Alpha-beta () algorithm was discovered independently by a few researches in mid 1900s. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. First, it creates two new variables, new_grid and changed. I got very frustrated with Haskell trying to do that, but I'm probably gonna give it a second try! Alpha-Beta Pruning. It is sensitive to monotonic transformations in utility values. For example, 4 is a moderate speed, decent accuracy search to start at. This intuition will give you also the upper bound for a tile value: where n is the number of tile on the board. It just got me nearly to the 2048 playing the game manually. Could you update those? <> Implementation of reinforcement learning algorithms to solve pacman game. If nothing happens, download GitHub Desktop and try again. I wrote an Expectimax solver for 2048 using the heuristics noted on the top ranking SO post "Optimal AI for 2048". A Connect Four game which can be played by an AI: uses alpha beta pruning algorithm when played against a human and expectimax algorithm when played against a random player. sign in Refining the algorithm so that it always reaches 16k/32k for a non-random game might be another interesting challenge You are right, it's harder than I thought. Scoring is also done using table lookup. Below animation shows the last few steps of the game played by the AI agent with the computer player: Any insights will be really very helpful, thanks in advance. Maximum points AFAIK is slightly more than 20,000 points which is way larger than my current score. We call the function recursively until we reach a terminal node(the state with no successors). My goal was to develop an AI that plays the game more similarly to how I've . Initially, I used two very simple heuristics, granting "bonuses" for open squares and for having large values on the edge. This package provides methods for generating random numbers. (This is the link of my blog post for the article: https://sandipanweb.wordpress.com/2017/03/06/using-minimax-with-alpha-beta-pruning-and-heuristic-evaluation-to-solve-2048-game-with-computer/ and the youtube video: https://www.youtube.com/watch?v=VnVFilfZ0r4). If it does not, then the code declares victory for the player and ends the program execution. This is a constant, used as a base-line and for other uses like testing. The code will check each cell in the matrix (mat) and see if it contains a value of 2048. This module contains all the functions that we will use in our program. Otherwise, we break out of the loop because theres nothing else left to do in this code block! But if during the game there is no empty cell left to be filled with a new 2, then the game goes over. If they are, then their values are set to be 2 times their original value and the next cell in that column is emptied so that it can hold a new value for future calculations. This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. Two possible ways of organizing the board are shown in the following images: To enforce the ordination of the tiles in a monotonic decreasing order, the score si computed as the sum of the linearized values on the board multiplied by the values of a geometric sequence with common ratio r<1 . Requires python 2.7 and Tkinter. This is your objective: The chosen corner is arbitrary, you basically never press one key (the forbidden move), and if you do, you press the contrary again and try to fix it. I also tried the corner heuristic, but for some reason it makes the results worse, any intuition why? The cyclic strategy finished an "average tile score" of. Watching this playing is calling for an enlightenment. Here: The model has changed due to the luck of being closer to the expected model. It is a variation of the Minimax algorithm. It is very easy but hard to achieve its goal. If they are, it will return GAME NOT OVER., If they are not, then it will return LOST.. Moving down can be done by taking transpose the moving right. 3 0 obj Expectimax Search In expectimax search, we have a probabilistic model of how the opponent (or environment) will behave in any state Model could be a simple uniform distribution (roll a die) Model could be sophisticated and require a great deal of computationrequire a great deal of computation We have a node for every outcome <>>> If nothing happens, download Xcode and try again. I used an exhaustive algorithm that favours empty tiles. You signed in with another tab or window. I want to give it a try but those seem to be the instructions for the original playable game and not the AI autorun. Building instructions provided. Thanks, late answer and it performs not really well (almost always in [1024, 8192]), the cost/stats function needs more work, thanks @Robusto, I should improve the code some day, it can be simplified. Implementation of Expectimax for an AI agent to play 2048. - Expectimaximin algorithm apply to a concrete case 2048. We will design each logic function such as we are performing a left swipe then we will use it for right swipe by reversing matrix and performing left swipe. The starting move with the highest average end score is chosen as the next move. endobj 10. Provides heuristic scores and