Imagine i have 12 marbles, all identical in every aspect except that 11 of them have exactly the same weight but you do not know the weight of the 12th one (it may be bigger ot smaller than the standard weight). You are provided with a physical balance, or a balance with two sides or trays, and your task is to find out the defective marble and its relation by weight to the others (whether its weight was more or less than the conventional weight) by just using the physical balance 3 times.
I have worked out a little bit:
I am going to divide 12 marbles into three subsets of 4 each and put a set on one side each. If they come out to be equal, the defective one lies in the 4 i kept. Now i have to deduce from 4 to 1 in two steps. I keep one out, and put the rest three in one side. On the second side, i put any three marbles which i verified to be correct (the ones i tested). If they come out to be equal, the one i left is the one and i can use my third step to determine either it was heavier or not. If they come out to be unequal, the incorrect one must be in the 3 left. I compare only two of them. If they are equal, the one i left is mine and if unequal, the trend shown by the 3 in the previous step determines my answer.
Now i am not able to solve for the unequal part. Any help would be appreciated.
Best Answer
If the first four are heavier than the second four, say, you are left with 8 possible outcomes: One of $A,B,C,D$ is heavy or one of $E,F,G,H$ is light. Make your second weighing so that equaloty occurs in oly two cases, for example compare $ABE$ vs. $CDF$ (so equality means $G$ or $H$ is light; left heavy means $A$ heavy or $B$ heavy or $F$ light; left ligt means $E$ light or $C$ heavy or $D$ heavy). The last weighing is easily set up.
For generalizations see e.g. http://arxiv.org/pdf/0906.0693