# Tic Tac Toe Part 3: The Minimax Algorithm

In two previous posts, I presented code to teach R to play the trivial game of Tic Tac Toe. I thought this was an unbeatable algorithm. Alas, a comment from Alberto shattered my pride as he was able to beat my code.

The reason for the demise of my code was that I didn’t implement a full minimax algorithm, but instead looked only two moves ahead. I thought the common strategy rules (start in the centre and occupy the corners) would make the program unbeatable. When I simulated the game by instructing the computer to play against itself, Alberto’s strategy never arose because the code forces the centre field. Alberto’s code shows that you need to look at least three moves ahead for a perfect game. He has been so kind to share his code and gave me permission to publish it.

Alberto recreated two functions, for completeness I have added the complete working code that merges his improvements with my earlier work. The first two functions are identical to the previous post. These functions draw the game board and process the human player’s move by waiting for a mouse click.

```# Draw the game board
draw.board <- function(game) {
xo <- c("X", " ", "O") # Symbols
par(mar = rep(1,4))
plot.new()
plot.window(xlim = c(0,30), ylim = c(0,30))
abline(h = c(10, 20), col="darkgrey", lwd = 4)
abline(v = c(10, 20), col="darkgrey", lwd = 4)
text(rep(c(5, 15, 25), 3), c(rep(25, 3), rep(15,3), rep(5, 3)), xo[game + 2], cex = 4)
# Identify location of any three in a row
square <- t(matrix(game, nrow = 3))
hor <- abs(rowSums(square))
if (any(hor == 3))
hor <- (4 - which(hor == 3)) * 10 - 5
else
hor <- 0
ver <- abs(colSums(square))
if (any(ver == 3))
ver <- which(ver == 3) * 10 - 5
else
ver <- 0
diag1 <- sum(diag(square))
diag2 <- sum(diag(t(apply(square, 2, rev))))
# Draw winning lines
if (all(hor > 0))
for (i in hor) lines(c(0, 30), rep(i, 2), lwd = 10, col="red")
if (all(ver > 0))
for (i in ver) lines(rep(i, 2), c(0, 30), lwd = 10, col="red")
if (abs(diag1) == 3)
lines(c(2, 28), c(28, 2), lwd = 10, col = "red")
if (abs(diag2) == 3)
lines(c(2, 28), c(2, 28), lwd = 10, col = "red")
}

# Human player enters a move
move.human <- function(game) {
text(4, 0, "Click on screen to move", col = "grey", cex=.7)
empty <- which(game == 0)
move <- 0
while (!move %in% empty) {
coords <- locator(n = 1) # add lines
coords\$x <- floor(abs(coords\$x) / 10) + 1
coords\$y <- floor(abs(coords\$y) / 10) + 1
move <- coords\$x + 3 * (3 - coords\$y)
}
return (move)
}
```

Alberto rewrote the functions that analyse the board and determine the move of the computer. The ganador (Spanish for winning) function assesses the board condition by assigning -10 or + 10 for a winning game and 0 for any other situation.

```ganador <- function(juego, player) {
game <- matrix(juego, nrow = 3, byrow = T)
hor <- rowSums(game)
ver <- colSums(game)
diag <- c(sum(diag(game)), sum(diag(apply(game, 1, rev))))
if (-3 %in% c(hor, ver, diag))
return(-10)
if (3 %in% c(hor, ver, diag))
return(10)
else
return(0)
}
```

The next function is the actual minimax algorithm. If the computer starts then the first move ($9!= 362880$ options to assess) takes a little while. The commented lines can be used to force a corner and make the games faster by forcing a random corner.

