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The evaluation function tries to keep the rows and columns monotonic either all decreasing or increasing while minimizing the number of tiles on the grid. There is also a discussion on Hacker News about this algorithm that you may find useful. More spaces makes the state more flexible, we multiply by which is the median since a grid filled with faces is an optimal impossible state. Here we evaluate faces that have the possibility to getting to merge, by evaluating them backwardly, tile 2 become of value , while tile is evaluated 2.

This is a simplified check of the possibility of having merges within that state, without making a look-ahead. This is not a direct answer to OP's question, this is more of the stuffs experiments I tried so far to solve the same problem and obtained some results and have some observations that I want to share, I am curious if we can have some further insights from this. I just tried my minimax implementation with alpha-beta pruning with search-tree depth cutoff at 3 and 5. 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. If I assign too much weights to the first heuristic function or the second heuristic function, both the cases the scores the AI player gets are low. I played with many possible weight assignments to the heuristic functions and take a convex combination, but very rarely the AI player is able to score Most of the times it either stops at or I also tried the corner heuristic, but for some reason it makes the results worse, any intuition why?

Also, I tried to increase the search depth cut-off from 3 to 5 I can't increase it more since searching that space exceeds allowed time even with pruning and added one more heuristic that looks at the values of adjacent tiles and gives more points if they are merge-able, but still I am not able to get 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 or I am not sure whether I am missing anything.

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. The following animation shows the last few steps of the game played where the AI player agent could get scores, this time adding the absolute value heuristic too:. The following figures show the game tree explored by the player AI agent assuming the computer as adversary for just a single step:.

And that the new tile is not random, but always the first available one from the top left. This variant is also known as Det I used an exhaustive algorithm that favours empty tiles. It performs pretty quickly for depth , but on depth 5 it gets rather slow at a around 1 second per move. Below is the code implementing the solving algorithm. The grid is represented as a length array of Integers. And scoring is done simply by counting the number of empty squares. I thinks it's quite successful for its simplicity.

The result it reaches when starting with an empty grid and solving at depth 5 is:. This algorithm is not optimal for winning the game, but it is fairly optimal in terms of performance and amount of code needed:. Many of the other answers use AI with computationally expensive searching of possible futures, heuristics, learning and the such. These are impressive and probably the correct way forward, but I wish to contribute another idea. Read the squares in the order shown above until the next squares value is greater than the current one.

This presents the problem of trying to merge another tile of the same value into this square. To resolve this problem, their are 2 ways to move that aren't left or worse up and examining both possibilities may immediately reveal more problems, this forms a list of dependancies, each problem requiring another problem to be solved first.

I think I have this chain or in some cases tree of dependancies internally when deciding my next move, particularly when stuck. The whole approach will likely be more complicated than this but not much more complicated. It could be this mechanical in feel lacking scores, weights, neurones and deep searches of possibilities.

The tree of possibilities rairly even needs to be big enough to need any branching at all. Learn more. What is the optimal algorithm for the game ? Ask Question. Asked 6 years, 11 months ago. Active 2 years ago. Viewed k times. My current algorithm: while! Improve this question. This might help! K Mar 12 '14 at It's a worst-case assumption, but might be useful.

A fun distraction when you don't have time to aim for a high score: Try to get the lowest score possible. In theory it's alternating 2s and 4s. Discussion on this question's legitimacy can be found on meta: meta. Show 4 more comments. Active Oldest Votes. Performance The AI in its default configuration max search depth of 8 takes anywhere from 10ms to ms to execute a move, depending on the complexity of the board position.

Here's the screenshot of the best run: This game took moves over 96 minutes, or an average of 4. Implementation My approach encodes the entire board 16 entries as a single bit integer where tiles are the nybbles, i. Heuristics Several heuristics are used to direct the optimization algorithm towards favorable positions. Improve this answer. David Greydanus 2, 1 1 gold badge 20 20 silver badges 40 40 bronze badges.

Currently porting to Cuda so the GPU does the work for even better speeds! Cool to watch, without the need to compile and everything In Firefox, performance is quite good Theoretical limit in a 4x4 grid actually IS not However that requires getting a 4 in the right moment i. Show 30 more comments. Smoothness The above heuristic alone tends to create structures in which adjacent tiles are decreasing in value, but of course in order to merge, adjacent tiles need to be the same value.

Free Tiles And finally, there is a penalty for having too few free tiles, since options can quickly run out when the game board gets too cramped. Edit: Here's a demonstration of the power of this approach. You can treat the computer placing the '2' and '4' tiles as the 'opponent'. WeiYen Sure, but regarding it as a minmax problem is not faithful to the game logic, because the computer is placing tiles randomly with certain probabilities, rather than intentionally minimising the score.

