2048 is (PSPACE) Hard, but Sometimes Easy

We prove that a variant of 2048, a popular online puzzle game, is PSPACE-Complete. Our hardness result holds for a version of the problem where the player has oracle access to the computer player's moves. Specifically, we show that for an $n \times n$ game board $\mathcal{G}$, computing a sequence of moves to reach a particular configuration $\mathbb{C}$ from an initial configuration $\mathbb{C}_0$ is PSPACE-Complete. Our reduction is from Nondeterministic Constraint Logic (NCL). We also show that determining whether or not there exists a fixed sequence of moves $\mathcal{S} \in \{\Uparrow, \Downarrow, \Leftarrow, \Rightarrow\}^k$ of length $k$ that results in a winning configuration for an $n \times n$ game board is fixed-parameter tractable (FPT). We describe an algorithm to solve this problem in $O(4^k n^2)$ time.Comment: 13 pages, 11 figure

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

Threes!, Fives, 1024!, and 2048 are Hard

We analyze the computational complexity of the popular computer games Threes!, 1024!, 2048 and many of their variants. For most known versions expanded to an m x n board, we show that it is NP-hard to decide whether a given starting position can be played to reach a specific (constant) tile value.Comment: 14 pages, 9 figure

2048 Without New Tiles Is Still Hard

We study the computational complexity of a variant of the popular 2048 game in which no new tiles are generated after each move. As usual, instances are defined on rectangular boards of arbitrary size. We consider the natural decision problems of achieving a given constant tile value, score or number of moves. We also consider approximating the maximum achievable value for these three objectives. We prove all these problems are NP-hard by a reduction from 3SAT. Furthermore, we consider potential extensions of these results to a similar variant of the Threes! game. To this end, we report on a peculiar motion pattern, that is not possible in 2048, which we found much harder to control by similar board designs

Homomorphism Reconfiguration via Homotopy

We consider the following problem for a fixed graph H: given a graph G and two H-colorings of G, i.e. homomorphisms from G to H, can one be transformed into the other by changing one color at a time, maintaining an H-coloring throughout.This is the same as finding a path in the Hom(G,H) complex. For H=K_k this is the problem of finding paths between k-colorings, which was recently shown to be in P for kleq 3 and PSPACE-complete otherwise (Bonsma and Cereceda 2009, Cereceda et al. 2011). We generalize the positive side of this dichotomy by providing an algorithm that solves the problem in polynomial time for any H with no C_4 subgraph. This gives a large class of constraints for which finding solutions to the Constraint Satisfaction Problem is NP-complete, but paths in the solution space can be found in polynomial time. The algorithm uses a characterization of possible reconfiguration sequences (that is, paths in Hom(G,H)), whose main part is a purely topological condition described in terms of the fundamental groupoid of H seen as a topological space

An analysis of key generation efficiency of RSA cryptosystem in distributed environments

Thesis (Master)--Izmir Institute of Technology, Computer Engineering, Izmir, 2005Includes bibliographical references (leaves: 68)Text in English Abstract: Turkish and Englishix, 74 leavesAs the size of the communication through networks and especially through Internet grew, there became a huge need for securing these connections. The symmetric and asymmetric cryptosystems formed a good complementary approach for providing this security. While the asymmetric cryptosystems were a perfect solution for the distribution of the keys used by the communicating parties, they were very slow for the actual encryption and decryption of the data flowing between them. Therefore, the symmetric cryptosystems perfectly filled this space and were used for the encryption and decryption process once the session keys had been exchanged securely. Parallelism is a hot research topic area in many different fields and being used to deal with problems whose solutions take a considerable amount of time. Cryptography is no exception and, computer scientists have discovered that parallelism could certainly be used for making the algorithms for asymmetric cryptosystems go faster and the experimental results have shown a good promise so far. This thesis is based on the parallelization of a famous public-key algorithm, namely RSA

Making Change in 2048

The 2048 game involves tiles labeled with powers of two that can be merged to form bigger powers of two; variants of the same puzzle involve similar merges of other tile values. We analyze the maximum score achievable in these games by proving a min-max theorem equating this maximum score (in an abstract generalized variation of 2048 that allows all the moves of the original game) with the minimum value that causes a greedy change-making algorithm to use a given number of coins. A widely-followed strategy in 2048 maintains tiles that represent the move number in binary notation, and a similar strategy in the Fibonacci number variant of the game (987) maintains the Zeckendorf representation of the move number as a sum of the fewest possible Fibonacci numbers; our analysis shows that the ability to follow these strategies is intimately connected with the fact that greedy change-making is optimal for binary and Fibonacci coinage. For variants of 2048 using tile values for which greedy change-making is suboptimal, it is the greedy strategy, not the optimal representation as sums of tile values, that controls the length of the game. In particular, the game will always terminate whenever the sequence of allowable tile values has arbitrarily large gaps between consecutive values