The Twelve Pennies Puzzle

HINT: Since you only get three tests, the first two tests must eliminate at least nine of the twelve pennies. Only two or three pennies can be left for the last test.

Divide the pennies into three stacks of four and name the pennies a1, a2, a3, a4, b1, b2, b3, b4, c1, c2, c3, c4.

Weigh the 'a' stack (a1, a2, a3, a4) against the 'b' stack (b1, b2, b3, b4).

1. If the two stacks are equal, you have eliminated all the pennies in both stacks and the unique penny is in the 'c' stack (c1, c2, c3, or c4); weigh two of the known equal pennies (say a1, a2) against two of the pennies from the 'c' stack (c1, c2).
   1. If equal, the unique penny is one of the two remaining pennies in the 'c' stack (c3 or c4); weigh a known penny (c1) against one of the unknown (c3).
      1. If equal, the unique penny is c4.
      2. If unequal, the unique penny is c3.
   2. If unequal, the unique penny is one of the two 'c'-stack pennies just weighed (c1 or c2); weigh a known penny (a1) against one of them (c1).
      1. If equal, the unique penny is c2.
      2. If unequal, the unique penny is c1.
2. If first stack is heavier, you have eliminated all the pennies in the third stack and the unique penny is either one of the 'a'-stack pennies (a1, a2, a3, a4)--if the unique penny is heavier, or one of the 'b'-stack pennies (b1, b2, b3, b4)--if the unique penny is lighter. Set aside two pennies from the heavier 'a' stack (a1, a2) and one from the lighter 'b' stack (b1). Weigh a stack containing one of the other 'a' pennies (a3) and two of the other 'b' pennies (b2, b3) against a stack containing a known regular penny (c1), the last 'a' penny (a4) and the last 'b' penny (b4).
   1. If equal, all these are eliminated and the unique penny is a1, a2, or b1; weigh a1 against a2.
      1. If equal, the unique penny is b1.
      2. If unequal, the heavier penny is the unique one, because the original a-stack was heavier.
   2. If the first stack is heavier, b2 and b3 are eliminated because they cannot be heavier, and a4 is eliminated because it cannot be lighter; the unique penny is a3 or b4; weigh c1 against a3.
      1. If equal, the unique penny is b4.
      2. If unequal, the unique penny is a3.
   3. If the first stack is lighter, a3 is eliminated because it cannot be lighter, and b4 is eliminated because it cannot be heavier; the unique penny is b2, b3, or a4; weigh b2 against b3.
      1. If equal, the unique penny is a4.
      2. If unequal, the lighter penny is the unique one, because the original b-stack was lighter.
3. If the first stack is lighter, just reverse a and b in the previous process.