The Proof of at Least Seven Rolls

This following is a mathematical-style proof that any game of Monopoly MUST consist of at least seven rolls (assuming that each player is trying to win and not breaking the rules). I chose to attack the problem by eliminating the possibility of winning a game in six rolls…one color group at a time.

Let’s start by looking at the dark purples, which should be pretty easy:

1. Dark Purples

Med. then Baltic -> Shortest Acquisition (Player 1): 2 turns, 4 rolls (7->adv to bw | 2, 2, any)
Shortest Required Pre-Hit (Player 2): 1 turn, 1 roll (7->adv to go)
Max Damage with Remaining Roll (Player 2): -450 (hotel on baltic)
MAXIMUM NET DAMAGE AT 6 ROLLS -250

Baltic then Med. -> Shortest Acquisition: 2 turns, 4 rolls (3 | 4->adv to bw, 2, any)
Shortest Required Pre-Hit (Player 2): 1 turn, 1 roll (7->adv to go)
Max Damage with Remaining Roll (Player 2): -450 (hotel on baltic)
MAXIMUM NET DAMAGE AT 6 ROLLS -250

We can quickly see a pattern forming here!

In any case where player one is required 4 rolls (and hence 2 turns) to complete a monopoly, the maximum times that player two can land on a built-on property is once. This is because player two must expend 1 roll (minimum) before the opposing monopoly is completed, which at most can only cause $150 in damage (income tax space). This leaves one roll to cause the remaining $1350 in damage!

There are only three properties that can pack that much punch in one shot: Pennsylvania Avenue (hotel=1400), Park Place (hotel=1500), and Boardwalk (3 houses=1400).

This simplifies the task of disproving 6 rolls for the remaining monopolies considerably: we must show that it requires at least 4 rolls to complete any monopoly through the yellows…

<<<< BEGIN FOUR-ROLLS PROOF >>>>

I have already shown that it takes at least 4 rolls for the dark purples (see above). [ DARK PURPLES ]

All of the remaining monopolies (through yellow) contain adjacent properties, which means that a chance or community chest card must have a way to send a token to one of the adjacent properties to complete a monopoly in three rolls! [ STATEMENT A ]

From statement A, we can immediately eliminate the light blues, the light purples, and the yellows from contention, since none of their adjacent properties can be reached via card. [ LIGHT BLUES, LIGHT PURPLES, YELLOWS ]

Furthermore, each one roll must be used to obtain exactly one of the three properties of a given monopoly group, since you can only end up on one property per roll (whether via card or not). This means that player one’s very first roll MUST obtain a property of the desired group. [ STATEMENT B ]

From statement B, we can eliminate the oranges from contention, since they are neither within 12 spaces of GO nor directly reachable via any card when starting at GO. [ ORANGES ]

The only red property reachable on the first roll is Illinois, but it is then impossible to reach another red on the following roll (too far away), so the reds are eliminated from contention. [ REDS ]

<<<< END FOUR-ROLLS PROOF >>>>

The easiest way to eliminate the greens is to look at building costs. The LEAST amount of money required to obtain a hotel on Pennsylvania Avenue is 300+300+320+2600 = $3520! It is clearly impossible to obtain anywhere near the amount of money required in four rolls (or even many more for that matter). [ GREENS ]

This leaves the dark blues. Can they be obtained in less than four rolls?

Since Boardwalk is two spaces away from Park Place, reaching Park Place in two rolls is NOT good enough (assuming we try to get Park Place before Boardwalk). This is because reaching Boardwalk requires rolling doubles once we are on Park Place, and a “waste roll” is needed afterward.

And since it is simple to see that Park Place is untouchable in one roll from GO, we cannot do better than four rolls this way.Let’s look at the opposite property order.

We can reach Boardwalk in one roll, but can we reach Park Place in two rolls from Boardwalk?

Since there is no card that can advance us to Park Place, our final roll must LAND us there.

Therefore, we must find all “advanceable” spaces within 12 of Park Place, which leaves us with Water Works and B&O Railroad. Both spaces require landing on the chance space near the reds, however, which is not within 12 of Boardwalk.

So, we have shown that it requires at least four rolls WITH the help of Advanced to Boardwalk to complete the dark blue monopoly. And since this spans two turns, we are again left with ONE roll for player two to land on Park Place or Boardwalk.

And since Advanced to Boardwalk has already been used by player one, this is impossible! [ DARK BLUES ]