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www/statics/euler/Euler_Problem-084_description.md

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+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.