Solution:

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

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.

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.

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.

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

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

If L1 < C then L1 is fake and
light

If L1 = C then H4 is fake and heavy