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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Better Gaming With Math and Statistics!

[Image courtesy of ThreeSixtyOne.gr.]

Statistical analysis is changing the world. The wealth of available data on the Internet these days, combining with our ever-increasing ability to comb through that data efficiently using computers, has spawned something of a golden age in data mining.

You don’t need to look any further than the discovery of Timothy Parker’s plagiaristic shenanigans for USA Today and Universal Uclick to see how impactful solid analysis can be.

But it’s also having an impact on how we play games. Statistical analysis is taking some of the mystery out of games you’d never expect, making players more efficient and capable than ever.

We discussed this previously with the game Monopoly — specifically how some spaces are far more likely to be landed on than others — and today, we’re looking at two more examples: Guess Who? and Hangman.

Guess Who? gives you a field of 24 possible characters, and you have to figure out which character your opponent has before she figures out the identity of your character. Usually, if you end up with a woman or someone with glasses, your odds of winning are low, because some aspects are simply less common than others.

But is there an optimal way to pare down the options? Absolutely.

Mathematician Rafael Prieto Curiel has devised a strategy for playing Guess Who?, based on an analysis of the notable features of each character, breaking it down into 22 possible questions to ask your opponent:

Based on this data, he has even created a flowchart of questions to ask to maximize your chances of victory. The first question? “Does your person have a big mouth?”

Yes, not exactly a great first-date question, but one that yields the best possible starting point for you to narrow down your opponent’s character.

It’s certainly better than my first instinct, which is always to ask, “Does your person look like a total goon?”

Now, when it comes to Hangman, the name of the game is letter frequency. Just like a round of Wheel of Fortune, you’re playing the odds at first to find some anchor letters to help you spell out the entire answer.

But, as it turns out, letter frequency is not the same across all word lengths. For instance, E is the most common letter in the English language, but it is NOT the most common letter in five-letter words. That honor belongs to the letter S.

In four-letter words, the most common letter is A, not E. And it can change, depending on the presence — or lack thereof — of other letters.

From How to Win Games and Beat People by Tom Whipple:

“E might be the most common letter in six-letter words, and S the second most common, but what if you guess E and E is not in it?” In six-letter words without an E, S is no longer the next best letter to try. It is A.

In fact, Facebook data scientist Nick Berry has created a chart with an optimal calling order based on the length of the blank word.

For one-letter words through 4-letter words, start with A. For five-letter words, start with S. For six-letter words through twelve-letter words, use E. And for words thirteen letters and above, start I.

Of course, if you’re the one posing the word to be guessed, “jazz” is statistically the least-likely word to be guessed using this data. And your opponent will surely hate you for choosing it.


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Let’s Make a Deal!

It’s a scenario every game show fan knows well. You’ve got three doors to choose from, and one of those doors will open to reveal a fabulous prize.

After you’ve made your choice (let’s say Door #2), our affable host Monty Hall plays Devil’s Advocate by opening one of the doors you didn’t choose (let’s say Door #1), revealing a goat or other lackluster result.

And then, Monty offers you a chance to change your mind. Will you stick with the door you initially chose, or will you switch to the other unopened door (Door #3)?

The average player sees two choices, Door #2 and Door #3, which on the surface sounds like a 50/50 shot, a coin flip. So would it surprise you to learn that people who switched from one door to the other doubled their chances to win the fabulous prize?

This is known as the Monty Hall Problem, an example of how statistics aren’t always what they seem, and it has puzzled people for decades.

It’s counterintuitive, isn’t it? I mean, you have two choices, so the odds should be 50/50. But you’re forgetting that third door that Monty eliminated. That third door makes all the difference, statistically speaking.

Let’s break it down. Your initial choice is between 3 doors, meaning you have a 1 in 3 chance of picking the correct door, and a 2 in 3 chance of picking the wrong one.

When Monty opens that other door, the odds haven’t changed. Only the number of options available has changed. Your door is still a 1 in 3 chance of being correct and a 2 in 3 chance of being wrong. But the remaining door now has a 2 in 3 chance of being correct!

So what appeared to be a coin flip between sticking with your choice and switching is now heavily weighted toward switching!

There have been several real-world tests of the Monty Hall Problem, and all of them have consistently shown that the people who switch were twice as likely to open the winning door!

The real puzzle here is how we fool ourselves. We take the numbers at face value — 3 doors become 2 doors, so a 1 in 3 chance becomes a 1 in 2 chance — and actually hurt our chances with those seemingly simple assumptions.

Being able to reconsider your assumptions is a major tool in the puzzler’s solving kit. Plenty of tricky crossword clues depend on you associating the clue with one thing, when the answer is something quite different.

After all, if you saw the clue “Unlocked” for a four-letter entry, you’d probably try OPEN before you tried BALD. Clever constructors are counting on that.

So be sure to remember Monty Hall and his three-door conundrum the next time you’re stumped on a puzzle. Maybe the answer is as simple as trying another door.

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