57 lines
2.9 KiB
Markdown
57 lines
2.9 KiB
Markdown
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In the game, Monopoly, the standard board is set up in the following way:
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~~~
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GO A1 CC1 A2 T1 R1 B1 CH1 B2 B3 JAIL
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H2 C1
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T2 U1
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H1 C2
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CH3 C3
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R4 R2
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G3 D1
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CC3 CC2
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G2 D2
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G1 D3
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G2J F3 U2 F2 F1 R3 E3 E2 CH2 E1 FP
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~~~
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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.
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Without any further rules we would expect to visit each square with equal probability: `2.5%`.
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However, landing on G2J (Go To Jail), CC (community chest), and CH (chance) changes this distribution.
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In addition to G2J, and one card from each of CC and CH, that orders the player to go directly to jail,
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if a player rolls three consecutive doubles, they do not advance the result of their 3rd roll.
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Instead they proceed directly to jail.
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At the beginning of the game, the CC and CH cards are shuffled.
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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.
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There are sixteen cards in each pile, but for the purpose of this problem we are only concerned with cards that order a movement;
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any instruction not concerned with movement will be ignored and the player will remain on the CC/CH square.
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Community Chest (2/16 cards):
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- Advance to GO
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- Go to JAIL
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Chance (10/16 cards):
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- Advance to GO
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- Go to JAIL
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- Go to C1
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- Go to E3
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- Go to H2
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- Go to R1
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- Go to next R (railway company)
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- Go to next R
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- Go to next U (utility company)
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- Go back 3 squares.
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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.
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For this reason it should be clear that, with the exception of G2J for which the probability of finishing on it is zero,
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the CH squares will have the lowest probabilities, as `5/8` request a movement to another square,
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and it is the final square that the player finishes at on each roll that we are interested in.
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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",
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assuming that they pay to get out on their next turn.
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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.
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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`.
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So these three most popular squares can be listed with the six-digit modal string: 102400.
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If, instead of using two 6-sided dice, two 4-sided dice are used, find the six-digit modal string.
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