Even though the AI is randomly placing the tiles, the goal is not to lose. Getting unlucky is the same thing as the opponent choosing the worst move for you. The "min" part means that you try to play conservatively so that there are no awful moves that you could get unlucky. I had an idea to create a fork of , where the computer instead of placing the 2s and 4s randomly uses your AI to determine where to put the values.

The result: sheer impossibleness. Can be tried out here: sztupy. SztupY Wow, this is evil. Reminds me of qntm. Show 24 more comments. AI Algorithm I found a simple yet surprisingly good playing algorithm: To determine the next move for a given board, the AI plays the game in memory using random moves until the game is over. See it in action The best achieved score is shown here: An interesting fact about this algorithm is that while the random-play games are unsurprisingly quite bad, choosing the best or least bad move leads to very good game play: A typical AI game can reach points and last moves, yet the in-memory random play games from any given position yield an average of additional points in about 40 extra moves before dying.

Improvements After implementing this algorithm I tried many improvements including using the min or max scores, or a combination of min,max,and avg. Ronenz Ronenz 1, 2 2 gold badges 15 15 silver badges 7 7 bronze badges. As an AI student I found this really interesting. Will take a better look at this in the free time. This is amazing! I just spent hours optimizing weights for a good heuristic function for expectimax and I implement this in 3 minutes and this completely smashes it.

Nice use of Monte Carlo simulation. Watching this playing is calling for an enlightenment. This blows all heuristics and yet it works. By far, the most interesting solution here. This is your objective: This is the model I chose by default. Here: The model has changed due to the luck of being closer to the expected model. So it will press right, then right again, then right or top depending on where the 4 has created then will proceed to complete the chain until it gets: So now the model and chain are back to: 64 4 8 16 32 X X x x x x x x Second pointer, it has had bad luck and its main spot has been taken.

It is likely that it will fail, but it can still achieve it: Here the model and chain is: O O O O 8 16 32 64 4 x x x When it manages to reach the it gains a whole row is gained again: O x x x x x x x x x x. Daren Daren 3, 3 3 gold badges 19 19 silver badges 33 33 bronze badges. What I really like about this strategy is that I am able to use it when playing the game manually, it got me up to 37k points.

Show 5 more comments. I copy here the content of a post on my blog The solution I propose is very simple and easy to implement. Algorithm Heuristic scoring algorithm 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.

Nicola Pezzotti Nicola Pezzotti 2, 13 13 silver badges 25 25 bronze badges. K Apr 8 '14 at Are you sure the instructions provided in the github page apply to your project? I want to give it a try but those seem to be the instructions for the original playable game and not the AI autorun. Could you update those? Add a comment. Not sure why this doesn't have more upvotes.

It's really effective for it's simplicity. How did you weight the empty spaces? Pretty impressive result. However could you possibly update the answer to explain roughly, in simple terms I'm sure the full details would be too long to post here how your program achieves this?

As in a rough explanation of how the learning algorithm works? Please see the code below: while! Vincent Lecrubier Vincent Lecrubier 4 4 silver badges 9 9 bronze badges. I ran , games testing this versus the trivial cyclic strategy "up, right, up, left, The cyclic strategy finished an "average tile score" of Do you have a guess why that might be? I'm thinking it does too many ups, even when left or right would merge a lot more.

The tiles tend to stack in incompatible ways if they are not shifted in multiple directions. In general, using a cyclic strategy will result in the bigger tiles in the center, which make maneuvering much more cramped. MultiplyByZer0 4, 3 3 gold badges 27 27 silver badges 46 46 bronze badges. This should be the top answer, but it would be nice to add more details about the implementation: e. For future readers: This is the same program explained by its author ovolve in the second-topmost answer here.

This answer, and other mentions of ovolve's program in this discussion, prompted ovolve to appear and write up how his algorithm worked; that answer now has a score of Algorithm while! Note: The constants can be tweaked.. K Khaled. K 5, 1 1 gold badge 28 28 silver badges 46 46 bronze badges. Why do you need a constant? If all you are doing is comparing scores, how does that affect the outcome of those comparisons?

K Jun 24 '15 at I applied convex combination tried different heuristic weights of couple of heuristic evaluation functions, mainly from intuition and from the ones discussed above: Monotonicity Free Space Available 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.

I have 4x4 grid for playing the game. Observation: If I assign too much weights to the first heuristic function or the second heuristic function, both the cases the scores the AI player gets are low. 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. Sandipan Dey Sandipan Dey I wrote a solver in Haskell, mainly because I'm learning this language right now.

As a consequence, this solver is deterministic. Try to extend it with the actual rules. It's a good challenge in learning about Haskell's random generator! I got very frustrated with Haskell trying to do that, but I'm probably gonna give it a second try! I did find that the game gets considerably easier without the randomization.

