- Tic Tac Toe Google Game
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The AI's smarts for playing Tic Tac Toe will follow a simple algorithm. An algorithm is a finite series of instructions to compute a result. A single program can make use of several different algorithms. An algorithm can be represented with a flow chart. The Tic Tac Toe AI's algorithm will compute the best move to make, as shown in Figure 10-4.
Note! This article is has also been translated to Japanese and Portuguese. I really appreciate the readers that reached out to me and translated this article.
- Tic Tac Toe Pro offers a host of exciting features, including:. Turn-based network play over Wi-Fi or 3G. Great graphics and exciting sound effects. Configurable player names and score tracking. Undo function. Automatic save when you get a phone call or exit the application Tic Tac Toe on the iPhone and iPod Touch has never been so exciting.
- Try to place at first 3 Xs / 3 Os in a horizontal, vertical or diagonal row. Click on the player to change the name. Click on the computer to change the game strength.
I recently built an unbeatable game of tic tac toe. It was a fun and very humbling project that taught me a ton. If you want to get totally schooled, give the tic tac toe game a shot here.
In order to make the game unbeatable, it was necessary to create an algorithm that could calculate all the possible moves available for the computer player and use some metric to determine the best possible move. After extensive research it became clear that the Minimax algorithm was right for the job.
Tic Tac Toe Google Game
It took a little while to really fundamentally understand the algorithm and implement it in my game. I found many code examples and explanations, but none that really walked a simpleton like me through the ins and outs of the process. I hope this post will help some of you to appreciate the elegance of this algorithm.
Describing a Perfect Game of Tic Tac Toe
Radagast mac os. To begin, let's start by defining what it means to play a perfect game of tic tac toe:
If I play perfectly, every time I play I will either win the game, or I will draw the game. Furthermore if I play against another perfect player, I will always draw the game.
How might we describe these situations quantitatively? Let's assign a score to the 'end game conditions:'
- I win, hurray! I get 10 points!
- I lose, shit. I lose 10 points (because the other player gets 10 points)
- I draw, whatever. I get zero points, nobody gets any points.
So now we have a situation where we can determine a possible score for any game end state.
Looking at a Brief Example
To apply this, let's take an example from near the end of a game, where it is my turn. I am X. My goal here, obviously, is to maximize my end game score. https://minddiamondgamefree-betrackxfjh.peatix.com.
If the top of this image represents the state of the game I see when it is my turn, then I have some choices to make, there are three places I can play, one of which clearly results in me wining and earning the 10 points. If I don't make that move, O could very easily win. And I don't want O to win, so my goal here, as the first player, should be to pick the maximum scoring move.
But What About O?
What do we know about O? Well we should assume that O is also playing to win this game, but relative to us, the first player, O wants obviously wants to chose the move that results in the worst score for us, it wants to pick a move that would minimize our ultimate score. Let's look at things from O's perspective, starting with the two other game states from above in which we don't immediately win:
The choice is clear, O would pick any of the moves that result in a score of -10.
Describing Minimax
- Tic Tac Toe Pro offers a host of exciting features, including:. Turn-based network play over Wi-Fi or 3G. Great graphics and exciting sound effects. Configurable player names and score tracking. Undo function. Automatic save when you get a phone call or exit the application Tic Tac Toe on the iPhone and iPod Touch has never been so exciting.
- Try to place at first 3 Xs / 3 Os in a horizontal, vertical or diagonal row. Click on the player to change the name. Click on the computer to change the game strength.
I recently built an unbeatable game of tic tac toe. It was a fun and very humbling project that taught me a ton. If you want to get totally schooled, give the tic tac toe game a shot here.
In order to make the game unbeatable, it was necessary to create an algorithm that could calculate all the possible moves available for the computer player and use some metric to determine the best possible move. After extensive research it became clear that the Minimax algorithm was right for the job.
Tic Tac Toe Google Game
It took a little while to really fundamentally understand the algorithm and implement it in my game. I found many code examples and explanations, but none that really walked a simpleton like me through the ins and outs of the process. I hope this post will help some of you to appreciate the elegance of this algorithm.
Describing a Perfect Game of Tic Tac Toe
Radagast mac os. To begin, let's start by defining what it means to play a perfect game of tic tac toe:
If I play perfectly, every time I play I will either win the game, or I will draw the game. Furthermore if I play against another perfect player, I will always draw the game.
How might we describe these situations quantitatively? Let's assign a score to the 'end game conditions:'
- I win, hurray! I get 10 points!
- I lose, shit. I lose 10 points (because the other player gets 10 points)
- I draw, whatever. I get zero points, nobody gets any points.
So now we have a situation where we can determine a possible score for any game end state.
Looking at a Brief Example
To apply this, let's take an example from near the end of a game, where it is my turn. I am X. My goal here, obviously, is to maximize my end game score. https://minddiamondgamefree-betrackxfjh.peatix.com.
