Among 11 coins of the same denomination, 1 is know to be counterfeit. Using a simple 2-pan scale or balance, how would you determine in 2 weighings if the fake coin is heavier or lighter than a real coin?

Take the 2 heaviest, and then weigh them. Take the two lightest, and then weigh them. After this you'll know which one is fake.

Not quite.

Put 5 coins in each pan.. (in theory) they should balance out.
Replace any coin in either pan with 11th coin and scale should once again balance out. If not.. fake coin is of diff weight.

Very good idea but it still doesn't determine if the fake coin is heavier or lighter.

I guess I didn't read the question well enough.

Facts: Eleven coins are in front of me. One of the coins is a counterfeit. I don't know which coin is the counterfeit coin, but I do know that the counterfeit coin is of a different weight.

Question: Using a simple 2-pan scale or balance, how would I determine in 2 weighings if the fake coin is heavier or lighter than a real coin?

Answer:

I'll put 5 coins in each pan.

scenario 1: The pans will balance out because all the 10 coins in the pans are real. Therefore, the 11th coin is the fake coin. I'll take all the coins out of the pans and weigh the fake coin against one of the real coins and I will know if the fake coin weights more or less than a real coin.

scenario 2: One of the pans will weigh more than the other. Therefore I will know that the 11th coin is a real coin. Selecting to continue my process of elimination with the heavier pan, I'll take the five coins in the lighter pan and set them aside. I'll take the 11th coin and the 5 coins in the heavier pan (6 coins in total) and put three in each pan.

scenario 2.1: The pans do not balance out, letting me know that the fake coin must be in this heavier group of coins and therefore is HEAVIER than a real coin.

scenario 2.2: The pans balance out. Therefore I will know that the fake coin must be in the lighter group (the 5 I set aside). So the counterfeit is LIGHTER than a real coin.


So the real question is... From scenario 2.2, in exactly two more weighings, how can you tell exactly which coin is the counterfeit coin?

my brain hurts 

First determine which is the fake coin. Then weigh the fake coin against one real coin and you'll get your answer.

Without even weighing the coins I can tell which is a one is a counterfeit. Anyone can tell me how I would know? Or what process I would take to find out.?

i would  utilizing it.

if the store owner spots it, then it's countefeit.

if the store owner doesn't, then it's not. 

Here's what I think:

I'd add another coin to the bunch to make it 12.  I'd know that my new coin was real since I selected it.  Now that I have twelve coins, I'd put 4 in each side of the scale (8 coins out of the 12).  Hold the other 4 in my hand.

1.  If the scale balances, then that means the fake coin is one of the ones in my hand.  Just swap those 4 with one set on the scale and see if what happens.  It should go up or down.  If the new coins rise, then my fake coin is lighter.  If it sinks, then the fake coin is heavier.

2.  If the scales don't balance, there's one up side and one down side.  The fake coin is in one of those sides.  That means the 4 coins in my hand are all REAL.  Replace the coins on the upside of the scale with my 4 real coins.  If the scale balances, then that means the fake coin is lighter.  If the scale does not balance, then the fake coin is heavier.

In two tries, I'd find out whether or not my fake coins was lighter or heavier.  Probably not the solution you were looking for since it involves adding another coin, but that's how I'd do it. 

Among 11 coins of the same denomination, 1 is know to be counterfeit. Using a simple 2-pan scale or balance, how would you determine in 2 weighings if the fake coin is heavier or lighter than a real coin?

Several scenarios can happen.

Scenario 1: fake coin in hand/not on scale
Weigh 1: Weigh each scale w/ 5 coins each. If leveled, then the fake coin in your hand.
Weigh 2: Replace one coin from either scale.  If replaced scale goes down, fake coin is heavier and if goes up, fake coin is lighter.

Scenario 2: fake coin on the scale, but not on scale selected for 2nd weighing
Weigh 1: Weigh each scale w/ 5 coins each. If not level, then the fake coin is somewhere on the scale.
Take coins from one side of the scale out, say lowered/heavier scale, and keep coins from higher/lighter scale.
Weigh 2: Weigh each scale with three coins (including one on hand/not used on weigh 1).  If level, then fake coin is heavier.  (Same process to determine if lighter using coins from other scale.) 

Scenario 3: fake coin on the scale and on scale selected for 2nd weighing
Weigh 1: Weigh each scale w/ 5 coins each. If not level, then the fake coin is somewhere on the scale.
Take coins from one side of the scale out, say lowered/heavier scale, and keep coins from higher/lighter scale.
Weigh 2: Stuck….any bright ideas

Since we know the coin on hand/not used in weighing 1 is not fake, we can weigh two coins on each scale, but that would not isolate the fake coin and need an additional weighing if the scale did not balance.  Same thing, if we were to weigh each side with three coins (including the one on hand).  It would not isolate the fake coin and need another weighing.

Stuck…yep yep, stuck.
 

From Mr. Complicated's idea..  Since this doesn't include adding another real coin to the mix like my original idea.

Just place 5 coins on each side of the pan. 

1.  If it balances, then the one in your hand is fake.  Replace one of the coins with the fake one and if that side rises, the fake is lighter, if it sinks, the fake is heavier.

2.  If scale does not balance.  Remove one coin from each side of the scale.  Continue removing coins, two at a time, until the scale balances.  When this happens, then the two you just removed, one is fake and one is real.  Now, remove all the other coins from the scale.  Choose one of the two coins in your hand.  You should've also noted, by now, which side of the scale they came from.  For example, coin 1 came from the side of the scale what was up while coin 2 came from the scale that was down.  Place coin 1 in the pan and in the other pan, use the 11th coin which you didn't use originally (this one, you know to be a real coin).  If coin 1 rises, then you know that the fake coin is lighter.  If coin 1 and the 11th coin balance, then you know that the one left in your hand is fake and since it came from the down side of the scale, it is heavier than the real coins. 

each time you remove a coin, i believe, is considered as a weighing already

  Didn't say nothing about how you remove the coins though.