**Question:**

Go through the previous question about gold coins. Because one of it is of less weight, you may have solved it easily. But if one of it is of different weight (you don’t know whether the odd coin is of more weight or less weight) then the question becomes more complicated. Think whether you can solve it. If not, see the answer. If you are confused with the answer also, then go to the next question and try to understand it and answer. Revert back to this question again, to understand the technique. Don’t leave it frustrated.

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**Answer:** If you take 6 and 6 as you have done for the previous question, you can never derive the correct answer, because you don’t know whether the said odd coin (culprit) weighs less or more. First let us divide those twelve coins into three groups as you suggested, consisting 1, 2, 3 and 4 in first group 5, 6, 7 and 8 in second group and remaining 9, 10, 11, 12 in third group. Now let us compare first group and second group, if these two groups are equal it’s easy to find the defective one as you explained. When if these two are unequal, let us suppose first group (1, 2, 3, and 4) is lighter and second group (5, 6, 7, and 8) is heavier, from this we can conclude that — 9,10,11,12 are not defective. The lighter one among 1, 2, 3, and 4 might be defective or the heavier one among 5, 6, 7, 8 might be defective. STEP II. Now let us put 1,2,3 and 12 aside and compare 4,5,6,7 and 8, 9, 10, 11

STEP II. Now let us put 1,2,3 and 12 aside and compare 4,5,6,7 and 8, 9, 10, 11 where 9, 10 and 11 are standard(not defective) ones. Here, if these two groups are equal, we can confirm that lighter one among 1, 2, 3 is defective and that can be found out in next one chance, that is an easy task But if not equal, need not be baffled….we have the way, it’s proved that 1,2,3 are not defective and the defective one is among 4, 5, 6, 7, 8 and also remember that it can be lighter 4 or can be heavier among 5, 6, 7, 8. If NOT EQUAL, other possible two

If NOT EQUAL, other possible two thing are, group of 4,5,6,7 can be heavier or can be lighter. If 4, 5, 6, 7 group is heavier, we can infer that defective one is heavier among 5, 6, 7 but not 4 or 8 because in the earlier comparison 4 being in lighter group it can’t be heavier now and can’t be defective and likewise 8 also can’t be defective as earlier it’s in heavier group and now it’s in lighter group. So we can the defective heavier one among 5, 6, 7 in next one chance. And f

So we can the defective heavier one among 5, 6, 7 in next one chance. And finally if 4, 5, 6, 7 group is lighter, here we can conclude that 5, 6, 7 are not defective because these are in heavier group in earlier comparison. Hence lighter 4 can be defective or heavier 8 can be defective. In the next chance compare 4 (or 8) with standard one and if it is equal heavier 8 is defective. If not equal, it can not be heavier but must be lighter and lighter 4 is defective.

If NOT EQUAL, other possible two thing are, group of 4,5,6,7 can be heavier or can be lighter. If 4, 5, 6, 7 group is heavier, we can infer that defective one is heavier among 5, 6, 7 but not 4 or 8 because in the earlier comparison 4 being in lighter group it can’t be heavier now and can’t be defective and likewise 8 also can’t be defective as earlier it’s in heavier group and now it’s in lighter group. So we can the defective heavier one among 5, 6, 7 in next one chance. And finally if 4, 5, 6, 7 group is lighter, here we can conclude that 5, 6, 7 are not defective because these are in heavier group in earlier comparison. Hence lighter 4 can be defective or heavier 8 can be defective. In the next chance compare 4 (or 8) with standard one and if it is equal heavier 8 is defective. If not equal, it can not be heavier but must be lighter and lighter 4 is defective.

Here, if these two groups are equal, we can confirm that lighter one among 1, 2, 3 is defective and that can be found out in next one chance, that is an easy task But if not equal, need not be baffled….we have the way, it’s proved that 1,2,3 are not defective and the defective one is among 4, 5, 6, 7, 8 and also remember that it can be lighter 4 or can be heavier among 5, 6, 7, 8. If NOT EQUAL, other possible two thing are, group of 4,5,6,7 can be heavier or can be lighter. If 4, 5, 6, 7 group is heavier, we can infer that defective one is heavier among 5, 6, 7 but not 4 or 8 because in the earlier comparison 4 being in lighter group it can’t be heavier now and can’t be defective and likewise 8 also can’t be defective as earlier it’s in heavier group and now it’s in lighter group. So we can the defective heavier one among 5, 6, 7 in next one chance. And finally if 4, 5, 6, 7 group is lighter, here we can conclude that 5, 6, 7 are not defective because these are in heavier group in earlier comparison. Hence lighter 4 can be defective or heavier 8 can be defective. In the next chance compare 4 (or 8) with standard one and if it is equal heavier 8 is defective. If not equal, it can not be heavier but must be lighter and lighter 4 is defective.

If NOT EQUAL, other possible two thing are, group of 4,5,6,7 can be heavier or can be lighter. If 4, 5, 6, 7 group is heavier, we can infer that defective one is heavier among 5, 6, 7 but not 4 or 8 because in the earlier comparison 4 being in lighter group it can’t be heavier now and can’t be defective and likewise 8 also can’t be defective as earlier it’s in heavier group and now it’s in lighter group. So we can the defective heavier one among 5, 6, 7 in next one chance. And finally if 4, 5, 6, 7 group is lighter, here we can conclude that 5, 6, 7 are not defective because these are in heavier group in earlier comparison. Hence lighter 4 can be defective or heavier 8 can be defective. In the next chance compare 4 (or 8) with standard one and if it is equal heavier 8 is defective. If not equal, it can not be heavier but must be lighter and lighter 4 is defective.