By overthinking I mean spending a lot of time modeling the problem.
I wrote a little code to generate random Battleship boards, and counted where each of the ships appeared. I did this billions of times to get good statistics, and what I ended up with is a little interesting. You can see the results for yourself over at my
results exploration page by changing the radio buttons for the ship you are interested in, but I have some screen caps below.
This is an example of the failure of the linear model. All the linear model knows is that in the spots nearby misses there is a lower probability of the ship being there, but what it doesn’t know to do is look at the arrangement of misses and check to see whether there is any possible way the ship can fit. This is a nonlinear effect, involving information at more than one square at a time.
It is these kinds of effects that this theory will miss, but as you’ll notice, it still does pretty well.
I’m wondering if it does as well against human opponents, who would not place the targets randomly.