87 research outputs found
Games, puzzles, and computation
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (p. 147-153).There is a fundamental connection between the notions of game and of computation. At its most basic level, this is implied by any game complexity result, but the connection is deeper than this. One example is the concept of alternating nondeterminism, which is intimately connected with two-player games. In the first half of this thesis, I develop the idea of game as computation to a greater degree than has been done previously. I present a general family of games, called Constraint Logic, which is both mathematically simple and ideally suited for reductions to many actual board games. A deterministic version of Constraint Logic corresponds to a novel kind of logic circuit which is monotone and reversible. At the other end of the spectrum, I show that a multiplayer version of Constraint Logic is undecidable. That there are undecidable games using finite physical resources is philosophically important, and raises issues related to the Church-Turing thesis. In the second half of this thesis, I apply the Constraint Logic formalism to many actual games and puzzles, providing new hardness proofs. These applications include sliding-block puzzles, sliding-coin puzzles, plank puzzles, hinged polygon dissections, Amazons, Kohane, Cross Purposes, Tip over, and others.(cont.) Some of these have been well-known open problems for some time. For other games, including Minesweeper, the Warehouseman's Problem, Sokoban, and Rush Hour, I either strengthen existing results, or provide new, simpler hardness proofs than the original proofs.by Robert Aubrey Hearn.Ph.D
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
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
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
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