We have 13 beads which look identical, but 6 of them are made of bronze, while the others are made of silver. All the bronze are of the same weight and all the silver also are of the same weight but the weight of each bronze is by 1 gram different from the weight of each silver (we don't know whether it is heavier or lighter). We randomly choose one of the 13 beads and by using a balance scale we want to determine whether it is bronze or silver. The scale shows the weight difference in grams between her two sides. What is the maximum number of weighings we will need?
I was told this is a variation of an old IMO problem but I can't figure out any idea.
I know the basic concept of the weightings using a balance scale, where we need $3^n$ weighings (taking into account the 3 different possible outcomes from each use of the scale: Balance, one side is heavier, one side is lighter), but this is different!
Since you haven't edited the question for a month after I pointed out a somewhat trivial solution, I'll assume that you didn't intend to preclude it, so I'll post it as an answer:
Two weighings are enough: First put all beads on one side, then put only the bead of interest on one side. That gives you the mean weight per bead and the weight of the bead of interest. If it's bronze, it will deviate from the mean by $\frac7{13}\text g$; if it's silver, by $\frac6{13}\text g$.