>>You have 12 billiard balls.
>>
>>One of them is either lighter or heavier than the others.
>>
>>You can weigh the billiard balls 3 times on a balance scale.
>>
>>You need to find out which ball it is and whether it is heaver or lighter.
>
>Here is a similar problem:
>
>There are 6 balls: 2 red, 2 blue and 2 white. One of the balls of each color is slightly lighter than the second ball of the same color (you can name these two weighs as A & B). You need to sort out all the balls. You can weigh them only 2 times on a balance scale.

Sorry I didn't respond to this earlier.
I've never had a good chance to work on it.
I'm going to name each ball by their column and a number.
R1, R2
B1, B2
W1, W2


In weighing A:
I will weigh the following:
R1,B1 --- W1,B2

Case 1 - A EQUAL
We will take the easiest possibility first.
If weighing A is equal

I will then weigh B:
R2 --- W2

They cannot be equal because we know
1. Either B1 is lighter or B2 is lighter.
2. Since weighing A is equal:
a. Either R1 is lighter and off setting the lighter ball B2
b. Or W1 is lighter and off setting the lighter ball B1.
c. If both R1 and W1 were lighter the scale would have to tip either up
the 2 combined light balls R1 AND B1 or W1 AND B2 would be lighter than
the combination on the other side.

This leaves only 2 possibilities from weighing B:
The left side (R2) is lighter.

This means, since R2 is obviously the lighter ball that R1 is normal.
That means as described above. That W1 must be lighter and be offsetting ball B1.

This leaves us with R2, W1, and B1 being the lighter balls.


The other possibility is that the right side is lighter in Weighing B:
The right side (W2) is lighter.

This means, since R1 is obviously the lighter ball that R2 is normal.
That means as described above, that R1 must be lighter and be offsetting ball B2.

This leaves us with R1, W2, B2 being the lighter balls.




CASE 2 - A - RIGHT SIDE LIGHTER
Now we will tackle the harder cases.
In weighing A ( R1,B1 --- W1,B2 )
The Right side is lighter ( W1, B2 ).

We know the following:
1. B1 is normal.
2. B2 is lighter.
If B1 was lighter, the only way for the right side to be lighter is for both W1 and
B2 to both be lighter. This is not possible, because both B1 and B2 cannot be lighter.

In weighing B we will:
W1,B1 --- W2,R1

If B is equal:
1. W2 can't be lighter because W2 can't be offset by W1 or B1 which is normal.
2. This means that W1 is lighter and is being offset by the R1.

This leaves us with R1, W1, B2 as the lighter balls.


If in B the left side is lighter ( W1, B1 )
1. W1 must be lighter, since B1 is normal.
2. Both W2 and R1 are normal, because neither offset W1.

This leaves us with R2, W1, B2 as the lighter balls.

If in B the right side is lighter ( W2, R1 )
1. W2 must be lighter and R1 must be normal.
a. If both W2 and R1 was lighter then that would make W1 normal
and in weighing A B2 would have been offset by R1 making both
sides equal.
b. If R1 was lighter and W2 was normal then in weighing B R1 would
have been offset by W1 and both sides would be equal.

This leaves us with R2, W2, B2 as the lighter balls.




CASE 3 - A - LEFT SIDE LIGHTER
This is the same as in case 2 but in reverse
and will only be included for completeness.
In weighing A ( R1,B1 --- W1,B2 )
The Left side is lighter ( R1, B1 ).

We know the following:
1. B1 is lighter.
2. B2 is normal.
If B2 was lighter, the only way for the left side to be lighter is for both R1 and
B1 to be lighter. This is not possible, because both B1 and B2 cannot be lighter.

In weighing B we will:
W1,R2 --- R1,B2

If B is equal:
1. R2 can't be lighter because R2 can't be offset by R1 or B2 which is normal.
2. This means that R1 is lighter and is being offset by the W1.

This leaves us with R1, W1, B1 as the lighter balls.



If in B the right side is lighter ( R1, B2 )
1. R1 must be lighter since B2 is normal.
2. Both W1 and R2 are normal, because neither offset R1.

This leaves us with R1, W2, B1 as the lighter balls.


If in B the left side is lighter ( W1, R2 )
1. R2 must be lighter and W1 must be normal.
a. If both W1 and R2 was lighter then that would make R1 normal.
and in weighing A B1 would have been offset by W1 making both
sides equal.
b. If W1 was lighter and R2 was normal then in weighing B W1 would
have been offset by R1 and both sides would be equal.

This leaves us with R2, W2, B1 as the lighter balls.