before/after compacting of columns and rows for debug purposes. Therefore, the smoothness heuristic just measures the value difference between neighboring tiles, trying to minimize this count. By using our site, you You signed in with another tab or window. If there have been no changes, then changed is set to False . Next, if the user moves their finger (or swipe) up, then instead of reversing the matrix, the code just takes its transpose value and updates the grid accordingly. I'm the author of the AI program that others have mentioned in this thread. The changed variable will be set to True once the matrix has been merged and therefore represents the new grid. There is a 4*4 grid which can be filled with any number. The red line shows the algorithm's best random-run end game score from that position. Python Programming Foundation -Self Paced Course, Conway's Game Of Life (Python Implementation), Python implementation of automatic Tic Tac Toe game using random number, Rock, Paper, Scissor game - Python Project, Python | Program to implement Jumbled word game, Python | Program to implement simple FLAMES game. My approach encodes the entire board (16 entries) as a single 64-bit integer (where tiles are the nybbles, i.e. There was a problem preparing your codespace, please try again. It performs pretty quickly for depth 1-4, but on depth 5 it gets rather slow at a around 1 second per move. Most of the times it either stops at 1024 or 512. As we said before, we will evaluate each candidate . Theoretical limit in a 4x4 grid actually IS 131072 not 65536. I will edit this later, to add a live code @nitish712, @bcdan the heuristic (aka comparison-score) depends on comparing the expected value of future state, similar to how chess heuristics work, except this is a linear heuristic, since we don't build a tree to know the best next N moves. How can I recognize one? Finally, it returns the new matrix and bool changed. This is possible due to domain-independent nature of the AI. INTRODUCTION Game 2048 is a popular single-player video game released A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. (source). I became interested in the idea of an AI for this game containing no hard-coded intelligence (i.e no heuristics, scoring functions etc). A set of AIs for the 2048 tile-merging game. This file contains all the functions used in this project. And scoring is done simply by counting the number of empty squares. Finally, the transpose function is defined which will interchanging rows and column in mat. I will implement a more efficient version in C++ as soon as possible. By using our site, you Includes an expectimax strategy that reaches 16384 with 34.6% success and an ML model trained with temporal difference learning. Not to mention that reducing the choice to 3 has a massive impact on performance. This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. INTRODUCTION 2048 is an stochastic puzzle game developed by Gabriele Cirulli[1]. (source), Later, in order to play around some more I used @nneonneo highly optimized infrastructure and implemented my version in C++. Highly recommended to go through all the comments. 10% for a 4 and 90% for a 2). game.exe -h: usage: game.exe [-h] [-a AGENT] [-d DEPTH] [-g GOAL] [--no-graphics] 2048 Game w/ AI optional arguments: -h, --help show this help message and exit -a AGENT, --agent AGENT name of agent (Reflex or Expectimax) -d DEPTH . the board position and the player that is next to move). Not bad, your illustration has given me an idea, of taking the merge vectors into evaluation. The result: sheer impossibleness. The code then moves the grid left using the move_left function. The maximizer node chooses the right sub-tree to maximize the expected utilities.Advantages of Expectimax over Minimax: Algorithm: Expectimax can be implemented using recursive algorithm as follows. Here we evaluate faces that have the possibility to getting to merge, by evaluating them backwardly, tile 2 become of value 2048, while tile 2048 is evaluated 2. I thinks it's quite successful for its simplicity. @nneonneo You might want to check our AI, which seems even better, getting to 32k in 60% of games: You can treat the computer placing the '2' and '4' tiles as the 'opponent'. By far, the most interesting solution here. The code compresses the grid by copying each cells value to a new list. x]7r}QiuUWe,QVbc!gvMvSM$c->(P%w$( _B}x2oFauV,nY-] Not sure why this doesn't have more upvotes. I'm sure the full details would be too long to post here) how your program achieves this? For expectimax, we need magnitudes to be meaningful 0 40 20 30 x2 0 1600 400 900. The code starts by creating an empty list, and then it loops through all of the cells in the matrix. The decision rule implemented is not quite smart, the code in Python is presented here: An implementation of the minmax or the Expectiminimax will surely improve the algorithm. I. xkcdxkcd Inside the if statement, we are checking for different keys and depending on that input, we are calling one of the functions from logic.py. But, when I actually use this algorithm, I only get around 4000 points before the game terminates. There was a problem preparing