Winning at Monopoly with Math

I am a complete fool for articles like _Business Insider_’s [“How To Use Math To Crush Your Friends At Monopoly Like You’ve Never Done Before”][bi]. I like that the math involved can range from the simple — how the distribution of dice rolls affects where most people will land, given a particular starting point — to the complex. If the slide decks length puts you off — there’s sixty plus slides in there!, you can always scroll to the end and read the half dozen concluding slides that tell you what you should do. But, really, the fun is in the careful working through of the numbers.


On Games and Rules

We were talking with our daughter today at lunch about what rules are: that there are different kinds of rules, or structures, with different levels of importance. “Don’t play with fire” is a different kind of rule than “your coat must be navy or gray” (one of the rules at her school). The conversation got around to rules in games, which is where it got interesting.

Now, the first thing to keep in mind is that the term *games* in her and her friends’ parlance covers a wide range of activities, something she herself made clear:

> I understand that when you are playing *Chutes and Ladders*, for example [yes, she said “for example”], that you can’t really change the rules, but when you are playing a mind game, a game you’re making up, then you should be able to change them.

“Mind games” here means what my wife calls “pretend play.” It can cover a lot of territory: building things with Legos (and then playing with them), playing with various kinds of figures (stuffed animals, fairies, Lego people), as well as dressing up and role playing.

We have been trying to encourage our daughter to be more “open” when playing with others. She’s not terrible. I’ve heard almost all her playmates be adamant that they want to play one thing, or play one way, and not another, but we are trying to build a foundation for her that will serve her well in the future. We have framed it in the past in terms of improvisational theater’s concept of always saying yes, but today Lily proffered her own metaphor:

> It’s like you’re in a room, and there’s a door, or a thousand doors, really, and you want to see what’s behind those doors but the other person has the key, and they need to give it to you.

Some of the examples she drew upon were based on her relationship with her best friend, who is a bit more procedurally-oriented — whether that’s a function of a stage or just how he’s built is not yet clear — and who tells her no, or that her way is the wrong way, or that there is only way to play a particular game. And so we are trying to equip with her not only with ways for her herself to be more open but to make it possible for others to be open as well. It’s fascinating to watch her play. On the one hand, she is an extremely creative person, quick to imagine and re-imagine whatever world within which she is operating. On the other hand, she can too quickly get fixed on her vision of things and thus negate the creativity of others. How to navigate the span of multiple visions is a difficult task. I don’t know of any easy solution, but beginning that conversation with her is one way for us to think about it. (I would argue that becoming a parent was one of the best things ever for my teaching.)

QR Code for JL.o

So I went to one of those on-line QR code generators and generated a QR code for JL.o’s URL:

QR Code for

Not much to say about it. I suppose I could put this on a business card, but is scanning the QR code really any easier than typing the ten letters of my name plus the four characters of the dot and the `org` or `com` extension? I’m guessing that the QR codes are really much more interesting in other contexts: what if each code were a clue for a scavenger hunt or a line of a poem? That could be a fun thing to set up.

Happy Towel Day! The BBC hosts a [Hitchhiker’s Guide game][hhgg] (Warning: Flash).


There are two games whose game play, or at the least the idea for it, inspires me. One is the all audio [Papa Sangre]( and the other is The Unfinished Swan:

I marvel at the imagination of the two, and I wish I were the one who came up with the idea. In sum, they inspire me.

[The King’s Gambit has been solved.]( According to ChessBase:

> Fifty years ago Bobby Fischer published a famous article, “A Bust to the King’s Gambit”, in which he claimed to have refuted this formerly popular opening. Now chess programmer IM Vasik Rajlich has actually done it, with technical means. 3000 processor cores, running for over four months, exhaustively analysed all lines that follow after 1.e4 e5 2.f4 exf4 and came to some extraordinary conclusions.

Rajlich’s response to the question of solving a chess gambit is fascinating:

> 1.e4 e5 2.f4 exf4. We now know the exact outcome of this position, assuming perfect play, of course. I know your next question, so I am going to pre-empt it: there is only one move that draws for White, and that is, somewhat surprisingly, 3.Be2. Every other move loses by force.

> *CB: How can you have worked that out, aren’t there gazillions of possible continuations?*

> Actually much more than “gazillions” – something in the order of 10^100, which is vastly more than the number of elementary particles in the universe. Obviously we could not go through all of them – nobody and nothing will ever be able to do that. But: you do not have to check every continuation. It’s similar to Alpha-Beta, which looks at a very small subset of possible moves but delivers a result that is identical to what you would get if you looked at every single move, down to the specified depth.

> *CB: But Alpha-Beta reduces the search to about the square root of the total number of moves. The square root of 10^100, however…*

> Yes, I know. But think about it: you do not need to search every variation to mate. We only need to search a tiny fraction of the overall space. Whenever Rybka evaluates a position with a score of +/– 5.12 we don’t need to search any further, we have our proof that in the continuation there is going to be a win or loss, and there is a forced mate somewhere deep down in the tree. We tested a random sampling of positions of varying levels of difficulty that were evaluated at above 5.12, and we never saw a solution fail. So it is safe to use this assumption generally in the search.

This notion of a “problem space” or a “solution space” is something worth exploring more, both for what it might do for humanities research but also for possible research into as a phenomenon itself.

Check out this year’s Humble Bundle. (For those not in the know, the Humble Bundle is a collection of great games that are cross-platform and DRM free. Every year there’s at least one game in the bundle that makes it worthwhile, especially when the developers split the proceeds with charities.) Even if you don’t want to buy it, click the link and enjoy the promotional video. Terrible voice impressions done right.