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This is an entry into the 29th fortnightly topic challenge - Retrograde Analysis.

Two friends and I were playing a game of Monopoly when we were forced to stop. It's been a long time since we started, but we would now like to continue from where we left off. The only problem is that we are not sure who is supposed to go next. Here is the current board:

Current Board

Text version:

  • 3 pieces

    • Battleship: currently on Mediterranean Avenue

    • Racecar: currently on Connecticut Avenue

    • Top hat: currently on Pennsylvania RR

  • 3 seats

    • Left: $1,380, Connecticut Avenue

    • Right: $860, Water Works, Mediterranean Avenue, Illinois Avenue, Electric Company, Pennsylvania Avenue

    • Bottom: $1,020, Pennsylvania RR

Here's what we can remember of the game:

  • Battleship moved first
  • No Chance/Community Treasure cards had been drawn
  • No one had gone to jail
  • We did not stop in the middle of someone's turn
  • No auctions were held

We've agreed to continue playing from the minimum number of turns it would take to get to the current setup. So, how did we get to this scenario and who should roll next?

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  • $\begingroup$ I suggest adding that you remember that no auctions were held. Auctions don't actually reduce the number of turns required (since battleship needs two turns to get to Mediterranean Avenue), but they add many alternative ways of getting to the money and property distribution. $\endgroup$ – thelem Apr 3 '17 at 16:17
  • $\begingroup$ @thelem good point, added. $\endgroup$ – David Starkey Apr 3 '17 at 16:30
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It is currently:

the hat's turn

because

The battleship moved first (\$1500), rolled two 12s and a 4. Allowing him/her to buy the electric company (\$1350), Illinois Avenue (\$1110) and the water company (\$960).

Then the hat went next (\$1500) and rolled a four (doubles) and landed on income tax (\$1300) and then rolled an 8 and landed on the electric company costing him \$80 (\$1220) and finally he rolled a 3 to land on the railroad and purchased it (\$1020).

Finally the car (\$1500) rolled a 9 to land on the Connecticut Avenue (\$1380).

For his second turn, the Battleship (\$1040) rolled a 6 (doubles) (\$720) and then a 7, passing GO (\$920), buying both properties along the way (\$860).

All that means it is now the

hat's turn.

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  • $\begingroup$ My apologies, that 1300 was a typo. 1380 is what it should have been. I've updated the question. $\endgroup$ – David Starkey Apr 3 '17 at 15:55
  • $\begingroup$ @DavidStarkey Bah. Well, that makes a huge difference. :) $\endgroup$ – Rubio Apr 3 '17 at 16:25
  • $\begingroup$ @Rubio It made me want to find a way that my typo could have produced a solution, though. I accidentally made a puzzle for myself, apparently. $\endgroup$ – David Starkey Apr 3 '17 at 16:29
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    $\begingroup$ @DavidStarkey Well. I did find a solution for the typo version. :) $\endgroup$ – Rubio Apr 3 '17 at 16:30
  • $\begingroup$ grats on 5k!!!! $\endgroup$ – Beastly Gerbil Apr 3 '17 at 20:03
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This solution is no longer valid. The original posting of the problem had a typo (and lacked the restriction of "no auctions"); with the typo, at least one auction was required to solve, and this answer provided such a solution.
For the corrected problem, @dcfyj's post provides the correct answer.

It is Player 1: Battleship's turn

Here's how the game plays out:

Player 1 - Battleship
  Turn 1:  rolls 12 → Electric Company; buys for 150
           rolls 12 → Illinois Avenue; buys for 240
           rolls  5 → Water Works; buys for 150
           (started 1500; paid 150+240+150; received 0; total 960)

Player 2 - Hat
  Turn 1:  rolls 2+2 → Income Tax; pays 200
           rolls 8 → Electric Company; pays rent of 80 to Player 1
           (started 1500; paid 200+80; received 0; total 1220)

Player 3 - Car
  Turn 1:  rolls 5+5 → Just Visiting
           rolls 5+5 → Free Parking
           rolls 8 → Water Works; pays rent of 80 to Player 1
           (started 1500; paid 80; received 0; total 1420)

Player 1 - Battleship
  Turn 2:  rolls 3+3 → Pennsylvania Ave; buys for 300 at auction
           rolls 2+2 → Luxury Tax; pays 100
           rolls  3 → Mediterranean Ave (passes go, +200); buys for 60
           (started 960+80+80; paid 300+60+100; received 200; total 860)

Player 2 - Hat
  Turn 2:  rolls 3 → Pennsylvania RR; buys for 200
           (started 1220; paid 200; received 0; total 1020)

Player 3 - Car
  Turn 2:  rolls 12 → Go (+200)
           rolls 2+2 → Income Tax; pays 200
           rolls 5 → Connecticut Ave; buys for 120
           (started 1420; paid 200+120; received 200; total 1300)

The only way this game works as described is ...

for some player to have underpaid by 20 on one property:

The total cash amongst the players is 3180.
The total face-value purchase price of properties purchased is 1240.
Together, those account for 4420 of the players' initial funds of 4500.

The only ways to add or remove money from the players collectively, without hitting Chance or Community Chest or Jail, are:
  - purchasing properties
  - hitting Income Tax or Luxury Tax
  - passing Go
The latter two only happen in increments of 100 or 200, so they cannot account for 4500 being reduced to 4420. Thus there had to be 20 "gained" somehow, and the only way that can happen is for a property to be purchased for 20 less than its face value.

The solution with the minimum number of moves that involves a property being auctioned happens if Player 1 is the one who lands on, and buys, a property via auction. Depending on the players, one might argue for either Pennsylvania Ave (because its high price gives the most room for an auction scenario to be beneficial), or Mediterranean Ave (a pretty crap property overall, which I could see selling below price) to be the most likely to go to auction. I've arbitrarily picked Pennsylvania Ave, but really any of Player 1's property purchases could have been made via auction, for a net saving of 20.

(Another possibility is simple error. I might guess that OP miscalculated the cost for Pennsylvania Ave, since before I actually looked at a board I misremembered its cost as 300 instead of 320. If Player 1 has, say, North Carolina Ave instead of Pennsylvania Ave, it also works. Or it's also possible one player's cash on hand is simply off by 20.)

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