From The Heart of Mathematics, by Edward B. Burger and Michael Starbird (language streamlined somewhat):
1) You have a balance scale and nine stones identical in appearance. Eight of the stones are also identical in weight; the ninth weighs slightly more. Using the scale twice, find the heavy stone.
Hint: Start by comparing one collection of stones with another collection of stones.
Answer: Divide the stones into three groups of three. Place one group on one side of the balance scale and another group on the other side. If one side is heavier than the other, then the heavy group contains the heavy stone. If the two groups weigh the same, then the heavy stone is in the third group, the group not on the scale.
Take two stones from the group containing the heavy stone and place one stone on each side of the scale. If one stone weighs more than the other, then you've found the heavy stone. If they weigh the same, then the heavy stone is the third of the group, the one not on the scale.
2) You have a balance scale and twelve stones identical in appearance. Eleven of the stones are also identical in weight; the twelfth weighs slightly more or less. Using the scale three times, find the unique stone and determine whether it's heavier or lighter than each of the others.
Hint: Divide the stones into three groups of four. Put one group on each side of the scale. If they weigh the same, the unique stone must be in the third group.
If the first two groups don't weigh the same, then the unique stone is either lighter than the rest, and one of the "light" group, or heavier than the rest, and among the "heavy" group. Keep track of the stones and consider putting potentially light stones with potentially heavy ones on the same side of the scale during the next weighing.
Answer: Divide the stones into three groups of four and weigh one group against another.
If the two groups weigh the same, then the unique stone is in the third group.
Weigh three stones from the third group against three stones from the first two groups (all eight stones of which weigh the same).
If the three stones from the third group are heavier (lighter) than the three stones that weigh the same, then the unique stone is heavier (lighter) than each of the other eleven and is among the three stones from the third group. Place one stone from the heavier (lighter) group on each side of the scale. If they weigh the same, then the third stone is the unique stone. If they don't weigh the same, then the heavier (lighter) stone is the unique stone.
If the three stones from the third group weigh the same as the other three stones, then the unique stone is the one stone not yet weighed. Weigh it against any of the other eleven stones to determine whether it's heavy or light.
If the two groups of four don't weigh the same, then the unique stone is among those eight stones, four of which are "light" and four "heavy." Keep track of which stones come from the "light" group and which come from the "heavy" group.
Place two "light" stones and two "heavy" stones on one side of the scale. On the other side place one "light" stone, one "heavy" stone, and two stones from among the four unweighed stones, all of which must weigh the same (because the unique stone isn't among them).
If the two sides weigh the same, then the unique stone must be one of the two stones not yet weighed, one of which is "heavy," one "light." Weigh either against any of the other ten stones. If they weigh the same, then the unique stone is the one still unweighed. If they don't weigh the same, then the unique stone is the "light" or "heavy" stone on the scale.
If the side containing two "heavy" stones and two "light" stones is heavier than the other, then two possibilities exist: either one of the "heavy" stones is the unique stone, or the "light" stone on the other side is the unique stone. Place one of the two "heavy" stones on each side of the scale. If they weigh the same, then the "light" stone from the other side is the unique stone. If they don't weigh the same, then the heavy one is the unique stone.
If the side containing two "light" stones and two "heavy" stones is lighter than the other, then two possibilities exist: either one of the "light" stones is the unique stone, or the "heavy" stone on the other side is the unique stone. Place one of the two "light" stones on each side of the scale. If they weigh the same, then the "heavy" stone from the other side is the unique stone. If they don't weigh the same, then the light one is the unique stone.
I hope these brainteasers give you as much frustration pleasure as they gave me.
*If any part of this post—question, hint, answer—is unclear, please let me know. Ambiguity in puzzles is unforgivable.
