Logical Question:

Answer #1
Firstly divide the 12balls into three groups of 4balls numbered 1,2,3

Weigh group1(g1) against group2(g2), g2 vs g3, g3 vs g1.

One of these would be equal. Note that the other group will have the ball we are in search of. Suppose take it as g2.

Now take a ball from either g1 or g3 and weigh the 4balls against g2.

Now, which ever weighs more or less than the ball from g1 or g2.

Answer #2
You can do this by using the balance a maximum of 3 times:
Separate the 12 balls into 3 groups of 4:
Group A: A1, A2, A3, A4
Group B: B1, B2, B3, B4
Group C: C1, C2, C3, C4
Balance 1: Weight Group A vs Group B:
Case 1: Group A and B are equal:
Group C contains the odd ball.
Balance 2: C1, C2, C3 vs A1, A2, A3:
Case 1a: Balance is equal:
C4 is the odd ball out
Balance 3: C4 vs any other ball:
Case 1ai: C4 is higher than the other ball
C4 is the odd ball out and is lighter
Case 1aii: C4 is lower than the other ball
C4 is the odd ball out and is heavier
Case 1b: C1, C2, C3 is higher than A1, A2, A3
Either C1, C2, or C3 is lighter
Balance 3: C1 vs C2
Case 1bi: C1 and C2 are equal
C3 is odd and lighter
Case 1bii: C1 is lighter than C2
C1 is odd and lighter
Case 1biii: C1 is heavier than C2
C2 is odd and lighter
Case 1b: C1, C2, C3 is lower than A1, A2, A3
Either C1, C2, or C3 is heavier
Balance 3: C1 vs C2
Case 1bi: C1 and C2 are equal
C3 is odd and heavier
Case 1bii: C1 is lighter than C2
C2 is odd and heavier
Case 1biii: C1 is heavier than C2
C1 is odd and heavier
Case 2: Group A is lighter than group B:
Group C contains all generic balls
Switch B1, B2, B3 out for C1, C2, C3, and flip A4 and B4 therefore...
Balance 2: A1, A2, A3, B4 vs C1, C2, C3, A4
Case 2a: Balance remains in the same state as it was after Balance 1
Either A1, A2, or A3 is odd and lighter
Balance 3: A1 vs A2
Case 2ai: A1 and A2 are the same
A3 is the odd ball out and lighter
Case 2aii: A1 is lighter than A2
A1 is the odd ball out and lighter
Case 2aiii: A1 is heavier than A2
A2 is the odd ball out and lighter
Case 2b: The two groups are of equal weight
Either B1, B2, or B3 is odd and heavier
Balance 3: B1 vs B2
Case 2bi: B1 and B2 are the same
B3 is the odd ball out and heavier
Case 2bii: B1 is lighter than B2
B2 is the odd ball out and heavier
Case 2biii: B1 is heavier than B2
B1 is the odd ball out and heavier
Case 2c: The second group is now lighter (i.e. C1, C2, C3, A4 is lighter than A1, A2, A3, B4)
Either A4 or B4 are the odd balls out
Balance 3: A4 vs B1
case 2ci: A4 and B1 are equal
B4 is the odd ball out and is heavier
case 2cii: A4 is lighter than B1
A4 is the odd ball out and is lighter
Case 3: Group A is heavier than group B:
Group C contains all generic balls
Switch B1, B2, B3 out for C1, C2, C3, and flip A4 and B4 therefore...
Balance 2: A1, A2, A3, B4 vs C1, C2, C3, A4
Case 2a: Balance remains in the same state as it was after Balance 1
Either A1, A2, or A3 is odd and heavier
Balance 3: A1 vs A2
Case 2ai: A1 and A2 are the same
A3 is the odd ball out and heavier
Case 2aii: A1 is heavierthan A2
A1 is the odd ball out and heavier
Case 2aiii: A1 is lighter than A2
A2 is the odd ball out and heavier
Case 2b: The two groups are of equal weight
Either B1, B2, or B3 is odd and lighter
Balance 3: B1 vs B2
Case 2bi: B1 and B2 are the same
B3 is the odd ball out and lighter
Case 2bii: B1 is lighter than B2
B1 is the odd ball out and lighter
Case 2biii: B1 is heavier than B2
B2 is the odd ball out and lighter
Case 2c: The second group is now heavier (i.e. C1, C2, C3, A4 is heavier than A1, A2, A3, B4)
Either A4 or B4 are the odd balls out
Balance 3: A4 vs B1
case 2ci: A4 and B1 are equal
B4 is the odd ball out and is lighter
case 2cii: A4 is lighter than B1
A4 is the odd ball out and is heavier