An intuitive reason why the "Rule 30" CA is random/pseudorandom? I'm a little bit hesitant to ask this here, so please notice the tag.  My hope is that someone will have a more satisfying answer than what I've heard before...
A long time ago I read (perhaps 'browsed' is a better word) Wolfram's "A New Kind of Science".  There are many many references to the "Rule 30" CA - http://mathworld.wolfram.com/Rule30.html.  However, no intuitive reasoning for it's random/pseudorandom behavior is provided prior to a digression to the importance of the discovery.  I was recently reminded of this when I heard that Mathematica (a program I use quite frequently) uses certain outputs from this CA as its random number generator.  
So my question is - beyond 'numerical phenomenology' is there an intuitive understanding why Rule 30 should behave in this random/pseudorandom manner?
 A: Well, here are some thoughts of mine.
You want triangular shape, so you fix 000 -> 0 ; 100 -> 1 ; 001 -> 1.
Now you don't want symmetry, so the only way for you is to assign different values to 011 and  110, say 011 -> 1 and 110 -> 0.
Now we kind of want lots of 111s to disappear fast, which is kind of required for random generators, so 111 -> 0.
We're left with choices for 101 and 010, of which there are 4: 26, 30, 58, 62. Chances are at least one of the choices will be interesting. Indeed, 30 looks pseudorandom, while others do not. 
A: If you look at the results of elementary cellular automata from mathworld, most of them seem to have some kind of symmetry.  (I don't want to make "symmetry" formal here; what I mean is that you get nice patterns of some sort.)
But I suspect that in general, the results of cellular automata are psuedorandom.  (I haven't looked at this too closely.)
So if I had to guess, I would say that the answer is just that most CAs on small neighborhoods are "nice", and most CAs on large neighborhoods aren't.  Rule 30 is not sporadic, but it's the first example of a family that eventually predominates.  
A: As I understand it Mathematica in fact uses rule 30 to generate pseudorandom numbers by just reading down a single column: so what it's exploiting is not just that rule 30 behaves pseudorandomly but that columns look more or less like strings of independent uniform random bits.  If we label the four cells on which the rule is defined

a b c
  d

it seems a lot easier to achieve this if (b,d) takes on all its possible values equally often, as rule 30 does.  
In fact, the stronger property is true that rule 30 can be run leftwards.  a is a (total) function of (b,c,d), therefore any two columns of cells can be completed uniquely to a halfplane extending leftwards satisfying rule 30.  This is suggestive at the very least -- if more things could appear to the left of some columns than others, the columns probably wouldn't be nice and uniform and all that.  
