## 22 May 2010

### PAC-MAN Turns Thirty

It's the thirtieth anniversary of Pac-Man, and there seem to be articles about it all over the internet. Is it too late for me to jump on the bandwagon?

I've been trying to think of something to say about the mathematics of Pac-Man, and it is not obvious that there is much of anything very mathematical about it. I never played it much, and was never particularly good at it, with space shooter games like Asteroids getting most of my attention (and quarters). I do recall that the good players had patterns they followed to get through each level, and that a well executed pattern would always work. As a kid who liked writing his own games on my then state-of-the-art Apple II+, I always assumed that patterns in Pac-Man worked because the game used simple psuedo random number sequences that always generated the same sequence of events in response to the player's actions. Well it turns out that idea was giving the games too much credit: I did a bit of research today and quickly found James Pitman's Pac-Man Dossier, which explains how the monsters give chase in a completely deterministic fashion. {And aside to James Pitman, "Well Done!")

So if there is nothing random about Pac-Man, what is left? I thought about trying to frame it as sort of a differential game like The Homicidal Chaffeur, where the driver of a not-too-maneuverable car tries to run down a nimble but slow pedestrian. That's a difficult approach though, and just seems like way too much trouble for a game like this. Instead I tried to think about ways to simply Pac-Man down to its basic elements ...

And those basic elements are choices. Every toggle of the joystick in Pac-Man is a choice, and it leads to one of three outcomes:

1. You make a good choice, maybe get to eat some dots (bringing you closer to finishing the level), and live to make another choice.
2. You make a neutral choice, which gains you nothing, but you don't get eaten by a ghost and live to make another choice.
3. You make a bad choice and get eaten by ghost.

This make Pac-Man a sort of puzzle. Outcomes one and two describe all puzzles - you simply keep working at them until you figure it out. The third outcome of Pac-Man is a sort of timer, where a bad choice will lose a life, and maybe the game. The thing about puzzles is that once you figure them out, once you know the pattern, they are easy to solve. Pac-Man is a fancy sort of puzzle with flashing lights and sound effects, but it's a puzzle at heart.

So congratulation to Pac-Man, the game that took the world by storm for thirty years ago, and still continues to entertain us.