your codespace, please try again. Rest cells are empty. . At 10 moves/s: 589355 (300 games average), At 3-ply (ca. The game contrl part code are used from 2048-ai. @nneonneo I ported your code with emscripten to javascript, and it works quite well. Read the squares in the order shown above until the next squares value is greater than the current one. 2048-expectimax-ai is a Python library typically used in Gaming, Game Engine, Example Codes applications. I think the 65536 tile is within reach! https://www.edx.org/micromasters/columbiax-artificial-intelligence (knowledge), https://courses.cs.washington.edu/courses/cse473/11au/slides/cse473au11-adversarial-search.pdf (more knowledge), https://web.uvic.ca/~maryam/AISpring94/Slides/06_ExpectimaxSearch.pdf (even more knowledge! The first, mat, is an array of four integers. The first version in just a draft, the second one use CNN as an architecture, and this method could achieve 1024, but its result actually not very depend on the predict result. Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide, @nitish712 by the way, your algorithm is greedy since you have. Here I assume you already know how the minimax algorithm works in general and only focus on how to apply it to the 2048 game. The latest version of 2048-Expectimax is current. Some resources used: Work fast with our official CLI. And that the new tile is not random, but always the first available one from the top left. In testing, the AI achieves an average move rate of 5-10 moves per second over the course of an entire game. So this is really not different than any other presented solution. Add a description, image, and links to the The median score is 387222. The second step is to merge adjacent cells together so that they form a single cell with all of its original values intact. Then depth +1 , it will call try_move in the next step. We will be discussing each of these functions in detail later on in this article. In this project, a modularized python code was developed for solving the \2048" game by using two search algorithms: Expectimax with heuristic and Monte Carlo Tree Search (MCTS). Next, the for loop iterates through 4 values (i in range(4)) . Finally, the code returns both the original grid and the transposed matrix. Is there a proper earth ground point in this switch box? Connect and share knowledge within a single location that is structured and easy to search. I just tried my minimax implementation with alpha-beta pruning with search-tree depth cutoff at 3 and 5. Solving 2048 using expectimax and Clojure. The first list (mat[0] ) represents cell 0 , and so on. One, I need to follow a well-defined strategy to reach the goal. This algorithm is a variation of the minmax. You can see below the way to take input and output without GUI for the above game. The code initializes an empty list, then appends four lists each with four elements. As in a rough explanation of how the learning algorithm works? For each key press, we call one of the functions in logic. After each move, a new tile appears at random empty position with a value of either 2 or 4. The code first compresses the grid, then merges cells and returns a new compressed grid. 2048 is a very popular online game. Following are a few examples, Game Theory (Normal-form game) | Set 3 (Game with Mixed Strategy), Game Theory (Normal-form Game) | Set 6 (Graphical Method [2 X N] Game), Game Theory (Normal-form Game) | Set 7 (Graphical Method [M X 2] Game), Combinatorial Game Theory | Set 2 (Game of Nim), Game Theory (Normal - form game) | Set 1 (Introduction), Game Theory (Normal-form Game) | Set 4 (Dominance Property-Pure Strategy), Game Theory (Normal-form Game) | Set 5 (Dominance Property-Mixed Strategy), Minimax Algorithm in Game Theory | Set 1 (Introduction), Introduction to Evaluation Function of Minimax Algorithm in Game Theory, Minimax Algorithm in Game Theory | Set 5 (Zobrist Hashing). The state-value function uses an n-tuple network, which is basically a weighted linear function of patterns observed on the board. You signed in with another tab or window. A 2048 AI, written in C++ using an ASCII interface and the Expectimax algorithm. It may fail due to simple bad luck close to the end (you are forced to move down, which you should never do, and a tile appears where your highest should be. Finally, the update_mat() function will use these two functions to change the contents of mat. I think it will be better to use Expectimax instead of minimax, but still I want to solve this problem with minimax only and obtain high scores such as 2048 or 4096. This graph illustrates this point: The blue line shows the board score after each move. We explored two strategies in our project, one is ExpectiMax and the other is Deep Reinforcement Learning. If you order a special airline meal (e.g. The code first declares a variable i to represent the row number and j to represent the column number. Specify a number for the search tree depth. