It is always possible to find odd ball in 3 weighings and in most of the cases it is possible to tell whether the odd ball is heavier or lighter. Only in one case, it is not possible to tell the odd ball is whether heavier or lighter.
1. Take 8 balls and weigh 4 against 4.
o If both are not equal, goto step 2
o If both are equal, goto step 3
2. One of these 8 balls is the odd one. Name the balls on heavier side of the scale as H1, H2, H3 and H4. Similarly, name the balls on the lighter side of the scale as L1, L2, L3 and L4. Either one of H's is heavier or one of L's is lighter. Weigh (H1, H2, L1) against (H3, H4, X) where X is one ball from the remaining 5 balls in intial weighing.
o If both are equal, one of L2, L3, L4 is lighter. Weigh L2 against L3.
? If both are equal, L4 is the odd ball and is lighter.
? If L2 is light, L2 is the odd ball and is lighter.
? If L3 is light, L3 is the odd ball and is lighter.
o If (H1, H2, L1) is heavier side on the scale, either H1 or H2 is heavier. Weight H1 against H2
? If both are equal, there is some error.
? If H1 is heavy, H1 is the odd ball and is heavier.
? If H2 is heavy, H2 is the odd ball and is heavier.
o If (H3, H4, X) is heavier side on the scale, either H3 or H4 is heavier or L1 is lighter. Weight H3 against H4
? If both are equal, L1 is the odd ball and is lighter.
? If H3 is heavy, H3 is the odd ball and is heavier.
? If H4 is heavy, H4 is the odd ball and is heavier.
3. One of the remaining 5 balls is the odd one. Name the balls as C1, C2, C3, C4, C5. Weight (C1, C2, C3) against (X1, X2, X3) where X1, X2, X3 are any three balls from the first weighing of 8 balls.
o If both are equal, one of remaining 2 balls is the odd i.e. either C4 or C5. Weigh C4 with X1
? If both are equal, C5 is the odd ball. But you can not tell whether it is heavier or lighter.
? If C4 is heavy, C4 is the odd ball and is heavier.
? If C4 is light, C4 is the odd ball and is lighter.
o If (C1, C2, C3) is heavier side, one of C1, C2, C3 is the odd ball and is heavier. Weigh C1 and C2.
? If both are equal, C3 is the odd ball and is heavier.
? If C1 is heavy, C1 is the odd ball and is heavier.
? If C2 is heavy, C2 is the odd ball and is heavier.
o If (C1, C2, C3) is lighter side, one of C1, C2, C3 is the odd ball and is lighter. Weigh C1 and C2.
? If both are equal, C3 is the odd ball and is heavier.
? If C1 is light, C1 is the odd ball and is lighter.
? If C2 is light, C2 is the odd ball and is lighter.