The minimax function returns a list with the move and its valuation through the ganador function. The function works recursively until it has filled the board and retains the best scoring move using the minimax method. To avoid the computer always playing the same move in the same situation random variables are added.

```minimax <- function(juego, player) {
free <- which(juego == 0)
if(length(free) == 1) {
juego[free] <- player
return(list(move = free, U = ganador(juego, player)))
}
poss.results <- rep(0, 9)
for(i in free) {
game <- juego
game[i] <- player
}
mm <- ifelse(player == -1, "which.min", "which.max")
if(any(poss.results == (player * 10))) {
move <- do.call(mm, list(poss.results))
return(list(move = move, U = poss.results[move]))
}
for(i in free) {
game <- juego
game[i] <- player
poss.results[i] <- minimax(game, -player)\$U
}
random <- runif(9, 0, 0.1)
poss.results[-free] <- 100 * -player
poss.results <- poss.results + (player * random)
move <- do.call(mm, list(poss.results))
return(list(move = move, U = poss.results[move]))
}
```

This final function stitches everything together and lets you play the game. Simply paste all functions in your R console and run them to play a game. The tic.tac.toe function can take two parameters, “human” and/or “computer”. The order of the parameters determines who starts the game.

```# Main game engine
tic.tac.toe <- function(player1 = "human", player2 = "computer") {
game <- rep(0, 9) # Empty board
winner <- FALSE # Define winner
player <- 1 # First player
players <- c(player1, player2)
draw.board(game)
while (0 %in% game & !winner) { # Keep playing until win or full board
if (players[(player + 3) %% 3] == "human") # Human player
move <- move.human(game)
else { # Computer player
move <- minimax(game, player)
move <- move\$move
}
game[move] <- player # Change board
draw.board(game)
winner <- max(eval.game(game, 1), abs(eval.game(game, -1))) == 6 # Winner, winner, chicken dinner?
player <- -player # Change player
}
}

tic.tac.toe()
```

This is my last word on Tic Tac Toe but now that the minimax conundrum is solved I could start working on other similar games such as Connect Four, Draughts or even the royal game of Chess.

# R-Cade Games: Simulating the Legendary Game of Pong

Pong is one of the earliest arcade games on the market, first released in 1972. From the day I first saw this miracle box, I wanted to know more about computers.

I learnt how to write code from the 1983 book Dr. C. Wacko’s Miracle Guide to Designing and Programming your own Atari Computer Arcade Games. This book explains in a very clear and humorous way how to write computer games in Atari basic. I devoured this book and spent many hours developing silly games. This article is an ode to Dr Wacko, a computer geek’s midlife-crisis and an attempt to replicate the software I developed thirty years ago.

I showed in a previous post that R can be used for board games. The question is whether we create arcade games in R. My challenge is to recreate the look and feel of 1980s arcade games, or R-Cade games, using R? The code shown below simulates the legendary game of pong.

## Playing Pong in R

The code is based on the Wacko’s Boing Program in the above-mentioned book. The R code is fully commented and speaks for itself. Please note that the animation is very clunky when you run it in RStudio. Only the native R Terminal displays the animation correctly.

Perhaps somebody can help me perfect this little ditty. I love to know how to read real-time USB input to control the game, so we get a step closer to the first R-Cade game.

