So I came across this fun online game yesterday called Tower Bloxx (via Jay is Games). It's a pretty simple game, but hard to master: you have to build towers, and you place segments of the tower from a swinging crane at the top of the screen. Since the crane is swinging, it makes it difficult to position segments precisely on top of one another, which is where the game playing comes in. The straighter the tower you make, the higher the population you get in the tower. Your citizens come in on flying umbrellas, too!

There's a "Build City" mode in Tower Bloxx, which lets you place towers you make in a city block, subject to certain rules. There are four types of towers, residential, commercial, industrial, and luxury, increasing in potential population in that order.

Here are the rules: blue residential towers can be placed anywhere. Red commercial towers must be placed adjacent to at least one blue tower. Green industrial towers must be placed adjacent to one blue and one red tower. Yellow luxury towers must have one blue, one red, and one green tower neighbor. (In the game, you must place towers according to these rules, but you may then demolish the required neighboring towers without a problem. I'm ignoring this loophole.) Here's a sample city block that follows all the rules:



Now, aside from the fact that it's strangely amusing and compelling making towers from a swinging crane, here's the real question that's been bothering me.


There are multiple ways to interpret this question. Feel free to answer whichever one interests you the most. Certain variations I can come up with:

• Maximize the number of yellow towers.
• Yellow towers are worth 4 points, green towers 3 points, red towers 2 points, and blue towers 1 point. Maximize the number of points in your city.
• Maximize the population in your city. (Of course this depends on your skill at creating towers, and it also depends on the maximum population each tower can hold. I don't know the maximum populations, so I can't provide the necessary information. If you can figure these numbers out, feel free to use this interpretation.)

If you find what you think is an optimal placement of towers in the city block, can you prove that it is optimal according to whatever interpretation of the question you are choosing? (I have my own optimal placement already, but I'm not sure I can prove it.)