Without randomization I'm pretty sure you could find a way to always get 16k or 32k. Either do it explicitly, or with the Random monad. You are right, it's harder than I thought. Show 1 more comments. Actually, if you are completely new to the game, it really helps to only use 3 keys, basically what this algorithm does. So not as bad as it seems at first sight.

Yes, it is based on my own observation with the game. Until you have to use the 4th direction the game will practically solve itself without any kind of observation. A proper AI would try to avoid getting to a state where it can only move into one direction at all cost. Using only 3 directions actually is a very decent strategy! It just got me nearly to the playing the game manually. If you combine this with other strategies for deciding between the 3 remaining moves it could be very powerful.

Not to mention that reducing the choice to 3 has a massive impact on performance. Show 2 more comments. Model the sort of strategy that good players of the game use. For example: 13 14 15 16 12 11 10 9 5 6 7 8 4 3 2 1 Read the squares in the order shown above until the next squares value is greater than the current one. Tile needs merging with neighbour but is too small: Merge another neighbour with this one. An efficient implementation of the controller is available on github.

In a separate repo there is also the code used for training the controller's state evaluation function. The training method is described in the paper. The controller uses expectimax search with a state evaluation function learned from scratch without human expertise by a variant of temporal difference learning a reinforcement learning technique. The state-value function uses an n-tuple network , which is basically a weighted linear function of patterns observed on the board.

It involved more than 1 billion weights , in total. My attempt uses expectimax like other solutions above, but without bitboards. In my case, this depth takes too long to explore, I adjust the depth of expectimax search according to the number of free tiles left:.

The scores of the boards are computed with the weighted sum of the square of the number of free tiles and the dot product of the 2D grid with this:. I think I found an algorithm which works quite well, as I often reach scores over , my personal best being around My solution does not aim at keeping biggest numbers in a corner, but to keep it in the top row.

There is already an AI implementation for this game here. The algorithm is iterative deepening depth first alpha-beta search. The evaluation function tries to keep the rows and columns monotonic either all decreasing or increasing while minimizing the number of tiles on the grid.

There is also a discussion on Hacker News about this algorithm that you may find useful. More spaces makes the state more flexible, we multiply by which is the median since a grid filled with faces is an optimal impossible state. Here we evaluate faces that have the possibility to getting to merge, by evaluating them backwardly, tile 2 become of value , while tile is evaluated 2.

This is a simplified check of the possibility of having merges within that state, without making a look-ahead. This is not a direct answer to OP's question, this is more of the stuffs experiments I tried so far to solve the same problem and obtained some results and have some observations that I want to share, I am curious if we can have some further insights from this.

I just tried my minimax implementation with alpha-beta pruning with search-tree depth cutoff at 3 and 5. 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.

If I assign too much weights to the first heuristic function or the second heuristic function, both the cases the scores the AI player gets are low. I played with many possible weight assignments to the heuristic functions and take a convex combination, but very rarely the AI player is able to score Most of the times it either stops at or I also tried the corner heuristic, but for some reason it makes the results worse, any intuition why?

Also, I tried to increase the search depth cut-off from 3 to 5 I can't increase it more since searching that space exceeds allowed time even with pruning and added one more heuristic that looks at the values of adjacent tiles and gives more points if they are merge-able, but still I am not able to get 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 or I am not sure whether I am missing anything.

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. The following animation shows the last few steps of the game played where the AI player agent could get scores, this time adding the absolute value heuristic too:. The following figures show the game tree explored by the player AI agent assuming the computer as adversary for just a single step:. And that the new tile is not random, but always the first available one from the top left.

This variant is also known as Det I used an exhaustive algorithm that favours empty tiles. It performs pretty quickly for depth , but on depth 5 it gets rather slow at a around 1 second per move. Below is the code implementing the solving algorithm. The grid is represented as a length array of Integers. And scoring is done simply by counting the number of empty squares. I thinks it's quite successful for its simplicity. The result it reaches when starting with an empty grid and solving at depth 5 is:.

This algorithm is not optimal for winning the game, but it is fairly optimal in terms of performance and amount of code needed:. Many of the other answers use AI with computationally expensive searching of possible futures, heuristics, learning and the such. These are impressive and probably the correct way forward, but I wish to contribute another idea.

Read the squares in the order shown above until the next squares value is greater than the current one. This presents the problem of trying to merge another tile of the same value into this square. To resolve this problem, their are 2 ways to move that aren't left or worse up and examining both possibilities may immediately reveal more problems, this forms a list of dependancies, each problem requiring another problem to be solved first.

