# Winning Monopoly With Math!

I’m always on the hunt for tips to make myself a better puzzler and gamer. Sometimes you stumble across those tips in unexpected places.

For instance, I was reading, of all things, a book about mathematics and Christmas — The Indisputable Existence of Santa Claus: The Mathematics of Christmas — and inside, I found a statistical analysis of the best strategy for winning a game of Monopoly.

Yes, we’ve discussed this topic before, but even that previous deep-dive into the mechanics of the game wasn’t as thorough or as revealing as the work by Dr. Hannah Fry and Dr. Thomas Oleron Evans in this Christmas-fueled tome of facts and figures.

They started with a breakdown of how your first turn could go, based on dice rolls. This is the same breakdown as in our previous post, but with some important differences. For instance, they also considered the chances of going to jail after multiple doubles rolls.

Also, they covered the statistical impact of how landing on card spaces can affect where you land on your first turn. The Community Chest is a curveball, because of the possible sixteen cards, three will send you somewhere on the board: Go, Mediterranean Ave, and Jail.

A simple statistical analysis is complicated even further by the Chance cards — nearly half of the sixteen cards send you elsewhere: Go, Income Tax, St. Charles Place, Pennsylvania Railroad, Illinois Ave., Jail, and Boardwalk.

If you extrapolate forward from this point, you uncover some interesting patterns:

The orange property set benefits from all the ex-cons leaving their cells, and after their next turn the reformed criminals will likely end up somewhere between the reds and yellows… Illinois Avenue, with its own dedicated Chance card directing people to it, gets an extra boost, making it the second most visited square on the board.

The property that is visited least frequently is Park Place, where players spend just 2.1% of their time.

Check out this graph. This shows potential earnings from each complete color set, with the dotted line marking the point where your purchase of the property is canceled out by how much the property has earned in rent thus far. Everything above that is profit.

As you can see, blue and brown properties start close to the dotted line, because they’re affordable to buy and build on. The standouts on this graph are New York Avenue (which earns \$30 a roll up through thirty rolls statistically) and Boardwalk, which is an expensive investment, but pays off handsomely down the line, remaining the top earning spot past thirty rolls.

Of course, that’s only single properties, and you can’t build on single properties. Let’s look at a chart for full color set revenue:

Some of our previous findings change radically. Boardwalk’s rating drops significantly, because of Park Place’s relative infrequency of being landed on (as we mentioned above).

So which properties should you nab to give yourself the best chance of winning? Well, that depends on how long the game lasts.

The average game of Monopoly takes approximately thirty turns per player, so the larger the number of players, the longer the game will last.

So, for a two-player game, your best bet is to go after the light blue or orange sets, since they’re better in the short term, and the odds are in your favor if the game stays short.

In a three- or four-player game, the orange and red sets are better, because the game is likely to last a while.

And if five or more people are playing, you’re really playing the long game, so the green set becomes your best chance for success.

What about building on those properties? Well, Fry and Evans considered that as well. If you’re playing against multiple opponents and know you’ll be in for a long game, then you definitely want to buy and place houses. But don’t fear if the first house takes a long time to start paying for itself.

As it turns out, your best strategy is to put three houses on your properties as quickly as possible, because the third house is the fastest to recoup on investment. So once the three houses are in place on each property, you can rest for a bit and regenerate your bank before investing further.

And there you have it. Better gaming through mathematics! The only thing better would be, well, playing practically any other game.

Kidding! (But not really.)

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# Keep your Get Out of Jail Free card handy.

Recently, friend of the blog Chris Begley sent me an article about some interesting math facts. Most of them were about probability and how many people misinterpret the likelihood of various events happening based on bad assumptions about probability.

For instance, since you have a 50/50 chance of heads or tails when you flip a coin, it seems logical that if you flipped a coin ten times, you’d get heads five times and tails five times, whereas in reality, it’s common to have runs of one result or the other that fly in the face of that simple 50/50 assumption.

But that wasn’t the fact that caught my eye. I thought it was much more intriguing that not all spaces on a Monopoly board have an equal likelihood of being landed on.

And that can affect how you play. For instance, if you believe each spot has an equal chance of being landed on — 1 in 40, given the 40 squares on the board — you might opt to buy all three colors in a given area to give yourself a 3/40 chance (7.5%), or you might go for all 4 railroads to give yourself a 4/40 chance (10%).

[A breakdown of spaces and likelihood of landing, based on the UK version.
(Chance and Community Chest cards differ between UK and US versions,
though probabilities for spaces in the US version are quite similar.)]

But that’s not how Monopoly actually works. Some spaces are far more likely than others. This is partly due to rolling two dice every time you move (which makes 6, 7, or 8 spaces the most likely results). There are also rule cards that make some squares more likely than others.

The most common space to land on is Jail (due in no small part to the Go to Jail square and where Chance and Community Chest cards send you). The most common PROPERTY to land on is Illinois Avenue, followed by B&O Railroad, Tennessee Avenue, New York Avenue, and Reading Railroad.

[A breakdown of space probabilities for the US version of the game.]

On the flip side, Mediterranean Avenue is the least likely to be landed on, followed by Baltic Avenue, Luxury Tax, Park Place, and Oriental Avenue. (Again, the Go to Jail square comes into play, as Park Place is seven squares away and the most common dice roll is 7.)

I like that a little properly applied math might make you a better Monopoly player. (Though if I’m going to walk the Boardwalk, I’d rather be playing The Doom That Came to Atlantic City.)

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