How far could you drive with absolutely no signs?

[hotdogPi]

I'm confident I could end up anywhere.  Might not ever find home again though.

You will.

wikipedia.org/wiki/Random_walk

To visualize the two-dimensional case, one can imagine a person walking randomly around a city. The city is effectively infinite and arranged in a square grid of sidewalks. At every intersection, the person randomly chooses one of the four possible routes (including the one originally traveled from). Formally, this is a random walk on the set of all points in the plane with integer coordinates.

Will the person ever get back to the original starting point of the walk? This is the 2-dimensional equivalent of the level crossing problem discussed above. In 1921 George Pólya proved that the person almost surely would in a 2-dimensional random walk, but for 3 dimensions or higher, the probability of returning to the origin decreases as the number of dimensions increases. In 3 dimensions, the probability decreases to roughly 34%.

[kphoger]

But would he run out of gas money first?  That's the real question.

[TechZeke]

All I'd really need to do is get to a major interstate(which is easy to figure out living in a metro area), and I could probably figure what city I was going through based on how far I drive (odometer) and the surrounding scenery. I'm from SoCal and currently live in San Antonio, so if I went down Interstate 10 towards the west I'll eventually start recognizing places. I could probably navigate to exactly where I lived before moving.

East or South of San Antonio would be more tricky for me, though.

[SP Cook]

I'm really pretty confident.  I tend to know enough about rivers, city skylines, particular buildings, college towns, lakes, and the general physical geography of the continent to dope out pretty much where I am along any interstate.  I also certainly know which direction the sun rises and sets.  I believe I could free hand an 85% or better accurate map of the entire interstate system, and do equally well with the local major roads in my region. 

I believe there is certainly no major league level city I could not drive to from any other random spot in the country.

[US 41]

But would he run out of gas money first?  That's the real question.

No chance. I get 40 mpg.