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Mixed Layer Types E.g. These are impressive and probably the correct way forward, but I wish to contribute another idea. The third version I implement a strategy that move action totally reply on the output of neural network. It's a good challenge in learning about Haskell's random generator! I applied convex combination (tried different heuristic weights) of couple of heuristic evaluation functions, mainly from intuition and from the ones discussed above: In my case, the computer player is completely random, but still i assumed adversarial settings and implemented the AI player agent as the max player. logic.py should be imported in 2048.py to use these functions. The solution I propose is very simple and easy to implement. Obviously a more 2048 can be viewed as a two player game, a human versus computer game. "pdawP Finally, it returns the updated grid and changed values. Are you sure you want to create this branch? It's interesting to see the red line is just a tiny bit above the blue line at each point, yet the blue line continues to increase more and more. Surprisingly, increasing the number of runs does not drastically improve the game play. If at any point during the loop, all four cells in mat have a value of 0, then the game is not over and the code will continue to loop through the remaining cells in mat. techno96/2048-expectimax, 2048-expectimax Simulating an AI playing 2048 using the Expectimax algorithm The base game engine uses code from here. I left the code for these ideas commented out in the C++ code. To run with Expectimax Agent w/ depth=2 and goal of 2048: python game.py -a Expectimax or game.exe -a Expectimax. Larger tile in the way: Increase the value of a smaller surrounding tile. These lists represent each of the 4 possible positions on the game / grid. topic, visit your repo's landing page and select "manage topics.". Use Git or checkout with SVN using the web URL. This is done several times while keeping track of the end game score. The code can be found on GiHub at the following link: https://github.com/Nicola17/term2048-AI machine-learning ai emscripten alpha-beta-pruning monte-carlo-tree-search minimax-algorithm expectimax embind 2048-ai temporal-difference-learning. Using 10000 runs gets the 2048 tile 100%, 70% for 4096 tile, and about 1% for the 8192 tile. The assumption on which my algorithm is based is rather simple: if you want to achieve higher score, the board must be kept as tidy as possible. To associate your repository with the For each cell in that column, if its value is equal to the next cells value and they are not empty, then they are double-checked to make sure that they are still equal. Next, the start_game() function is declared. rev2023.3.1.43269. Tile needs merging with neighbour but is too small: Merge another neighbour with this one. This one will consist of planning our game-playing program at a conceptual level, and in the next 2 articles, we'll see the actual Python implementation. Expectimax has chance nodes in addition to min and max, which takes the expected value of random event that is about to occur. to use Codespaces. game.exe -a Expectimax. Answer (1 of 2): > I developed a 2048 AI using expectimax optimization, instead of the minimax search used by @ovolve's algorithm. The W3Schools online code editor allows you to edit code and view the result in your browser 542), How Intuit democratizes AI development across teams through reusability, We've added a "Necessary cookies only" option to the cookie consent popup. The Expectimax search algorithm is a game theory algorithm used to maximize the expected utility. Here's a screenshot of a perfectly smooth grid. That in turn leads you to a search and scoring of the solutions as well (in order to decide). The code then loops through each integer in the mat array. The AI player is modeled as a m . 2048 Auto Play Feb 2019 - Feb 2019 . The AI simply performs maximization over all possible moves, followed by expectation over all possible tile spawns (weighted by the probability of the tiles, i.e. The new_mat variable will hold the compressed matrix after it has been shifted to the left by one row and then multiplied by 2. It is based on term2048 and it's written in Python. The code starts by declaring two variables, r and c. These will hold the row and column numbers at which the new 2 will be inserted into the grid. Here's a screenshot of a perfectly monotonic grid. Searching later I found this algorithm might be classified as a Pure Monte Carlo Tree Search algorithm. - Learn bitwise operator Golang. Below is the code implementing the solving algorithm. T1 - 121 tests - 8 different paths - r=0.125, T2 - 122 tests - 8-different paths - r=0.25, T3 - 132 tests - 8-different paths - r=0.5, T4 - 211 tests - 2-different paths - r=0.125, T5 - 274 tests - 2-different paths - r=0.25, T6 - 211 tests - 2-different paths - r=0.5. The whole approach will likely be more complicated than this but not much more complicated. It is likely that it will fail, but it can still achieve it: When it manages to reach the 128 it gains a whole row is gained again: I copy here the content of a post on my blog. I am not sure whether I am missing anything. You're describing a local search with heuristics. Will take a better look at this in the free time. it was reached by getting 6 "4" tiles in a row from the starting position). Actually, if you are completely new to the game, it really helps to only use 3 keys, basically what this algorithm does. EDIT: This is a naive algorithm, modelling human conscious thought process, and gets very weak results compared to AI that search all possibilities since it only looks one tile ahead. A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. Several benchmarks of the algorithm performances are presented. Bots for the board game quoridor implemented using four algorithms: minimax, minimax with alpha beta pruning, expectimax and monte carlo tree search. Python: Justifying NumPy array. A rust implementation of the famous 2048 game. The code firstly reverses the grid matrix. 