## The Pong Code

```# Sound library
library(beepr)

# Game parameters
skill <- 0.87 # Skill (0-1)
score <- 0
high.score <- 0

# Define playing field
par(mar = rep(1,4), bg = "black")
plot.new()
plot.window(xlim = c(0, 30), ylim = c(0, 30))
lines(c(1, 30, 30, 1), c(0, 0, 30, 30), type = "l", lwd = 5, col = "white")

# Playing field boundaries (depend on cex)
xmin <- 0.5
xmax <- 29.4
ymin <- 0.5
ymax <- 29.4

# initial position
x <- sample(5:25, 1)
y <- sample(5:25, 1)
points(x, y, pch = 15, col = "white", cex = 2)

psize <- 4

# Set direction
dx <- runif(1, .5, 1)
dy <- runif(1, .5, 1)

# Game play
while (x > xmin - 1) {
sound <- 0 # Silence
Sys.sleep(.05) # Pause screen. Reduce to increase speed
points(x, y, pch = 15, col = "black", cex = 2) # Erase ball
# Move ball
x <- x + dx
y <- y + dy
# Collision detection
if (x > xmax) {
dx <- -dx * runif(1, .9, 1.1) # Bounce
if (x > xmin) x <- xmax # Boundary
sound <- 10 # Set sound
}
if (y < ymin | y > ymax) {
if (y < ymin) y <- ymin
if (y > ymax) y <- ymax
dy <- -dy * runif(1, .9, 1.1)
sound <- 10
}
if (x < xmin & (y > ypaddle - (psize / 2)) & y < ypaddle + (psize / 2)) {
if (x < xmin) x <- xmin
dx <- -dx * runif(1, .9, 1.1)
sound <- 2
score <- score + 1
}
# Draw ball
points(x, y, pch = 15, col = "white", cex = 2)
if (sound !=0) beep(sound)
if (runif(1, 0, 1) < skill) ypaddle <- ypaddle + dy # Imperfect follow
# Erase back line
lines(c(0, 0), c(0, 30), type = "l", lwd = 8, col = "black")
if (ypaddle > 30 - (psize / 2)) ypaddle <- 30 - (psize / 2)
lines(c(0, 0), c(ypaddle - (psize / 2), ypaddle + (psize / 2)), type = "l", lwd = 8, col = "white")
}
beep(8)
text(15,15, "GAME OVER", cex=5, col = "white")
s <- ifelse(score == 1, "", "s")
text(15,5, paste0(score, " Point", s), cex=3, col = "white")
```

# Tic Tac Toe War Games: The Intelligent Minimax Algorithm

In a previous post, I shared how to build a randomised Tic Tac Toe simulation. The computer plays against itself playing at random positions. In this post, I will share how to teach the computer to play the game strategically.

I love the 1983 classic movie War Games. In this film, a computer plays Tic Tac Toe against itself to learn that it cannot win the game to prevent a nuclear war.

Back in those days, I devoured the wonderful book Writing Strategy Games on your Atari by John White which contains an algorithm to play Tic Tac Toe War Games. This is my attempt to relive the eighties using R.

You can find the code on my GitHub page.

## Drawing the Board

A previous post describes the function that draws the Tic Tac Toe board. For completeness, the code is replicated below. The game board is a vector of length nine consisting of either -1 (X), 0 (empty field) or 1 (O). The vector indices correspond with locations on the game board:

1 2 3
4 5 6
7 8 9

```draw.board &lt;- function(board) { # Draw the board
xo &lt;- c(&quot;X&quot;, &quot; &quot;, &quot;O&quot;) # Symbols
par(mar = rep(0,4))
plot.new()
plot.window(xlim = c(0,30), ylim = c(0,30))
abline(h = c(10, 20), col=&quot;darkgrey&quot;, lwd = 4)
abline(v = c(10, 20), col=&quot;darkgrey&quot;, lwd = 4)
pieces &lt;- xo[board + 2]
text(rep(c(5, 15, 25), 3), c(rep(25, 3), rep(15,3), rep(5, 3)), pieces, cex = 6)
# Identify location of any three in a row
square &lt;- t(matrix(board, nrow = 3))
hor &lt;- abs(rowSums(square))
if (any(hor == 3))
hor &lt;- (4 - which(hor == 3)) * 10 - 5
else
hor &lt;- 0
ver &lt;- abs(colSums(square))
if (any(ver == 3))
ver &lt;- which(ver == 3) * 10 - 5
else
ver &lt;- 0
diag1 &lt;- sum(diag(square))
diag2 &lt;- sum(diag(t(apply(square, 2, rev)))) # Draw winning lines if (hor &gt; 0) lines(c(0, 30), rep(hor, 2), lwd=10, col=&quot;red&quot;)
if (ver &gt; 0) lines(rep(ver, 2), c(0, 30), lwd=10, col=&quot;red&quot;)
if (abs(diag1) == 3) lines(c(2, 28), c(28, 2), lwd=10, col=&quot;red&quot;)
if (abs(diag2) == 3) lines(c(2, 28), c(2, 28), lwd=10, col=&quot;red&quot;)
}
```

## Human Players

This second code snippet lets a human player move by clicking anywhere on the graphic display using the locator function. The click location is converted to a number to denote the position on the board. The entered field is only accepted if it has not yet been used (the empty variable contains the available fields).

```# Human player enters a move
move.human &lt;- function(game) {
text(4, 0, &quot;Click on screen to move&quot;, col = &quot;grey&quot;, cex=.7)
empty &lt;- which(game == 0)
move &lt;- 0
while (!move %in% empty) {
coords &lt;- locator(n = 1) # add lines
coords\$x &lt;- floor(abs(coords\$x) / 10) + 1
coords\$y &lt;- floor(abs(coords\$y) / 10) + 1
move &lt;- coords\$x + 3 * (3 - coords\$y)
}
return (move)
}
```

## Evaluate the Game

This code snippet defines the eval.game function which assesses the current board and assigns a score. Zero means no outcome, -6 means that the X player has won and +6 implies that the O player has won.

```# Evaluate board position
eval.game &lt;- function(game, player) {
# Determine game score
square &lt;- t(matrix(game, nrow = 3))
hor &lt;- rowSums(square)
ver &lt;- colSums(square)
diag1 &lt;- sum(diag(square))
diag2 &lt;- sum(diag(t(apply(square, 2, rev))))
eval &lt;- c(hor, ver, diag1, diag2)
# Determine best score
minimax &lt;- ifelse(player == -1, &quot;min&quot;, &quot;max&quot;)
best.score &lt;- do.call(minimax, list(eval))
if (abs(best.score) == 3) best.score &lt;- best.score * 2
return (best.score)
}
```

## Computer Moves

The computer uses a modified Minimax Algorithm to determine its next move. This article from the Never Stop Building blog and the video below explain this method in great detail.

The next function determines the computer’s move. I have not used a brute-force minimax algorithm to save running time. I struggled building a fully recursive minimax function. Perhaps somebody can help me with this. This code looks only two steps deep and contains a strategic rule to maximise the score.

The first line stores the value of the players move, the second remainder of the matrix holds the evaluations of all the opponents moves. The code adds a randomised variable, based on the strategic value of a field. The centre has the highest value because it is part of four winning lines. Corners have three winning lines and the rest only two winning lines. This means that the computer will, all things being equal, favour the centre over the corners and favour the other fields least. The randomised variables in the code ensure that the computer does not always pick the same field in a similar situation.

```# Determine computer move
move.computer &lt;- function(game, player) {
empty &lt;- which(game == 0)
eval &lt;- matrix(nrow = 10, ncol = 9, data = 0)
for (i in empty) {
game.tmp &lt;- game
game.tmp[i] &lt;- player
eval[1, i] &lt;- eval.game(game.tmp, player)
empty.tmp &lt;- which(game.tmp ==0)
for (j in empty.tmp) {
game.tmp1 &lt;- game.tmp
game.tmp1[j] &lt;- -player
eval[(j + 1), i] &lt;- eval.game(game.tmp1, -player)
}
}
if (!any(abs(eval[1,]) == 6)) { # When winning, play move
# Analyse opponent move
minimax &lt;- ifelse(player == -1, &quot;max&quot;, &quot;min&quot;) # Minimax
best.opponent &lt;- apply(eval[-1,], 1, minimax)
eval[1,] &lt;- eval[1,] * -player * best.opponent
}
# Add randomisation and strategic values
board &lt;- c(3, 2, 3, 2, 4, 2, 3, 2, 3) # Strategic values
board &lt;- sapply(board, function(x) runif(1, 0.1 * x, (0.1 * x) + 0.1)) # Randomise
eval[1, empty] &lt;- eval[1, empty] + player * board[empty] # Randomise moves
# Pick best game
minimax &lt;- ifelse(player == -1, &quot;which.min&quot;, &quot;which.max&quot;) # Minimax
move &lt;- do.call(minimax, list(eval[1,])) # Select best move
return(move)
}
```

This last code snippet enables computers and humans play each other or themselves. The players vector contains the identity of the two players so that a human can play a computer or vice versa. The human player moves by clicking on the screen.

