Bust-a-Move/Puzzle Bobble is NP-Complete

We prove that the classic 1994 Taito video game, known as Puzzle Bobble or Bust-a-Move, is NP-complete. Our proof applies to the perfect-information version where the bubble sequence is known in advance, and it uses just three bubble colors.Comment: 9 pages, 9 figures. Corrected mistakes in gadget

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

Depth, balancing, and limits of the Elo model

-Much work has been devoted to the computational complexity of games. However, they are not necessarily relevant for estimating the complexity in human terms. Therefore, human-centered measures have been proposed, e.g. the depth. This paper discusses the depth of various games, extends it to a continuous measure. We provide new depth results and present tool (given-first-move, pie rule, size extension) for increasing it. We also use these measures for analyzing games and opening moves in Y, NoGo, Killall Go, and the effect of pie rules

LaserTank is NP-complete

We show that the classical game LaserTank is $\mathrm{NP}$-complete, even when the tank movement is restricted to a single column and the only blocks appearing on the board are mirrors and solid blocks. We show this by reducing $3$-SAT instances to LaserTank puzzles.Comment: 5 page

Bejeweled, Candy Crush and other Match-Three Games are (NP-)Hard

The twentieth century has seen the rise of a new type of video games targeted at a mass audience of "casual" gamers. Many of these games require the player to swap items in order to form matches of three and are collectively known as \emph{tile-matching match-three games}. Among these, the most influential one is arguably \emph{Bejeweled} in which the matched items (gems) pop and the above gems fall in their place. Bejeweled has been ported to many different platforms and influenced an incredible number of similar games. Very recently one of them, named \emph{Candy Crush Saga} enjoyed a huge popularity and quickly went viral on social networks. We generalize this kind of games by only parameterizing the size of the board, while all the other elements (such as the rules or the number of gems) remain unchanged. Then, we prove that answering many natural questions regarding such games is actually \NP-Hard. These questions include determining if the player can reach a certain score, play for a certain number of turns, and others. We also \href{http://candycrush.isnphard.com}{provide} a playable web-based implementation of our reduction.Comment: 21 pages, 12 figure

Lemmings is PSPACE-complete

Lemmings is a computer puzzle game developed by DMA Design and published by Psygnosis in 1991, in which the player has to guide a tribe of lemming creatures to safety through a hazardous landscape, by assigning them specific skills that modify their behavior in different ways. In this paper we study the optimization problem of saving the highest number of lemmings in a given landscape with a given number of available skills. We prove that the game is PSPACE-complete, even if there is only one lemming to save, and only Builder and Basher skills are available. We thereby settle an open problem posed by Cormode in 2004, and again by Forisek in 2010. However we also prove that, if we restrict the game to levels in which the available Builder skills are only polynomially many (and there is any number of other skills), then the game is solvable in NP. Similarly, if the available Basher, Miner, and Digger skills are polynomially many, the game is solvable in NP. Furthermore, we show that saving the maximum number of lemmings is APX-hard, even when only one type of skill is available, whatever this skill is. This contrasts with the membership in P of the decision problem restricted to levels with no "deadly areas" (such as water or traps) and only Climber and Floater skills, as previously established by Cormode.Comment: 26 pages, 11 figure

Enhancing level difficulty and additional content in platform videogames through graph analysis

In this article we present a system that enhances content in platform game levels. This is achieved by adding particular gaming entities and adjusting their arrangement, causing consequent changes in the inherent difficulty and in path related aspects. This idea follows our prior work for the automatic creation of level environments. Starting with a primal level structure and a corresponding graph that sketches the user path, the system detects mandatory and optional path sections and adapts them in order to create more elaborate challenges to the user, forcing detours to gather specific objects or trigger certain events. Alternatively, a designer can create that base level structure and use the algorithm to adapt it to a certain profile. Also, some adjustments can be made to enhance multiplayer cooperative gaming for uneven skilled players, where the path is adapted to force a difficult route to one player and an easier one for the other player. Our experiments showed interesting results on some popular games, where it is possible to observe the previous principles put into practise. The approach is generic and can be expanded to other similar games