# Monopoly – Project Euler problem 84

This is basically a question of building a Monopoly simulator.

We begin at square GO. Roll two six-sided dice, and advance.

If we land on the “Go to jail” (G2J) square, we go to the JAIL square.

If we land on a CC square, we draw a card from the “community chest” stack of cards. There are 16 cards there, only two changes the position of the player: “Advance to GO” and “Go to Jail”

If we land on a CH square, we draw a card from the Chance-stack. There are 16 cards here, only ten changes the position:

• Go to JAIL
• Go to C1
• Go to E3
• Go to H2
• Go to R1
• Go to next R (railway company)
• Go to next R
• Go to next U (utility company)
• Go back 3 squares.

There are three CH squares, and three CC.

A player being sent to jail, is assumed to buy himself out immediately. And if a player rolls three consecutive doubles, he does not advance the result of the third roll, but goes directly to jail.

The two stacks of cards are shuffled at the beginning.

Find the probability of landing on a given square, and return the three squares that are most likely to land on.

Numbering the squares from 00 to 39, the answer is the concatenated string representing these three squares. For 6 sided dice, the most likely squares to end on after a roll of the dice, are squares 10, 24 and 00. The answer is 102400.

I am going to need some dice:

``````dice <- function() sample(1:6, 2, replace=TRUE)
``````

I’m gonna need a community chest function. And I am going to take a chance. That is, I am going to asume that the whole “shuffle the deck at the start, and place drawn card at bottom” is going to be equivalent to just picking a random card:

``````pool <- rep(0,14)
pool <- c(pool,1,11)
cc <- function(cs){
res <- sample(pool,1)
ifelse(res==0,cs,res)
}
``````

It takes the current square, builds a vector containing 14 repetitions of 0, one instance of “1” corresponding to GO, and one of “11”, corresponding to jail. Then it returns one of the values.

And a chance function. Again, I am going to make the same assumption as with the community chest function:

``````ch_pool <- rep(0,6)
ch_pool <- c(ch_pool,1:10)
ch <- function(cs){
chance <- sample(ch_pool,1)
if(chance==0) res <- cs
if(chance==1) res <- 1
if(chance==2) res <- 11
if(chance==3) res <- 12
if(chance==4) res <- 25
if(chance==5) res <- 40
if(chance==6) res <- 6
if(chance==7&cs==8) res <- 16
if(chance==7&cs==23) res <- 26
if(chance==7&cs==37) res <- 6
if(chance==8&cs==8) res <- 16
if(chance==8&cs==23) res <- 26
if(chance==8&cs==37) res <- 6
if(chance==9&cs==8) res <- 13
if(chance==9&cs==23) res <- 29
if(chance==9&cs==37) res <- 13
if(chance==10) res <- cs-3
return(res)
}
``````

This is basically the same as the community function. Just a bit longer.

I am also going to need a structure for the playing board:

``````hit <- numeric(40)
``````

A third chance I’m taking. That the part about hitting three doubles in a row, and then going to jail, will also disappear if I just run this long enough:

``````curr_square <- 1

for(i in 1:10000000){
curr_square <- ((curr_square + sum(dice())-1)%%40 + 1)
if(curr_square == 31) curr_square <- 11
if(curr_square %in% c(2,18,34)) curr_square <- cc(curr_square)
if(curr_square %in% c(8,23,37)) curr_square <- ch(curr_square)
hit[curr_square] <- hit[curr_square] + 1
}
``````

I begin at square 1. And then I am going to repeat the following 1.000.000 times:
Roll the dice, add the result to the current square. If the result is more than 40, do some simple math to continue from square 1.

If the square I’m landing on is square 31, set new square to 11 – that is the jail.

If it is 2, 18 or 34, run the cc – community chest function. That will return the square that we land on after drawing a card from that stack.

If it is 8, 23 or 37, we have landed on a chance square. Run the ch – chance function. That will return the square that we land on after drawing a card from that stack.

That should give me the frequencies with which the player have landed on each square.

Now I can rank the results:

``````freq <- rank(hit)
``````

That gives me the rank. Number 40 is the square that was hit most times.

I need to translate that to the numbering of the squares that is used for the result.

``````squares <- c("00","01","02","03","04","05","06","07","08","09",10:39)
paste(squares[freq %in% c(40,39,38)], collapse="")
``````
```## [1] "001024"
```

pasting together the squares hit most, second most, and third-most times. That is fortunately the result given in the problem.

Now lets try with a new set of dice:

``````dice <- function() sample(1:4, 2, replace=TRUE)
``````

And the same code:

``````hit <- numeric(40)
curr_square <- 1

for(i in 1:10000000){
curr_square <- ((curr_square + sum(dice())-1)%%40 + 1)
if(curr_square == 31){ curr_square <- 11}
if(curr_square %in% c(2,18,34)) curr_square <- cc(curr_square)
if(curr_square %in% c(8,23,37)) curr_square <- ch(curr_square)
hit[curr_square] <- hit[curr_square] + 1
}

freq <- rank(hit)

answer <- paste(squares[freq %in% c(40,39,38)], collapse="")

``````

And we are rewarded with the nice green tickmark.

This was not as difficult as I thought. But maybe it would be wise to look at the forum for the problem, and study some of the non-brute force solutions.

Oh – another lesson learned. In the code above, I run through 10.000.000 throw with the dice. I actually wanted to throw the dice 1.000.000 times. Always count your zeroes!

Also – 100.000 thow is enough to get the right result.