Q) Suppose we have a set of t coins, all but two of which have uniform weight $0$. and two counterfeit coins have different weights$>0$. If one can only use a spring scale, what is the best solution to the problem of finding the counterfeit coins? Find an algorithm by using binary or ternary BCH code for non adaptive search, where you cannot use previous results for designing the next step, we are forced to design all step before we start weighing.
can any one help me how to solve this problem?
Thank you
Here is a way that works for $t=16$. Divide the coins into four groups, $A,B,C,D$ and label the coins $A1, A2, A3, \dots D4$. Weigh each group against each other (six weighings). At least two groups have all true coins and those will be the only groups that balance. If the two false coins are in the same group, the other three will balance.
In the case the false ones are not together, say $A$ and $B$ are heavy, $C$ and $D$ balance. Then we know the $C$ and $D$ coins are true. Now stack all the $1$ coins together, all the $2$ coins together, the $3$'s, and the $4$'s. Again weigh each stack against each other. Say the $1$'s and $2$'s are heavy. Then the heavy ones are either $A1, B2$ or $A2, B1$. If $A vs B$ had the same result as $1 vs 2$, it is $A1, B2$, otherwise it is $A2, B1$
If the two false ones are in the same group, say it is $A$. Then when you weigh the next sets, you will find the two false coins and when you weigh the two sets with false coins you will find the relative wieghts. