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This video claims that the fastest game of monopoly involves 4 turns and 9 rolls of the dice. I am wondering if this is actually the shortest game or can we do better in theory? What is the probability of this game happening in real life?

https://www.youtube.com/watch?v=gHJkTz6Ej3U

For reference, the moves in the above video are as follows:

Player 1, Turn 1:
Roll: 6-6, Lands on: Electric Company
Action: None, Doubles therefore roll again
Roll: 6-6, Lands on: Illinois Avenue
Action: None, Doubles therefore roll again
Roll: 4-5, Lands on: Community Chest "Bank error in your favour, Collect \$200"
Action: Collects \$200 (now has \$1700)

Player 2, Turn 1:
Roll: 2-2, Lands on: Income Tax
Action: Pay \$200 (now has \$1300), Doubles therefore rolls again
Roll: 5-6, Lands on: Pennsylvania Rail Road
Action: None

Player 1, Turn 2:
Roll: 2-2, Lands on: Park Place
Action: Purchase (\$350, now has \$1350), Doubles therefore rolls again
Roll: 1-1, Lands on: Boardwalk
Action: Purchase (\$400, now has \$950), Doubles therefore rolls again
Roll: 3-1, Lands on Baltic Avenue
Action: Collect \$200 for passing GO (now has \$1150), Purchase 3 houses for Boardwalk, 2 for Park Place (\$1000, now has \$150)

Player 2, Turn 2:
Roll: 3-4, Lands on: Chance, “Advance to Boardwalk”
Action: Advance to Boardwalk, Rent is \$1400, only has \$1300 = Bankrupt

GAME OVER

Dmitry Kamenetsky
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    What version of monopoly? The UK version? The US version? –  May 25 '20 at 11:09
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    No idea if that is the shortest game but I would have said the probability for that game is 96/(36 ^ 9) * 1/(16 ^ 2) which is 96 / 25999348907114496 or one in 271 trillion - based on https://i.stack.imgur.com/7MP4F.png – Martin Smith May 25 '20 at 12:00
  • Any version of the game. – Dmitry Kamenetsky May 25 '20 at 13:01
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    I think the video answers it all, this link agrees: https://scatter.wordpress.com/2010/05/30/the-shortest-possible-game-of-monopoly-21-seconds/ –  May 25 '20 at 13:13
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    Can you provide a short summary of the moves in the video? If the video gets taken down or something, future users won’t be able to view it. – boboquack May 25 '20 at 13:58
  • Here is a slightly different version for the UK monopoly. The video is much slower and easier to understand in my opinion: https://www.youtube.com/watch?v=YxkhCw3xqII – Dmitry Kamenetsky May 26 '20 at 01:38
  • How do you define shorter, by turns or by rolls of the dice? – Jason Goemaat May 26 '20 at 04:20
  • Jason I would be interested to see the answer to both definitions of shortest. – Dmitry Kamenetsky May 26 '20 at 04:36
  • I'm with @MartinSmith, though he takes a few examples of to lowering odds with equivalent moves (allowing 6-3 for player 1's last roll on turn one for example). There are others that would finish in the same number of moves with basically the same game. Since P1 ends the game with $150, there are 4 other community chest cards that would give them enough money. Also the 1,3 to land on Baltic on his last turn to pass go seems arbitrary. The only 'bad' outcomes would be a 5 to pay income tax or doubles causing an extra turn or a bad chance draw. Could hit chance on turn 1 too. (66,55,56) – Jason Goemaat May 26 '20 at 04:37
  • @MartinSmith If I've counted right, the dice roll numerator should be 160, not 96. There are 20 ways for player 2's round 1 2nd roll and round 2 roll to add up to 18 with no doubles:
    • 2*4=8 if 11 then 9, via Marylebone station
    • 4*2=8 if 9 then 11, via Whitehall
    • 2*2=4 if 10 then 10 with no 5-5, via Northumberland Avenue

    So the dice roll probability is 4220/36^9 = 160/36^9.

     – Rosie F May 26 '20 at 05:31
  • Yeah, depends whether you regard games where the player lands on different properties but it doesn't materially affect the outcome as the "same game" I guess – Martin Smith May 26 '20 at 06:02
  • The "one in 271 trillion" or thereabouts is just for the card and dice rolls though. When you factor in the probability of the players making those actions in a real game I imagine the probability is basically zero. – Martin Smith May 26 '20 at 07:50
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    o.O I'm puzzled by the rules applied here! Are these the official ones? We always did it like that: no buying until you passed Go once. Buying houses only if you're sitting on your own street. Only one house at the time. Well, it seems I just learned, why Monopoly always took us hours and hours ^^ – Jessica May 26 '20 at 15:24
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    If player 2 is not deliberately trying to lose, it's a bizarre move to pay $200 for income tax on the first move of the game when it is a known fact that one's assets are worth $1,500 at the start of the game, and 10% of that is $150. That said, the extra $50 wouldn't change the outcome. – Daniel May 27 '20 at 03:22
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    @Jessica: No matter what rules you play by, Monopoly takes hours and hours. – Peter Shor May 27 '20 at 14:11
  • @Jessica - those are not rules of Monopoly, and most of the common house rules cause the game to last an awful long time! (Even more so) – komodosp May 27 '20 at 16:09
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    One problem with that game, it ignores the rule that unbought purchases are put up for auction. – komodosp May 27 '20 at 16:11
  • @colmde originally I thought that wouldn't affect anything but if player 2 buys Electric Company, Illinois Avenue, Baltic Avenue for a dollar each the mortgage value of 225 would be enough to save them – Martin Smith May 27 '20 at 18:35

5 Answers5

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If you are trying to lose the game as quickly as possible then you can do it in 1 turn and 2 rolls of the dice. This is using the UK version as I'm not sure about the US one.

Roll 1:

You get a double 3 (a six) which lands you on The Angel, Islington. You don't buy this but put it up for auction. You then go up to M1500 in the auction.

Roll 2 (Because of the doubles):

You roll 11 and this takes you to the community chest. Out of these you need to get Pay a M10 fine or take a chance card. You need to choose the chance card then you need the Pay school fees of M150. You will not be able to pay this as you have no money and your assets can only mortgage to give you M50. Your opponent wins the game.

The chance of this is:

Getting double 3s: 1 in 36
Getting the right community chest card: 1 in 16
Getting the right chance card: 1 in 16
Getting an 11: 1 in 18

So the total is: 1 in 165888 or 0.0006028% (which is a lot more likely that the example Dmitry Kamenetsky gave

Community
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William Pennanti
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  • I think you can get more likely (but no shorter) by rolling a double six to land on Electric Co,, doing the same action, and then a 10 to land directly on chance, bypassing the Community Chest. – DJClayworth May 25 '20 at 19:37
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    I think the video shows a game where the other player isn't intentionally trying to lose, which makes for a somewhat more complex puzzle. – Bass May 25 '20 at 20:49
  • Here is another version of intentionally trying to lose, found by Robert Manning: https://www2.stetson.edu/~efriedma/mathmagic/1112.html – Dmitry Kamenetsky May 26 '20 at 01:39
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    the chance aspect of this is complicated — what is the chance of the selling price being $1500? (Probably not very high) – JDL May 26 '20 at 08:17
  • @JDL that's the only bit of it which isn't down to chance. As per the rules, if you don't want to buy, you can put it up to auction. You then bid 1500 to buy it, thereby leaving you no cash, and just the mortage value to your name – Matt Taylor May 26 '20 at 13:23
  • Why bother with the Chance? Just bid $1501 and bankrupt yourself during the auction. – Nuclear Hoagie May 26 '20 at 14:03
  • @NuclearWang: I don't think that works. Most of the rules aren't in the rulebook anymore (to save on printing costs???) but we always played that bid just gets rolled back. – Joshua May 26 '20 at 19:03
  • @Joshua I don't think anything addressing this specifically was ever in the rule book, so I don't see any particular reason why this would allow you to cancel a bid, rather than simply going bankrupt as you would any other time you owe the bank more than you can pay. As a house rule I agree it's reasonable, but by the rulebook, there's nothing to differentiate auction debt from any other debt to the bank. – Nuclear Hoagie May 26 '20 at 19:41
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    @NuclearWang: It was an illegal action so it was rolled back like any illegal action. – Joshua May 26 '20 at 20:01
  • It doesn’t have to be only that chance card - any with a cost > 50 will work. Are there any others? That could change the probability. – Tim May 27 '20 at 09:31
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No, the game given in the OP is not the shortest possible Monopoly game (using the American edition). As with the UK edition solution given by William Pennanti, a two-player game can be lost in a single turn with two rolls of the dice:

Player 1, turn 1, roll 1:

The player starts with \$1500 and rolls a 3 and a 3, advancing to Oriental Avenue. The player declines to buy the property outright and so it goes up for auction. The bidding ends with Player 1 buying the property for \$1500, leaving them with \$0.

Player 1, turn 1, roll 2:

The player rolls a 5 and a 6, advancing to Community Chest. The player draws the "Pay hopsital \$100" card. The player is bankrupt because they have no money, and even mortgaging Oriental Avenue for \$50 would not be enough to pay the fee.

The probability of this game:

Rolling 3–3: 1 in 36
Rolling an 11: 1 in 18
Drawing the right Community Chest card: 1 in 16
Total: 1 in 10,368, or 0.00964506…%

Psychonaut
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  • It doesn’t have to be only that community chest card - it can be any with a cost > 50. Are there any others? That could change the probability. – Tim May 27 '20 at 09:28
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A two player game can be lost in a single dice roll. As long as the first player lands on a deeded property on their first roll, they can allow the property to go up for auction. Now they just have to bid more money than they have, and their own bid will force them into bankruptcy. Game over.

This can happen so long as the first player lands on a property that can be bought on their first turn, which excludes rolls of 2, 4, 7, and 10, which have a combined probability of 13/36. Therefore, the first player can bankrupt themselves on their first roll in 23/36 cases (63.8% of the time).

See: https://boardgames.stackexchange.com/questions/39455/in-monopoly-what-happens-if-the-auction-winner-cannot-pay-his-her-bid

Nuclear Hoagie
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    Is it possible to bid more than you have?! – Dmitry Kamenetsky May 26 '20 at 14:18
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    @DmitryKamenetsky There's nothing that would prevent it - you just incur a debt that you cannot afford to pay, just like any other way of going bankrupt. In practice, the only way to prevent this would be to have each player calculate their total worth before each auction in order to set an upper limit for each player, which would be awfully cumbersome, and isn't mentioned anywhere in the rules. – Nuclear Hoagie May 26 '20 at 14:25
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    This isn't the only way to prevent it. In practice people manage to play monopoly all the time without bankrupting themselves. It is only necessary to perform the cumbersome "exact" calculation if you are bidding something close to your net worth. Otherwise it just needs a cursory rough check – Martin Smith May 26 '20 at 14:52
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    @MartinSmith regardless, it sounds like the rules do not prevent it, and do not allow for backing out of a bid once made. – Ben Barden May 26 '20 at 15:40
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Technically, the shortest Monopoly game can be achieved with

ZERO rolls

by simply

forfeiting the game before even starting to cast the dice.

legrojan
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In the https://scatter.wordpress.com/2010/05/30/the-shortest-possible-game-of-monopoly-21-seconds/ someone found a faster way that doesn't use auctioning (or other non-playing scenarios):

P1,T1
(same as above; 3 turns)
P2,T1
3-4 -> Chance -> Advance to Illinois
Action: Purchase Illinois (-$240, but effective -$120 because of ability to mortgage: total: $1380)
P1,T2
2-2 Park Place, purchase
6-4 Chance (+$200 for passing GO), advance to Boardwalk, purchase property and houses
P2,T2
6-6 Chance
Action: Pay $50 to each player (total $1330)
1-2 Boardwalk; GAME
8 turns instead of the above 9
Zizy Archer
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