Related papers: Strongly Solving 2048 4x3

2048 is a single-player stochastic puzzle game. This intriguing and addictive game has been popular worldwide and has attracted researchers to develop game-playing programs. Due to its simplicity and complexity, 2048 has become an…

2048 is a single player video game, played by millions mostly on mobile devices. We prove rigorously for the first time that there is an algorithm with winning probability at least 0.99969, and that there is a strategy for achieving the 256…

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…

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…

We theoretically analyze the popular mobile app game `2048' for the first time in $n$-dimensional space. We show that one can reach the maximum value $2^{n_1n_2+1}$ and $2^{\left({\prod_{i=1}^{d} n_i}\right)+1}$ for the two dimensional…

We study the complexity of a particular class of board games, which we call `slide and merge' games. Namely, we consider 2048 and Threes, which are among the most popular games of their type. In both games, the player is required to slide…

2048 is an engaging single-player, nondeterministic video puzzle game, which, thanks to the simple rules and hard-to-master gameplay, has gained massive popularity in recent years. As 2048 can be conveniently embedded into the…

Delayed and sparse rewards present a fundamental obstacle for reinforcement-learning (RL) agents, which struggle to assign credit for actions whose benefits emerge many steps later. The sliding-tile game 2048 epitomizes this challenge:…

The game of 2048 is a highly addictive game. It is easy to learn the game, but hard to master as the created game revealed that only about 1% games out of hundreds million ever played have been won. In this paper, we would like to explore…

For any odd integer $n\geq3$ a board (of size $n$) is a square array of $n\times n$ positions with a simple rule of how to move between positions. The goal of the game we introduce is to find a path from the upper left corner of a board to…

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…

In recent years, reinforcement learning has seen interest because of deep Q-Learning, where the model is a convolutional neural network. Deep Q-Learning has shown promising results in games such as Atari and AlphaGo. Instead of learning the…

This paper revisits a classical problem of slotted multiple access with success, idle, and collision events on each slot. First, results of a 2-user multiple access game are reported. The game was conducted at the University of Southern…

Optimizing artificial intelligence (AI) for dynamic environments remains a fundamental challenge in machine learning research. In this paper, we examine evolutionary training methods for optimizing AI to solve the game 2048, a 2D sliding…

While the game Connect-Four has been solved mathematically and the best move can be effectively computed with search based methods, a strong solution in the form of a look-up table was believed to be infeasible. In this paper, we revisit a…

This is the first survey of highly excited eigenstates of a chaotic 3D billiard. We introduce a strongly chaotic 3D billiard with a smooth boundary and we manage to calculate accurate eigenstates with sequential number (of a 48-fold…

Retrograde analysis is used in game-playing programs to solve states at the end of a game, working backwards toward the start of the game. The algorithm iterates through and computes the perfect-play value for as many states as resources…

Before chess came to Northern Europe there was Tafl, a family of asymmetric strategy board games associated strongly with the Vikings. The purpose of this paper is to study the combinatorial state-space complexity of an Irish variation of…

There are 134,459 distinct initial hands at the video poker game Jacks or Better, taking suit exchangeability into account. A computer program can determine the optimal strategy (i.e., which cards to hold) for each such hand, but a complete…

Strong solving of perfect-information games certifies optimal play from every reachable position, but the required state-space coverage is often prohibitive. Weak solving is far cheaper, yet it certifies correctness only at the initial…