The loop keeps running until the board is full or a winner has been identified. A previous Tic Tac Toe post explains the draw.board function.

```# Main game engine
tic.tac.toe &lt;- function(player1 = &quot;human&quot;, player2 = &quot;computer&quot;) {
game &lt;- rep(0, 9) # Empty board
winner &lt;- FALSE # Define winner
player &lt;- 1 # First player
players &lt;- c(player1, player2)
draw.board(game)
while (0 %in% game &amp; !winner) { # Keep playing until win or full board
if (players[(player + 3) %% 3] == &quot;human&quot;) # Human player
move &lt;- move.human(game)
else # Computer player
move &lt;- move.computer(game, player)
game[move] &lt;- player # Change board
draw.board(game)
winner &lt;- max(eval.game(game, 1), abs(eval.game(game, -1))) == 6 # Winner, winner, chicken dinner?
player &lt;- -player # Change player
}
}
```

You can play the computer by running all functions and then entering tic.tac.toe().

I am pretty certain this simplified minimax algorithm is unbeatable—why don’t you try to win and let me know when you do.

## Tic Tac Toe War Games

Now that this problem is solved, I can finally recreate the epic scene from the WarGames movie. The Tic Tac Toe War Games code uses the functions explained above and the animation package. Unfortunately, there are not many opportunities to create sound in R.

```# WAR GAMES TIC TAC TOE
source(&quot;Tic Tac Toe/Tic Tac Toe.R&quot;)

# Draw the game board
draw.board.wargames &lt;- function(game) {
xo &lt;- c(&quot;X&quot;, &quot; &quot;, &quot;O&quot;) # Symbols
par(mar = rep(1,4), bg = &quot;#050811&quot;)
plot.new()
plot.window(xlim = c(0,30), ylim = c(0,30))
abline(h = c(10, 20), col = &quot;#588fca&quot;, lwd = 20)
abline(v = c(10, 20), col = &quot;#588fca&quot;, lwd = 20)
text(rep(c(5, 15, 25), 3), c(rep(25, 3), rep(15,3), rep(5, 3)), xo[game + 2], cex = 20, col = &quot;#588fca&quot;)
text(1,0,&quot;r.prevos.net&quot;, col = &quot;#588fca&quot;, cex=2)
# Identify location of any three in a row
square &lt;- t(matrix(game, nrow = 3))
hor &lt;- abs(rowSums(square))
if (any(hor == 3))
hor &lt;- (4 - which(hor == 3)) * 10 - 5
else
hor &lt;- 0
ver &lt;- abs(colSums(square))
if (any(ver == 3))
ver &lt;- which(ver == 3) * 10 - 5
else
ver &lt;- 0
diag1 &lt;- sum(diag(square))
diag2 &lt;- sum(diag(t(apply(square, 2, rev)))) # Draw winning lines if (all(hor &gt; 0)) for (i in hor) lines(c(0, 30), rep(i, 2), lwd = 10, col=&quot;#588fca&quot;)
if (all(ver &gt; 0)) for (i in ver) lines(rep(i, 2), c(0, 30), lwd = 10, col=&quot;#588fca&quot;)
if (abs(diag1) == 3) lines(c(2, 28), c(28, 2), lwd = 10, col = &quot;#588fca&quot;)
if (abs(diag2) == 3) lines(c(2, 28), c(2, 28), lwd = 10, col = &quot;#588fca&quot;)
}

library(animation)
player &lt;- -1
games &lt;- 100
saveGIF ({
for (i in 1:games) {
game &lt;- rep(0, 9) # Empty board
winner &lt;- 0 # Define winner
#draw.board.wargames(game)
while (0 %in% game &amp; !winner) { # Keep playing until win or full board
empty &lt;- which(game == 0)
move &lt;- move.computer(game, player)
game[move] &lt;- player
if (i &lt;= 12) draw.board.wargames(game)
winner &lt;- max(eval.game(game, 1), abs(eval.game(game, -1))) == 6
player &lt;- -player } if (i &gt; 12) draw.board.wargames(game)
}
},
interval = c(unlist(lapply(seq(1, 0,-.2), function (x) rep(x, 9))), rep(0,9*94)),
movie.name = &quot;wargames.gif&quot;, ani.width = 1024, ani.height = 1024)
```

# Tic Tac Toe Simulation — Random Moves

Tic Tac Toe might be a futile children’s game but it can also teach us about artificial intelligence. Tic Tac Toe, or Naughts and Crosses, is a zero-sum game with perfect information. Both players know exactly what the other did and when nobody makes a mistake, the game will always end in a draw.

