is PSPACE-complete, strengthening the previous NP-hardness result (FUN 2014). Both our PSPACE-hardness and the previous NP-hardness use levels of arbitrary dimensions and require either arbitrarily large screens or a game engine that remembers the state of off-screen sprites. We also analyze the complexity of the less general case where the screen size is constant, the number of on-screen sprites is constant, and the game engine forgets the state of everything substantially off-screen, as in most, if not all, Super Mario Bros. video games. In this case we prove that the game is solvable in polynomial time, assuming levels are explicitly encoded; on the other hand, if levels can be represented using run-length encoding, then the problem is weakly NP-hard (even if levels have only constant height, as in the video games). All of our hardness proofs are also resilient to known glitches in Super Mario Bros., unlike the previous NP-hardness proof

Restricted Power - Computational Complexity Results for Strategic Defense Games

We study the game Greedy Spiders, a two-player strategic defense game, on planar graphs and show PSPACE-completeness for the problem of deciding whether one player has a winning strategy for a given instance of the game. We also generalize our results in metatheorems, which consider a large set of strategic defense games. We achieve more detailed complexity results by restricting the possible strategies of one of the players, which leads us to Sigma^p_2- and Pi^p_2-hardness results

A quantum procedure for map generation

Quantum computation is an emerging technology that promises a wide range of possible use cases. This promise is primarily based on algorithms that are unlikely to be viable over the coming decade. For near-term applications, quantum software needs to be carefully tailored to the hardware available. In this paper, we begin to explore whether near-term quantum computers could provide tools that are useful in the creation and implementation of computer games. The procedural generation of geopolitical maps and their associated history is considered as a motivating example. This is performed by encoding a rudimentary decision making process for the nations within a quantum procedure that is well-suited to near-term devices. Given the novelty of quantum computing within the field of procedural generation, we also provide an introduction to the basic concepts involved.Comment: To be published in the proceedings of the IEEE Conference on Game

I Am Error

I Am Error is a platform study of the Nintendo Family Computer (or Famicom), a videogame console first released in Japan in July 1983 and later exported to the rest of the world as the Nintendo Entertainment System (or NES). The book investigates the underlying computational architecture of the console and its effects on the creative works (e.g. videogames) produced for the platform. I Am Error advances the concept of platform as a shifting configuration of hardware and software that extends even beyond its ‘native’ material construction. The book provides a deep technical understanding of how the platform was programmed and engineered, from code to silicon, including the design decisions that shaped both the expressive capabilities of the machine and the perception of videogames in general. The book also considers the platform beyond the console proper, including cartridges, controllers, peripherals, packaging, marketing, licensing, and play environments. Likewise, it analyzes the NES’s extension and afterlife in emulation and hacking, birthing new genres of creative expression such as ROM hacks and tool-assisted speed runs. I Am Error considers videogames and their platforms to be important objects of cultural expression, alongside cinema, dance, painting, theater and other media. It joins the discussion taking place in similar burgeoning disciplines—code studies, game studies, computational theory—that engage digital media with critical rigor and descriptive depth. But platform studies is not simply a technical discussion—it also keeps a keen eye on the cultural, social, and economic forces that influence videogames. No platform exists in a vacuum: circuits, code, and console alike are shaped by the currents of history, politics, economics, and culture—just as those currents are shaped in kind

Large Peg-Army Maneuvers

Despite its long history, the classical game of peg solitaire continues to attract the attention of the scientific community. In this paper, we consider two problems with an algorithmic flavour which are related with this game, namely Solitaire-Reachability and Solitaire-Army. In the first one, we show that deciding whether there is a sequence of jumps which allows a given initial configuration of pegs to reach a target position is NP-complete. Regarding Solitaire-Army, the aim is to successfully deploy an army of pegs in a given region of the board in order to reach a target position. By solving an auxiliary problem with relaxed constraints, we are able to answer some open questions raised by Cs\'ak\'any and Juh\'asz (Mathematics Magazine, 2000). To appreciate the combinatorial beauty of our solutions, we recommend to visit the gallery of animations provided at http://solitairearmy.isnphard.com.Comment: Conference versio

NP-completeness of the game Kingdomino

Kingdomino is a board game designed by Bruno Cathala and edited by Blue Orange since 2016. The goal is to place $2 \times 1$ dominoes on a grid layout, and get a better score than other players. Each $1 \times 1$ domino cell has a color that must match at least one adjacent cell, and an integer number of crowns (possibly none) used to compute the score. We prove that even with full knowledge of the future of the game, in order to maximize their score at Kingdomino, players are faced with an NP-complete optimization problem