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Problem 084: Monopoly odds


In the game, Monopoly, the standard board is set up in the following way:

GO A1 CC1 A2 T1 R1 B1 CH1 B2 B3 JAIL
H2                               C1
T2                               U1
H1                               C2
CH3                              C3
R4                               R2
G3                               D1
CC3                              CC2
G2                               D2
G1                               D3
G2J  F3 U2 F2 F1 R3 E3 E2 CH2 E1 FP

A player starts on the GO square and adds the scores on two 6-sided dice to determine the number of squares they advance in a clockwise direction. Without any further rules we would expect to visit each square with equal probability: 2.5%. However, landing on G2J (Go To Jail), CC (community chest), and CH (chance) changes this distribution.

In addition to G2J, and one card from each of CC and CH, that orders the player to go directly to jail, if a player rolls three consecutive doubles, they do not advance the result of their 3rd roll. Instead they proceed directly to jail.

At the beginning of the game, the CC and CH cards are shuffled. When a player lands on CC or CH they take a card from the top of the respective pile and, after following the instructions, it is returned to the bottom of the pile. There are sixteen cards in each pile, but for the purpose of this problem we are only concerned with cards that order a movement; any instruction not concerned with movement will be ignored and the player will remain on the CC/CH square.

Community Chest (2/16 cards):

  • Advance to GO
  • Go to JAIL Chance (10/16 cards):
  • Advance to GO
  • 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.

The heart of this problem concerns the likelihood of visiting a particular square. That is, the probability of finishing at that square after a roll. For this reason it should be clear that, with the exception of G2J for which the probability of finishing on it is zero, the CH squares will have the lowest probabilities, as 5/8 request a movement to another square, and it is the final square that the player finishes at on each roll that we are interested in. We shall make no distinction between "Just Visiting" and being sent to JAIL, and we shall also ignore the rule about requiring a double to "get out of jail", assuming that they pay to get out on their next turn.

By starting at GO and numbering the squares sequentially from 00 to 39 we can concatenate these two-digit numbers to produce strings that correspond with sets of squares.

Statistically it can be shown that the three most popular squares, in order, are JAIL (6.24%) = Square 10, E3 (3.18%) = Square 24, and GO (3.09%) = Square 00. So these three most popular squares can be listed with the six-digit modal string: 102400.

If, instead of using two 6-sided dice, two 4-sided dice are used, find the six-digit modal string.

v   XX
  ========================================    v      <     v<
  ???????????????????????????????????????? v<<   v<<<|-"!":<|-+98:<
>040p"}}@"**58*v             v   _v# -1<  v<10   v 10> v    <    <|<
   v           <                  0    |:+<2?^#*4<2?^#$<         ^<
   >1-:2+0\1pv                            ^3<    ^3<               -
   |:        <     >>>v >>>v v<         >>>v   >>>v    >>>v        2
v $<               12 v 12 v #^p04<+55$<01 v   01 v    01 v >v     :
|                   >4^v<>4^ #v<       v<>3^    >3^     >3^
>$58vv             <v $<|-*94:<|-+*294:<|-7:  <    v p04+55$<      <
#v *<  >>>v   >>>v ^    <      <        <
:    $ 01 v   01 v  #>v   >v    >v   >v   >v   >v  #>v   >v   >v   > v
-    >#^?2>4*#^?2>+ :|>1-:|>1 -:|>1-:|>1-:|>1-:|>1-:|>1-:|>1-:|>1-:| #
1>1-::2v>3^    >3^   >55+v>56 +v>64*v>"'"v>5v  >0v  v    <    v    >40g3-   v
 |:p2+ <>::1+2g2+1g\ 2+2g 2+1 g \`                         v>v       #  >62*v
 >$v    ^_v#:<-1p2\g 05+2 :-1 p 2+2\ g2+1 :: p 05 g    2+2:_^#>40g492*+-|
   >0>:58* \`|  >$22 g.32 g.4 2 g.@                 >40g6%1+2/2*5*5+v$  >74*v
     ^+1  <  >#<^#       >#    >#   >#   ># >#   >#         #<      >       v
^<                  <        <                     <                 <  $p04<
Stack:   (0)


This is not really a mathematical problem (or at least not with my solution).

All I did was implement the rules in my befunge program and run a randomized game for 1 000 000 turns. This is called an Monte Carlo algorithm and if we have enough runs it becomes pretty improbable to get a wrong result.

Perhaps there are some fancy mathematical solutions out there, but this works too.

A note to the befunge code: It got pretty messy because of all the decisions we have to implement for the different monopoly rules, but all we needed as storage was an 40-element array. So it wasn't that hard to fit it all in the 80x25 space.

Interpreter steps: 146 519 794
Execution time (BefunExec): 19s (7.63 MHz)
Program size: 77 x 20 (fully conform befunge-93)
Solution: 101524
Solved at: 2015-09-16

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