This is my short version of the answer....
First: Let the coins be: A, B, C, D, E, F, G, H, J, K, L & M

The First Weighing will help us find at least 4 coins that are not fake.

Divide the coins into three groups: A+B+C+D, E+F+G+H, and J+K+L+M

Put A+B+C+D on the left side of the scale

Put E+F+G+H on the right side of the scale

Now there are 3 possible outcomes:

1. The scale balanced

2. The left side went down

3. The right side went down

*** So let's start with scenario #1(The scale balanced):

The balanced scale tells us that all eight coins on the scale are not fake, which means that the fake coin is not on the scale (J+K+L+M). We don't know if the fake is heavier or lighter than a "real" coin.

Since we know that the first eight coins are "real" let's weigh 3 "real" coins against three from the other group and see what happens.

The Second Weighing:

Put A+B+C on the left.

Put J+K+L on the right

Compare: A+B+C ^ J+K+L

Now, again, there are three possible outcomes:

1a The sides balance, A+B+C = J+K+L

1b Left side goes down, J+K+L are lighter

1c Right side goes down, J+K+L are heavier

*** So let's start with 1a ( The sides balance, A+B+C = J+K+L):

Since (A+B+C) balances with (J+K+L), and because we already know that (A+B+C) are all "real," we now know that (J+K+L) are also all "real." The fake coin must be one of the coins not on the scale.

The fake coin must also be one of the coins that was not on the scale during the first weighing. 11 coins have been weighed so far, only one coin was not weighed one time or the other, therefore, that not-yet-weighed coin is our fake coin.

The Third Weighing:

We should now know which coin is the "non-real": The only coin that has not yet been weighed. We use our 3rd weighing to figure out whether it is heavier or lighter than a genuine coin.

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*** So let's try now scenario #2 ( The left side went down):

We're back to the First Weighing in which A+B+C+D on*the left and E+F+G+H on the right. The left side of the scale went down.

Conclusion:

The left side going down tells us that (A+B+C+D) is heavier than (E+F+G+H). But, we still don't know if the fake is heavier than or lighter than a "real" coin. If the fake is heavier than a "real" coin, it is one of the group (A+B+C+D). If the fake is lighter than a "real" coin, it is in the group (E+F+G+H).

This also tells us that (J+K+L+M) are all "real" coins. Now we rotate three of the coins around by:

taking three coins (B+C+D) off the left side of the scale, and

moving three coins (F+G+H) from the right side of the scale to the left side, and

moving three coins (K+L+M) from the table onto the right side of the scale.

This, in essence, gives us four groups of three coins:

The three coins that got removed from the scale (B+C+D), which will be heavier fakes if the scales balance the second time

The three coins that changed sides on the scale (F+G+H), which might be lighter fakes if the left side goes up the second time instead of down

The three coins that never changed position (A+E+J). We will know that one of them is the fake if the balance doesn't change. We already know that J is Jenuine, but A could be a heavier fake, or E could be a lighter fake

The fourth group is the three coins that got added to the scale (K+L+M). They are definitely "real".

Second Weighing (under the scenario of the First Weighing that Left side went down):

Put A+F+G+H on the left.

Put E+K+L+M on the right

Compare: A+F+G+H ^ E+K+L+M

1b Left side still goes down :

Since the scale did not change its position, and still has the left side heavier, then the fake coin is one of the coins that did not change position. Either:

the fake coin is (A), and it is heavier than a real coin, or

the fake coin is (E), and it is lighter than a real coin.

Remember, we know that (J) is real.

Third Weighing: Can you figure out a method to determine, with ONE use of the scales, which of the above statements is true.

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***So let's try scenario #3. (The right side went down):

It tells us that (A+B+C+D) is lighter than (E+F+G+H). But, we still don't know if the fake is heavier than or lighter than a real coin. If the fake is lighter than a real coin, it is one of the group (A+B+C+D). If the fake is heavier than a real coin, it is in the group (E+F+G+H).

This also tells us that (J+K+L+M) are all real coins. When we rotate three of the coins around by:

taking three coins (B+C+D) off the left side of the scale, and

moving three coins (F+G+H) from the right side of the scale to the left side, and

moving three coins (K+L+M) from the table onto the right side of the scale.

This, in essence, gives us four groups of three coins:

The three coins that got removed from the scale (B+C+D), which will be lighter fakes if the scales balance the second time

The three coins that changed sides on the scale (F+G+H), which might be heavier fakes if the left side goes down the second time instead of up.

The three coins that never changed position (A+E+J). We will know that one of them is the fake if the balance doesn't change. We already know that J is Jenuine, but A could be a lighter fake, or E could be a heavier fake

The fourth group is the three coins that got added to the scale (K+L+M). They are definitely real.

Second Weighing (under the scenario of the First Weighing that Right side went down):

Put A+F+G+H on the left.

Put E+K+L+M on the right

Compare: A+F+G+H ^ E+K+L+M

OK, this goes on-and-on for the rest of the possibilities, but I think you get the idea. It is way too long to continue with it.

What do I get?