If the top of this image represents the state of the game I see when it is my turn, then I have some choices to make, there are three places I can play, one of which clearly results in me wining and earning the 10 points. If I don't make that move, O could very easily win. And I don't want O to win, so my goal here, as the first player, should be to pick the maximum scoring move.
But What About O?
What do we know about O? Well we should assume that O is also playing to win this game, but relative to us, the first player, O wants obviously wants to chose the move that results in the worst score for us, it wants to pick a move that would minimize our ultimate score. Let's look at things from O's perspective, starting with the two other game states from above in which we don't immediately win:
The choice is clear, O would pick any of the moves that result in a score of -10.
Describing Minimax
The key to the Minimax algorithm is a back and forth between the two players, where the player whose 'turn it is' desires to pick the move with the maximum score. In turn, the scores for each of the available moves are determined by the opposing player deciding which of its available moves has the minimum score. And the scores for the opposing players moves are again determined by the turn-taking player trying to maximize its score and so on all the way down the move tree to an end state.
A description for the algorithm, assuming X is the 'turn taking player,' would look something like:
- If the game is over, return the score from X's perspective.
- Otherwise get a list of new game states for every possible move
- Create a scores list
- For each of these states add the minimax result of that state to the scores list
- If it's X's turn, return the maximum score from the scores list
- If it's O's turn, return the minimum score from the scores list
You'll notice that this algorithm is recursive, it flips back and forth between the players until a final score is found.
Let's walk through the algorithm's execution with the full move tree, and show why, algorithmically, the instant winning move will be picked:
- It's X's turn in state 1. X generates the states 2, 3, and 4 and calls minimax on those states.
- State 2 pushes the score of +10 to state 1's score list, because the game is in an end state.
- State 3 and 4 are not in end states, so 3 generates states 5 and 6 and calls minimax on them, while state 4 generates states 7 and 8 and calls minimax on them.
- State 5 pushes a score of -10 onto state 3's score list, while the same happens for state 7 which pushes a score of -10 onto state 4's score list.
- State 6 and 8 generate the only available moves, which are end states, and so both of them add the score of +10 to the move lists of states 3 and 4.
- Because it is O's turn in both state 3 and 4, O will seek to find the minimum score, and given the choice between -10 and +10, both states 3 and 4 will yield -10.
- Finally the score list for states 2, 3, and 4 are populated with +10, -10 and -10 respectively, and state 1 seeking to maximize the score will chose the winning move with score +10, state 2.
That is certainly a lot to take in. And that is why we have a computer execute this algorithm.
##A Coded Version of Minimax Hopefully by now you have a rough sense of how th e minimax algorithm determines the best move to play. Let's examine my implementation of the algorithm to solidify the understanding:
Here is the function for scoring the game:
Simple enough, return +10 if the current player wins the game, -10 if the other player wins and 0 for a draw. You will note that who the player is doesn't matter. X or O is irrelevant, only who's turn it happens to be.
And now the actual minimax algorithm; note that in this implementation a choice
or move
is simply a row / column address on the board, for example [0,2] is the top right square on a 3x3 board.
When this algorithm is run inside a PerfectPlayer
class, the ultimate choice of best move is stored in the @choice
variable, which is then used to return the new game state in which the current player has moved.
##A Perfect but Fatalist Player Implementing the above algorithm will get you to a point where your tic tac toe game can't be beat. But and interesting nuance that I discovered while testing is that a perfect player must always be perfect. In other words, in a situation where the perfect player eventually will lose or draw, the decisions on the next move are rather fatalistic. The algorithm essentially says: 'hey I'm gonna lose anyway, so it really doesn't matter if I lose in the next more or 6 moves from now.'
I discovered this by passing an obviously rigged board, or one with a 'mistake' in it to the algorithm and asked for the next best move. I would have expected the perfect player to at least put up a fight and block my immediate win. It however, did not:
Let's see what is happening here by looking through the possible move tree (Note, I've removed some of the possible states for clarity):
- Given the board state 1 where both players are playing perfectly, and O is the computer player. O choses the move in state 5 and then immediately loses when X wins in state 9.
- But if O blocks X's win as in state 3, X will obviously block O's potential win as shown in state 7.
- This puts two certain wins for X as shown in state 10 and 11, so no matter which move O picks in state 7, X will ultimately win.
As a result of these scenarios, and the fact that we are iterating through each blank space, from left to right, top to bottom, all moves being equal, that is, resulting in a lose for O, the last move will be chosen as shown in state 5, as it is the last of the available moves in state 1. The array of moves being: [top-left, top-right, middle-left, middle-center].
What is a gosh-darn, tic tac toe master to do?
Fighting the Good Fight: Depth
The key improvement to this algorithm, such that, no matter the board arrangement, the perfect player will play perfectly unto its demise, is to take the 'depth' or number of turns till the end of the game into account. Basically the perfect player should play perfectly, but prolong the game as much as possible.
