# Tag Archives: Math

## Monopoly Math Theory

Like I said in the last post, I spent some time working on the actual probabilities for each square in Monopoly. First thing first: Chance/Chest cards suck.

About half of the Chance/Chest cards can be classified as “move cards”, as they instruct the player to move to a square. Lets say you draw the “Go to jail” card from Chance and move to jail. That card now goes to the bottom of the stack, so the probability of anyone getting that card is now 0 for the next 16 Chance draws(there are 17 Chance cards), and 1 on the 17th draw. Therefore the probability of the squares is affected by the initial order of the C/C cards, since its common for not all the cards to be drawn during shorter games.

The number of players also affects things; obviously more players will go through the C/C cards faster, but it also complicates the “get out of jail” cards. I never pay the \$50 to leave jail, therefore I either need a “get of jail” card or a doubles roll. If two other players already have the freedom cards, that means when I leave the jail it’ll be on a doubles roll. This means that I can’t land on squares 3,5,7,9, or 11 on a jail exit move. Though the effects are probably negligible in the grand scheme of things, they’re definitely present.

The only other factor, aside from standard two-dice probabilities, is that rolling three doubles in a row immediately moves the player to jail.

With all these factors considered, solving things gets pretty complex. Example: You’re on Water Works, and roll double 4s. You’re now on Chance. This Chance turns out to be a “Go Back 3 Spaces” card, putting you on Community Chest. The Community Chest card is an “Advance to Go” card. Since you had double 4s, you get to roll again; doing so gives you get double 1s, putting you on Community Chest. Go To Jail! At the end of it, you’ve visited four different squares on a single turn. On the other hand the same outcome could have occurred if double 1s were rolled from Water Works, as this would have put the player on the “Go To Jail” square.

Such complex move sequences are very rare, but are certainly possible. Therefore instead of trying to come up with an equation, I decided to calculate the potential pathways to each square and the probability of said pathways. I also took some liberties with C/C cards and simply treated them as a random draw. I’ve attached the final Excel sheet as usual:monopoly_final_paths

And, a screenshot of the final ordering and game board composition:

There are definitely some differences between the theory and the experimental results, but the general trends are fairly similar. Railways and orange still appear as the most popular properties. There’s a lot more “smoothness” in the theory though, especially in the middle range properties. The one last thing to do with this data would be to factor in costs of buying and rent, to see which are the best price wise. I’m sick of this game and data though, so maybe another day. 🙂

T

## Monopoly Math

So it was family game night tonight, and we decided to play Monopoly. Because the game can take a while, we generally set a time limit, something between 1.5 to 3 hours depending on how late we start the game. My strategy for playing is pretty straight forward: collect as many railways and utilities as possible, and go for orange, yellow, and dark blue property groups. It tends to work pretty well for me, so I decided to do some data recording this game. I tallied up every squared landed on in Excel, including Chance/Chest cards like “Go directly to jail”, “Advance to boardwalk”, “Go to go”, and so on. We don’t play with any custom rules that affect moving, so the data reflects standard Monopoly procedures.

Unfortunately the game only lasted around 1 hour 20 minutes (I had a lucky start & early win), so the dataset isn’t very big. I wasn’t about to ask for a rematch though and we probably won’t play Monopoly again for a week or two, so I decided to work with what I had. Here’s the full organized data and analysis; click for full-size of course:

Haha, go Orange and Railways! I knew I was backing the right horse. 🙂 I think the reasoning behind the high frequency of orange landings has to do with the fact that it’s outside of the jail. Some of the most probable rolls on two dice (6,8,9) will land you on orange. There are also Chance/Chest cards that take the player to the pink squares right before orange, and the railway right before orange. Railways/corners are obviously going to be popular because of the aforementioned cards, and because they’re well spread around the board. Same goes for Chance/Chest, which combined make up 6/40 or 15% of the game board. Also, I find it funny that Tax squares received the least hits.

There’s a lot more I could do with this as far as comparison to theory goes, but the problem is calculating theoretical probabilities for each square. Since there are a lot of different routes to get to squares, with the possibility of going to jail, getting a Chance/Chest card that moves the player, etc things get complicated. I do intend to sit down and work through the math later tonight/tomorrow, so I’ll post again with an update at that point. And, as always, if anyone wants the data to play with: Monopoly.xlsx. The first Sheet is the raw data, and the second is the analysis page shown above.

T

## A Penny for Your Thoughts?

I’ve been saving all my cent coins in a jar for the last few months, with the purpose of counting up the years on them to see what sort of distribution there is. Today I counted my jar of 185 pennies, and the results were not entirely what I expected. I should mention that these are Canadian coins, as I am Canadian after all.

I’d gone into this assuming that 2010 would have the most, then 2009, then 2008, and so on. Some sort of exponential function would make sense. I wasn’t sure about 2011 because I don’t know when the mints make coins, or if they’re consistently minted throughout the year or if there’s one day/week/month where the penny machines get cranked up to the 11 setting.

To start with, here’s the raw data in case anyone else wants to take a look: raw_penny_data. In addition to the year:count data it also includes a cumulative count column and a weighted average column. Feel free to do whatever you want with it, citing it if you use it somewhere else of course.

And here’s the resulting graph from Excel. I’ve add in some labels with Photoshop on some of the relative peaks:

I knew 2010 would be high, but holy shit that’s a big peak. And what’s up with 1994? I’ve heard before that after 5 years 50% of pennies from a year have dropped out of circulation. 1994 has held in there pretty well, and 1995 is right up there too. I wonder if there was a surge in penny production over that timeframe? Also, 1987 looks relatively large when compared with the other counts around it. The overall trend looks exponential though, as predicted.

After seeing the results, I think I’m going to continue saving my pennies and add the new data at the end of the year, to see what a larger sample size will produce. I’m curious if the 1994 spike will become absorbed with a larger sample size, or if there’s actually some reason for there to be more pennies from that area.

Also, not relating to the data, I found a single penny that completely lacked a date. I’ve never seen that before. It looked like a 1995-2005ish penny based on the level of wear and tear, and with the exception of the lack of a date it’s identical to all the other pennies. I couldn’t find any information on these type of pennies either, maybe it’s some sort of defect? At any rate, it’s not a common feature on Canadian coins so it might be worth something. I intend to hang onto this dateless penny.

Here’s a photo of the full penny pile:

And the dateless penny. The date is normally right about the CA in Canada, but is nowhere to be found on this coin:

That’s all for now.

T