Tic Tac Toe is a simple game but also the much more complex game of chess is a zero-sum game with perfect information.

In this two-part post, I will build an unbeatable Tic Tac Toe Simulation. This first part deals with the mechanics of the game. The second post will present an algorithm for a perfect game.

## Drawing the Board

This first code snippet draws the Tic Tac Toe simulation board. The variable xo holds the identity of the pieces and the vector board holds the current game. Player X is denoted with -1 and player O with +1. The first part of the function draws the board and the naughts and crosses. The second part of the code check for three in a row and draws the corresponding line.

```draw.board <- function(board) { # Draw the board
xo <- c("X", " ", "O") # Symbols
par(mar = rep(0,4))
plot.new()
plot.window(xlim = c(0,30), ylim = c(0,30))
abline(h = c(10, 20), col="darkgrey", lwd = 4)
abline(v = c(10, 20), col="darkgrey", lwd = 4)
pieces <- xo[board + 2]
text(rep(c(5, 15, 25), 3), c(rep(25, 3), rep(15,3), rep(5, 3)), pieces, cex = 6)
# Identify location of any three in a row
square <- t(matrix(board, nrow = 3))
hor <- abs(rowSums(square))
if (any(hor == 3))
hor <- (4 - which(hor == 3)) * 10 - 5
else
hor <- 0
ver <- abs(colSums(square))
if (any(ver == 3))
ver <- which(ver == 3) * 10 - 5
else
ver <- 0
diag1 <- sum(diag(square))
diag2 <- sum(diag(t(apply(square, 2, rev))))
# Draw winning lines
if (hor > 0) lines(c(0, 30), rep(hor, 2), lwd=10, col="red")
if (ver > 0) lines(rep(ver, 2), c(0, 30), lwd=10, col="red")
if (abs(diag1) == 3) lines(c(2, 28), c(28, 2), lwd=10, col="red")
if (abs(diag2) == 3) lines(c(2, 28), c(2, 28), lwd=10, col="red")
}
```

## Random Tic Tac Toe

The second part of the code generates ten random games and creates and animated GIF-file. The code adds random moves until one of the players wins (winner <> 0) or the board is full (no zeroes in the game vector). The eval.winner function checks for three in a row and declares a winner when found.

There are 255,168 possible legal games in Tic Tac Toe, 46,080 of which end in a draw. This implies that these randomised games result in a draw 18% of the time.

```eval.winner <- function(board) { # Identify winner
square <- t(matrix(board, nrow = 3))
hor <- rowSums(square)
ver <- colSums(square)
diag1 <- sum(diag(square))
diag2 <- sum(diag(t(apply(square, 2, rev))))
if (3 %in% c(hor, ver, diag1, diag2)) return (1)
else
if (-3 %in% c(hor, ver, diag1, diag2)) return (2)
else
return(0)
}

# Random game
library(animation)
saveGIF ({
for (i in 1:10) {
game <- rep(0, 9) # Empty board
winner <- 0 # Define winner
player <- -1 # First player
draw.board(game)
while (0 %in% game & winner == 0) { # Keep playing until win or full board
empty <- which(game == 0) # Define empty squares
move <- empty[sample(length(empty), 1)] # Random move
game[move] <- player # Change board
draw.board(game)
winner <- eval.winner(game) # Evaulate game
player <- player * -1 # Change player
}
draw.board(game)
}
},
interval = 0.25, movie.name = "ttt.gif", ani.width = 600, ani.height = 600)
```