This solution is no longer valid.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.)
