Divide the coins into groups of four (A, B and C).  Measure A against B.  There are three outcomes:

Weighing ONE: A = B, A > B, A < B.

EQUAL:

If A = B then we know the fake is in C.  Label the coins, C1, C2, C3 and C4.  Take three of the C coins and three coins that we know are real gold.

Weighing TWO: C1C2C3 against AAA.  If the outcome is they are equal then we know that C4 is the fake and we can weigh it against any real coin to tell if it is heavier or lighter for our third weighing.  If C1C2C3 is lighter than AAA then we know that C4 is real and the fake is either C1, C2 or C3. 

Weighing THREE; For our final weighing we can weigh C1 against C2.  If C1 < C2 then C1 is fake and lighter.  If they are equal then C3 is fake and lighter.  If C2 < C1 then C2 is fake and lighter.

UNEQUAL:

If A > B or A < B during our first weighing then we know that all C coins are real.  Since the solution is the same for A > B and B > A we will just discuss one of them.  Whichever group was heavier we will label H1, H2, H3 and H4; we will label the lighter group L1, L2, L3 and L4.

Weighing TWO:

Weigh H1H2H3L1 (left side of scale) against CCCH4 (right side of scale)

The outcome of this weighing: Equal, Left side heavy, and Right side heavy

If they are equal then we know all of the H’s are real coins and the fake is either L2, L3 or L4.

Equal:

Weighing THREE:  Weigh L2 against L3

If L2 = L3 then L4 is fake and light

If L2 < L3 then L2 is fake and light

If L3 < L2 then L3 is fake and light

We know that L2, L3 and L4 are real AND H1,H2 and H3 have a heavy coin

Weighing THREE: weigh H1 against H2

If H1 > H2 then H1 is fake and heavy

If H2 > H1 then H2 is fake and heavy

If H1 = H2 then H3 is fake and heavy

We know that L2, L3 and L4 are real AND L1 is light or H4 is heavy

Weighing THREE: weigh L1 against any real coin

If L1 < C then L1 is fake and light

If L1 = C then H4 is fake and heavy

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