Marble piece which is different, can be lighter or can be heavier. Solve for both cases

Try dividing into groups in a way that if compared groups in first attempt have equal weight then remaining marbles are less in count

Divide them into 3 groups of 4 each. Make sure to also handle case where first attempt result in equal weight on both sides

Try mix and match including using marbles for which we know that weight is not different

Let's divide 12 marbles into 3 groups of 4 marbles each and name them as following

Group 1 - A1, A2, A3, A4

Group 2 - B1, B2, B3, B4

Group 3 - C1, C2, C3, C4

Weighing 1 - Let's start by putting Group1 on one side of the scale and Group2 on another side. There are the following 3 outcomes.

1. Both groups have the same weight

2. Group1 is heavier than Group2 or we can read this as Group2 is lighter than Group1 as we don't know yet that marble having different weight is lighter or heavier than others

3. The reverse of above. We won't explain the solution to this as it is similar to case 2

Case 1: Group1 and Group2 have the same weight

Now we know that the marble with different weight is in group 3, however, we don't know yet that the marble is lighter or heavier.

Weighing 2 - We now take C1, C2, C3 (only 3 of the group) and put them on one side of the scale. We then take A1, A2, A3 and put them on the other side of the scale. This will give us three cases

1. Both sides have the same weight - this means C4 is outlier and now we just need to know if its heavier or lighter. This can be done by comparing it with A1 as A1 is standard weight

2. C1, C2, C3 are heavier than A1, A2, A3 - we already know that A1, A2, A3 are standard weight marbles, hence if C1, C2, C3 side is lower than we are sure that the outlier marble piece is among C1, C2, C3 and certainly is heavier than other marbles. We can put C1 on one side and C2 on the other side of scale and measure. If both are same weight then C3 is answer. If one is heavier than that is answer as we already know that outlier peice is heavier

3. C1, C2, C3 are lighter than A1, A2, A3 - solve using similar steps as we have done for heavier case except pick the lighter marble as answer.

Case 2: Group1 and Group2 have different weights

Let's say group 1 (A1, A2, A3, A4) were heavier then group 2(B1, B2, B3, B4). This means group3 (C1, C2, C3, C4) are of standard weight. Please note that other scenario where group 1 is lighter than group2 can be solved exacty in same way (just swap A nad B in solution) hence I will not be explaining that.

Weighing 2 -We compare A1, A2, A3, B4 and compare them against C1, C2, C3, A4

Few things to note -

1. If A4 is outlier then it can be only be heavier then others as we saw Group1 being heavier than Group2

2. If B4 is outlier then it can be only be lighter then others as we saw Group2 being lighter than Group1

There are following three cases when we weigh

1. A1, A2, A3, B4 is lighter than C1, C2, C3, A4 then B4 is answer and its lighter

2. C1, C2, C3, A4 is heavier than A1, A2, A3, B4 then A4 is answer and its heavier

3. If both A1, A2, A3, B4 and C1, C2, C3, A4 have same weight then only remaining candidates for outlier are B1, B2, B3 and we know that outlier is lighter than others. We put B1 on one side and B2 on other side and weigh. If any of these is lighter then that is our answer else B3 is our answer

Hope you had fun solving this.