I think I have this chain or in some cases tree of dependancies internally when deciding my next move, particularly when stuck. The whole approach will likely be more complicated than this but not much more complicated. It could be this mechanical in feel lacking scores, weights, neurones and deep searches of possibilities. The tree of possibilities rairly even needs to be big enough to need any branching at all.

Learn more. What is the optimal algorithm for the game? Ask Question. Asked 6 years, 9 months ago. Active 1 year, 11 months ago. Viewed k times. My current algorithm: while! This might help! K Mar 12 '14 at It's a worst-case assumption, but might be useful. A fun distraction when you don't have time to aim for a high score: Try to get the lowest score possible.

In theory it's alternating 2s and 4s. Discussion on this question's legitimacy can be found on meta: meta. Active Oldest Votes. Performance The AI in its default configuration max search depth of 8 takes anywhere from 10ms to ms to execute a move, depending on the complexity of the board position. Here's the screenshot of the best run: This game took moves over 96 minutes, or an average of 4. Implementation My approach encodes the entire board 16 entries as a single bit integer where tiles are the nybbles, i.

Heuristics Several heuristics are used to direct the optimization algorithm towards favorable positions. David Greydanus 2, 1 1 gold badge 20 20 silver badges 40 40 bronze badges. Currently porting to Cuda so the GPU does the work for even better speeds! Cool to watch, without the need to compile and everything In Firefox, performance is quite good Theoretical limit in a 4x4 grid actually IS not However that requires getting a 4 in the right moment i.

Smoothness The above heuristic alone tends to create structures in which adjacent tiles are decreasing in value, but of course in order to merge, adjacent tiles need to be the same value. Free Tiles And finally, there is a penalty for having too few free tiles, since options can quickly run out when the game board gets too cramped. Edit: Here's a demonstration of the power of this approach.

You can treat the computer placing the '2' and '4' tiles as the 'opponent'. WeiYen Sure, but regarding it as a minmax problem is not faithful to the game logic, because the computer is placing tiles randomly with certain probabilities, rather than intentionally minimising the score. Even though the AI is randomly placing the tiles, the goal is not to lose.

Getting unlucky is the same thing as the opponent choosing the worst move for you. The "min" part means that you try to play conservatively so that there are no awful moves that you could get unlucky. I had an idea to create a fork of , where the computer instead of placing the 2s and 4s randomly uses your AI to determine where to put the values. The result: sheer impossibleness.

Can be tried out here: sztupy. SztupY Wow, this is evil. Reminds me of qntm. AI Algorithm I found a simple yet surprisingly good playing algorithm: To determine the next move for a given board, the AI plays the game in memory using random moves until the game is over.

See it in action The best achieved score is shown here: An interesting fact about this algorithm is that while the random-play games are unsurprisingly quite bad, choosing the best or least bad move leads to very good game play: A typical AI game can reach points and last moves, yet the in-memory random play games from any given position yield an average of additional points in about 40 extra moves before dying. Improvements After implementing this algorithm I tried many improvements including using the min or max scores, or a combination of min,max,and avg.

Ronenz Ronenz 1, 2 2 gold badges 15 15 silver badges 7 7 bronze badges. As an AI student I found this really interesting. Will take a better look at this in the free time. This is amazing! I just spent hours optimizing weights for a good heuristic function for expectimax and I implement this in 3 minutes and this completely smashes it.

Nice use of Monte Carlo simulation. Watching this playing is calling for an enlightenment. This blows all heuristics and yet it works. By far, the most interesting solution here. This is your objective: This is the model I chose by default. Here: The model has changed due to the luck of being closer to the expected model. So it will press right, then right again, then right or top depending on where the 4 has created then will proceed to complete the chain until it gets: So now the model and chain are back to: 64 4 8 16 32 X X x x x x x x Second pointer, it has had bad luck and its main spot has been taken.

It is likely that it will fail, but it can still achieve it: Here the model and chain is: O O O O 8 16 32 64 4 x x x When it manages to reach the it gains a whole row is gained again: O x x x x x x x x x x. Daren Daren 3, 3 3 gold badges 18 18 silver badges 33 33 bronze badges. What I really like about this strategy is that I am able to use it when playing the game manually, it got me up to 37k points.

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Yes, the. Um, No offence but no one is going to fall for this. No owner is gonna take a plugin for a starter and use it. It would be shady as hell. This is useless. If they download it from here, and install it on their server, then you can. I offer this plugin, and many more can be found at BukkitDev.

This is actually helping me… Im on my bukkit server and cant op myself , not even with ops. Just a quick tip: if you start craftbukkit. Hmm… the only way I know of is trying to exploit Remote Bukkit with the default username and password. Unfortunately, dev. Let me know if you get anything useful from decompiling. Save my name, email, and website in this browser for the next time I comment.

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