OK, folks, here it is, my solution for the riddle and i found out on my own. And i cannot remember when i used the terms "equal" and "different balls" with so countless numberings this often the last time. ^_______^;;;;;;
I divided it into three spoilers so that, if maybe somewhere out there, someone
, lonely and curious, someone like me, who just needs a hint to do the rest alone.. uh.. can do so. I mean, you should really try it. It's F.U.N.!
Divide the balls into 3 groups A, B and C á 4 balls and weight groups A and B.
a) The balls of group A and B are equal: the sought-after ball is in goup C.
b) The balls of group A and B are different: the sought-after ball is in wether in group A or B.
case a) Take 3 balls of group C and replace them with 3 from group B on the balance.
a1) All balls on the balance are equal, again. The ball we're looking for is the left over one: now we already now which ball it is, but not wether if it's heavier or lighter.
a2) The balance shows a difference, so the sought-after ball is 1 of the 3 replaced balls: we don't know which ball exactly it is, but we know if it's heavier or lighter.
case b) Take 3 balls of group A from the balance, replace them with 3 balls from group B and fill group B up with 3 balls from group C (we know they're equal because the "right" ball is already on the balance).
b1) All balls are now equal. The ball we're looking for is 1 of the 3 from group A: as in result a2) we don't know which ball exactly it is, but we know if it's heavier or lighter.
b2) The weights are still different and the balance shows the same as in the first weighting: the 3 balls from group A we took out are equal to those from group B which replaced them and those again are equal the "new" ones from group C. The ball we're looking for must be 1 of the 2 from group A or B which haven't been exchanged. So we know it's 1 out of 2 and don't know if it would be heavier or lighter, so we have to keep in mind which one of the 2 balls is heavier (and which one lighter).
b3) The weights are still different, but have "changed sides": the ball we're looking for is 1 of the 3 from group B that have been moved from one side of the balance to the other. Again we don't know which ball exactly it is, but we know if it's heavier or lighter.
Case a2), b1), b3) Put 2 of the 3 balls on the balance.
I) Both balls are equal: the ball we're looking for is the third one. And according to the previous results we kow if it's heavier or lighter.
II) The balls are different: 1 of the balls on the balance is the sought-after and according to the previous results we kow if it's the heavier or lighter one.
Case a1) Take the already known to be different ball and weight it against an "equal one".
Result III): now we know if it's heavier or lighter.
Case b2) Take heavier one of the 2 balls and weight it against an "equal ball".
IV) The balls are equal: the left over ball is it. And it's lighter.
V) The balls weight different: it is the heavier ball. And it's err.. heavier.
You may now excuse me, I'll have a Martini or three to compensate my brain's rotation.