We know that Aunt Bee is not Barney's aunt, because he says so in Aunt Bee's 
Invisible Beau.  Barney says to Andy, "If Aunt Bee was my aunt ... I'd drive 
straight over to Mt. Pilot, I'd get that Hendricks out in the alley, and 
SOCKO!"  

It is true that in certain areas, particularly the American South, the term 
"cousin" is used somewhat loosely.  If we assume that Andy and Barney are FIRST 
cousins, then that means one of Andy's parents is a sibling of one of Barney's 
parents (and A and B share a common grandparent).  The only combination that 
fits all the facts we are given in various episodes is, Barney's father was the 
brother of Andy's mother, but that is neither here nor there.  If one of 
Barney's grandparents was a sibling to one of Andy's grandparents (and they 
shared a common great-grandparent, in other words one of A's parents was a 
first cousin of one of B's parents) then Andy and Barney would be second 
cousins.  So we really don't know how closely Andy and Barney are related, but 
we do know they are cousins, because either Andy or Barney says so in three 
different episodes.

(This is about to get complicated, so do try to keep up if you can, because you 
will need it later on).  Now, if you have a cousin and that cousin has a child, 
that child is your cousin once removed (not your second cousin).  If that child 
produced a child, the new child would be your cousin TWICE removed.

 If you have a child yourself and your first cousin has a child, then those 
children are second cousins to each other.  If THOSE two children each produced 
a child, then those children would be THIRD cousins to each other, etc.

So even if Andy and Barney are second cousins or third cousins or fiftieth 
cousins or 200th cousins, it makes no difference how distant, Opie is still 
Barney's cousin once removed, because he is the son of Barney's cousin Andy.

Paul Mulik

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