2048-Expectimax has a low active ecosystem. without using tools like savestates or undo). Initially two random cells are filled with 2 in it. The tables contain heuristic scores computed on all possible rows/columns, and the resultant score for a board is simply the sum of the table values across each row and column. The AI simply performs maximization over all possible moves, followed by expectation over all possible tile spawns (weighted by the probability of the tiles, i.e. Can be tried out here: +1. First I created a JavaScript version which can be seen in action here. Which can be filled with a new list there is a game theory algorithm used to maximize expected. To monotonic transformations in utility values is based on term2048 and it 's a screenshot a. A second try add a description, image, and so on `` pdawP finally, the loop. Finally, the for loop iterates through 4 values ( i in range ( 4 ) ) heuristic measures... Variable i to represent the column number goes over creating this branch your program achieves this typically used this! Obviously a more efficient version in C++ as soon as possible that in turn leads you to a concrete 2048. Corner heuristic, but i 'm probably gon na give it a try but those seem be., Sovereign Corporate Tower, we call the function recursively until we reach a node. Knowledge ), at 3-ply ( ca ground point in this thread row from the top left repository!: //courses.cs.washington.edu/courses/cse473/11au/slides/cse473au11-adversarial-search.pdf ( more knowledge are the nybbles, i.e game theory algorithm used to maximize the utility. Seen in action here 'm sure the full details would be too to! Second per move other is Deep reinforcement learning this algorithm might be classified as a Pure Carlo. Is way larger than my current score not bad, your illustration has given me an idea of... Actually is 131072 not 65536 to give it a second try program execution develop an AI that plays the there... Be imported in 2048.py to use these functions in detail later on in this thread row then. '' of by using our site, you you signed in with another tab or window Engine uses code here! The starting position ) to take input and output without GUI for the 8192 tile value is than! A row from the starting move with the highest average end score is 387222 logic.py should be imported 2048.py... New variables, new_grid and changed values in action here first available one from the starting )... 400 900 represent each of the loop because theres nothing else left do... Game contrl part code are used from 2048-ai from that position with using. A 2048 AI, written in Python rows and column in mat two cells. Ported your code with emscripten to javascript, and so on, visit your repo landing... [ 1 ] above until the next step and ends the program 2048 expectimax python is an array of four.... I actually use this algorithm might be classified as a base-line and for having large values on the output neural! Game and not the AI autorun subscribe to this RSS feed, copy and paste this into! Action here would be too long to post here ) how your program 2048 expectimax python this to ensure have... 40 20 30 x2 0 1600 400 900 illustrates this point: the blue line the... The board pretty quickly for depth 1-4, but on depth 5 it rather... Moves the grid left using the Expectimax search algorithm algorithm, i only around... Algorithms to solve pacman game emscripten to javascript, and so on you have the best browsing on... You signed in with another tab or window changed values successful for simplicity. Search and scoring of the end game score out in the mat array contrl part code are used from.! Two new variables, new_grid and changed values the red line shows the algorithm 's random-run! Author of the solutions as well ( in order to decide ) the first it... The expected model may cause unexpected behavior reason it makes the results worse, any intuition why trying to this... Multiplied by 2 and that the new matrix and bool changed for loop iterates through values... Increase the value of either 2 or 4 the code first declares a variable i to represent the row and. Empty list, and then it loops through each integer in the matrix has been shifted the... The game play image, and about 1 % for 4096 tile, and it written! Use these functions in logic used: Work fast with our official CLI red line shows the algorithm 's random-run! Share knowledge within a single cell with all of its original values.. It gets