In order to achieve this we will subtract the depth, that is the number of turns, or recursions, from the end game score, the more turns the lower the score, the fewer turns the higher the score. Updating our code from above we have something that looks like this:
So each time we invoke minimax, depth is incremented by 1 and when the end game state is ultimately calculated, the score is adjusted by depth. Let's see how this looks in our move tree:
This time the depth (Shown in black on the left) causes the score to differ for each end state, and because the level 0 part of minimax will try to maximize the available scores (because O is the turn taking player), the -6 score will be chosen as it is greater than the other states with a score of -8. And so even faced with certain death, our trusty, perfect player now will chose the blocking move, rather than commit honor death.
In Conclusion
I hope all of this discussion has helped you further understand the minimax algorithm, and perhaps how to dominate at a game of tic tac toe. If you have further questions or anything is confusing, leave some comments and I'll try to improve the article. All of the source code for my tic tac toe game can be found on github.
Play a simple version of the game right here!
I recommend playing this in a PC browser window
It may take a while to load, please be patient.
You can also play the game on itch.io here!
Tic Tac Toe Google
The Rules
No Quantumness — A.k.a Classical
At its core, Quantum TicTacToe is very similar to regular, classical, TicTacToe. In classical TicTacToe, two players X and O alternate turns and pick squares on which to put their symbol. Three X's or three O's in a row (or column or diagonal) results in a win for that player. If you've never played TicTacToe before, try it out in the game above and keep the quantumness-slider all the way to the left (No Quantum).
Quantum TicTacToe
If you're ready to move beyond the classical world and into the quantum realm, it's time to increase the quantumness! We'll explain all of the quantum moves below. Some of these moves are disabled depending on the quantumness level. For those who just want to start playing, here's how they can be performed:
- Single move: press a single square. If a square turns red, it can no longer be used for quantum moves. Single squares are colored blue if the can.
- Superposition: press one empty square, then drag and hold and release on another empty square.
- Entanglement: press on a square, drag and hold to another and then release. This is very similar to superpositions, but for this move the squares need not be empty!
- Measurement: click the measure button, and then click an occupied square to measure it. Resulting single squares turn red, and are fixed for the rest of the round. Only these squares count towards a win!
Minimal Quantumness
As a first step, let's try out the Minimal Quantum setting. This setting allows you do perform a quantum move that gives you possibilities that regular TicTacToe will never have. Namely, you can click&hold on an empty square and release on another empty square to create what quantum physicists call a superposition. Try it out in the game for yourself!
When two squares are in a superposition, it means that your X (or O!) is in both places at once, until it is forced to pick one over the other. This is very similar to a coin with heads or tails that is forever spinning on its side. Whilst spinning, we can equally well say it is both heads and tails simultaneously! Only when it stops spinning, because of friction or because we push it over, does the coin pick a side. The same is true with quantum superpositions. In the game, we force all the superpositions to pick a side when the board is full.
You can also force a superposition to decide by using the collapse button. You don't get to place an X or an O in that turn, but you get to pick a superposition instead. Once you select it, it will randomly choose one of the two squares.
Tic Tac Toe Template
Hint: At this level of quantumness, see if you can use superpositions to block your opponent from making a winning move!
Moderate Quantumness
At a moderate level of quantumness, three new quantum features become available on top of the Minimal Quantum settings.
The first thing you'll notice, is that when you place a single square it no longer turns red. Instead, the square stays blue, indicating that it has not yet been measured. To win the game, you need to have three measured squares in a row/column/diagonal. So having three blue X's in a row/column/diagonal does not mean you win!
To turn blue squares into red squares, you either need to measure them with the collapse button or wait for the board to get full (then, all squares are measured automatically).
But beware. As long as a square is blue, your opponent can entangle it. Garagequest mac os. This is the second quantum feature. Creating entanglement happens just as with a superposition, you drag&hold an empty square and release it on top of a blue square. Try it out! You'll see the two squares turn into a Q-symbol, which is a mix between the X and the O. This means that if one of the two squares is the X, the other is always the O. Which one is which can be found out only through measuring, or waiting until the board is full.
There is one more quantum feature in this mode: you can place another superposition on top of your own! This is where things become really strange. If you have two superpositions on top of each other, you'll notice that you only get to see half symbols. If you measure these squares, there is a chance they both become empty! The other option, is that both become your symbol. Which of these options happens, is determined with a 50/50 chance!
Hint 1: Use entanglement to get a shot at swapping your opponent's square with your own.
Tic Tac Toe (itch) (intsmite) Mac Osteoarthritis
Hint 2: Try to use two superpositions of your own squares as a chance for erasing a mistake.
High Quantumness
At this level of quantumness, even more features become available on top of the Moderate Quantumness level. If you can play this level of the game confidently, you're really starting to grasp the way the quantum realm works.
In this mode, you can entangle your square with part of an opponent's superposition. The resulting 3-square entangled state is quite complicated, and can result in four different outcomes when measured! See if you can find out which!
This mode is called mostly quantum, and not fully quantum, because the full quantum world would allow for more than 3-square entangled states. Visualizing this for a game is a very difficult task! (But that doesn't mean I won't try for future versions! So keep an eye out for that.)