Puzzle 11: 12 balls, one is defective

Puzzle 11: Suppose you have 12 balls. One of them is defective(whether light or heavy). You need to find out the defective one by using the weight scale 3 times only.
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Solution:


There may be more than one solution, however, I am describing here the one which struck me. We will divide balls in groups of four. Thus we will have three groups, A, B and C. Without loss of generality , put group A and B on comparison balance.

case 1 : A = B
In this case, we know that defective ball is in group C. Put (C1.C2,C3) and ( A1,A2,A3) on comparison balance.

case 1a : (C1,C2,C3) = ( A1,A2,A3)
Clearly, C4 is the defective ball. Compare it with A1 and find out whether it is heavier or lighter.

case 1b : (C1,C2,C3) < ( A1,A2,A3)
Now, we know that defective ball is lighter. Compare C1 and C2. if they are unequal, we know which one is defective depending upon which one is lighter. If they are equal, we know that C3 is the lighter defective ball.

case 1c : (C1,C2,C3) > (A1,A2,A3)
Now, it is obvious that defective ball is heavier. Carry out the same procedure as described in case 1b, to find out which one is heavier defective ball.

case 2 : A > B
From this observation we can deduce that one of the ball in group A is heavier or one of the ball in group B is lighter. Now, compare ( A4, B3,B4) and ( B1,B2,C1).

case 2a : ( A4,B3,B4) > ( B1,B2,C1)
This tells us that either A4 is heavier or one of B1-B2 is lighter. Compare B1 and B2. If they are equal then A4 is the heavier defective ball. If not, we know which one is defective depending on which one is lighter.

case 2b : (A4,B3,B4) < ( B1,B2,C1)
This tells us that one of B3 and B4 is lighter defective ball.Compare B3 and B4 and find out which one is defective.

case 2c : (A4,B3,B4) = (B1,B2,C1)
This is an easier case. Now we know that one of A1,A2,A3 is defective and heavier. Compare A1 and A2 if they are equal A3 is defective, if not, the heavier of the two is defective.

case 3 : A < B
Line of reasoning is exactly similar to case 2. ( replace heavier by lighter and vice versa ).