You have nine coins, supposedly made of silver. But one is counterfeit, and you do not know whether it is lighter or heavier than the rest—it is just different in weight somehow. You have three weighings on a 2-pan scale to figure out which is the counterfeit coin and whether it is lighter or heavier.
Call the coins a, b, c, and m, n, p, and x, y, z.
Lemma: If we know the odd coin is one of three possible coins, and we know whether the odd coin is light or heavy, then we can identify the odd coin in one weighing. Therefore we use two weighings to narrow the odd coin down to one of three, and to find whether it’s light or heavy.
1st weighing: abc against mnp.
Case 1: abc is heavy. This means xyz are OK, and either
(a) the odd coin is a, b, or c and it’s heavy, or
(b) the odd coin is m, n, or p and it’s light.
2nd weighing: abc against xyz.
Case A: abc is heavy. This means a, b, or c is heavy.
Use the lemma.
Case B: abc = xyz. Then m, n, or p is light. Lemma.
Case C: xyz is heavy. This is impossible because it would mean that a, b, or c is light, but we know that (a) and (b) above are the only possibilities.
Case 2: abc = mnp. This means the odd coin is x, y, or z.
2nd weighing: xyz against either abc or mnp to see whether xyz is heavy or light.
Then use the Lemma on x, y, z.
Case 3: mnp is heavy. This means xyz are OK, and either
(a) the odd coin is a, b, or c and it’s light, or
(b) the odd coin is m, n, or p and it’s heavy.
2nd weighing: abc against xyz, as in Case 1.
Case A: abc is light. Then a, b, or c is light. Lemma.
Case B: abc = xyz. Then m, n, or p is heavy. Lemma.
Case C: abc is heavy. This is impossible since abc is either OK or light.