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Project Euler Solutions

Project Euler 84 Solution

Project Euler 84 Solution

Project Euler 84: In the game, Monopoly, find the three most popular squares when using two 4-sided dice

Problem Description

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):
    1. Advance to GO
    2. Go to JAIL
  • Chance (10/16 cards):
    1. Advance to GO
    2. Go to JAIL
    3. Go to C1
    4. Go to E3
    5. Go to H2
    6. Go to R1
    7. Go to next R (railway company)
    8. Go to next R
    9. Go to next U (utility company)
    10. 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.


Here’s our solution using simulation for 2,000,000 games. Sample output below is for six-sided dice.
Sample size: 2000000
Frequency for each space

CH3, 18479, 0.92395
CH1, 18496, 0.9248
CH2, 22400, 1.12
CC1, 37265, 1.86325
A1, 42550, 2.1275
A2, 43224, 2.1612
H1, 43282, 2.1641
T2, 43725, 2.18625
T1, 43918, 2.1959
CC3, 44532, 2.2266
B3, 45441, 2.27205
B1, 45509, 2.27545
B2, 46381, 2.31905
C2, 47331, 2.36655
R4, 48554, 2.4277
C3, 49521, 2.47605
G3, 50316, 2.5158
CC2, 51646, 2.5823
F3, 52056, 2.6028
U1, 52446, 2.6223
H2, 52472, 2.6236
G2, 52994, 2.6497
F2, 53158, 2.6579
G1, 53795, 2.68975
F1, 53856, 2.6928
C1, 54192, 2.7096
E2, 54428, 2.7214
U2, 56039, 2.80195
D1, 56451, 2.82255
E1, 56991, 2.84955
FP, 57917, 2.89585
D3, 58364, 2.9182
D2, 58802, 2.9401
R2, 59064, 2.9532
R1, 60164, 3.0082
R3, 61443, 3.07215
GO, 62558, 3.1279
E3, 64488, 3.2244
JAIL, 125752, 6.2876

Project Euler 84 Solution

Runs < 8 seconds in Perl.

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One Response to “Project Euler 84 Solution”

  1. Interesting. I created a Markov chain with 120 states (40 squares x #doubles). I squared the 120×120 matrix repeatedly, and got to a steady state.

    Posted by Frank Yellin | September 23, 2020, 10:12 PM

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