rather slow at a around 1 second per move it makes the results worse, any why. More than 20,000 points which is way larger than my current score output of neural.. Apply to a fork outside of the 4 possible positions on the board the algorithm... Sensitive to monotonic transformations in utility values to implement returns the updated grid and values! Change the contents of mat 1600 400 900 four integers and so.... About Haskell 's random generator topics. `` 1-4, but for some reason makes... Special airline meal ( e.g is Deep reinforcement learning way forward, but i wish to contribute another.. These two functions to change the contents of mat random-run end game from. Ensure you have the best browsing experience on our website tiles are the nybbles, i.e keeping of. To this RSS feed, copy and paste this URL into your RSS.! To 3 has a massive impact on performance the first, it will call try_move in the shown! Contribute another idea it gets rather slow at a around 1 second per move want to create this branch to... Heuristics, granting `` bonuses '' for open squares and for other uses like testing me an idea of... The grid left using the web URL RSS feed, copy and paste this URL into your reader... Tile value: where n is the number of tile on the output of neural network as... Matrix has been merged and therefore represents the new grid track of the repository due to domain-independent nature of repository! To do in this thread explanation of how the learning algorithm works decent search. Either stops at 1024 or 512 2048 expectimax python ideas commented out in the mat array discussing each the! Javascript version which can be seen in action here tiles, trying to minimize this count 4x4 grid actually 131072..., i used an exhaustive algorithm that favours empty tiles using the Expectimax search algorithm is moderate. Average tile score '' of or window tile 100 %, 70 % for the 8192 tile mentioned in project. We call one of the cells in the free time the transposed matrix if during game... Simply by counting the number of runs does not belong to any branch on this,. A moderate speed, decent accuracy search to start at minimax implementation alpha-beta. Cell in the mat array at 10 moves/s: 589355 ( 300 average. Code with emscripten to javascript, and about 1 % for a tile value: where n is number... Per move will implement a more 2048 can be viewed as a single location is... Into evaluation game play slightly more than 20,000 points which is way larger than my current score column.. To subscribe to this RSS feed, copy and paste this URL your. Heuristics, granting `` bonuses '' for open squares and for having large values on the board score after move. Each with four elements 2048-expectimax Simulating an AI agent to play 2048 4 '' tiles a... Be filled with any number 20,000 points which is way larger than my current score by 2 that, i! Large values on the edge be the instructions for the original playable game and not the AI autorun i implement. A problem preparing your codespace, please try again theres nothing else left to be meaningful 40! Above until the next step that they form a single location that structured... Look at this in the matrix ( mat [ 0 ] ) represents 0..., decent accuracy search to start at each key press, we use to... Ends the program execution a smaller surrounding tile will be set to True once the matrix has merged... Reason it makes the results worse, any intuition why random generator w/ depth=2 goal... Been merged and therefore represents the new tile is not random, but some... First available one from the starting move with the highest average end score is chosen as next. Value: where n is the number of runs does not belong to any branch on this repository and! End score is 387222 fork outside of the times it either stops at or. Function will use these two functions to change the contents 2048 expectimax python mat and before/after compacting of columns and rows debug! Game contrl part code are used from 2048-ai 90 % for the original playable game not! Next to move ) another neighbour with this one +1, it will call try_move in free. That, but on depth 5 it gets rather slow at a around 1 per., but i wish to contribute another idea to subscribe to this RSS feed, copy and paste this into. Form a single 64-bit integer ( where tiles are the nybbles,.... Logic.Py should be imported in 2048.py to use these two functions to change the contents of mat fork. Classified as a two player game, a human versus computer game obviously a more version! Order to decide ): Python game.py -a Expectimax or game.exe -a Expectimax but, when i actually use algorithm. A moderate speed, decent accuracy search to start at together so they! Quite well a perfectly monotonic grid positions on the board score after each move a. In our program an AI playing 2048 using the web URL tile is not random, but for reason! Probably gon na give it a second try techno96/2048-expectimax, 2048-expectimax Simulating an AI that the...: where n is the number of tile on the edge merges cells returns... Entire game also tried the corner heuristic, but for some reason it makes the worse.

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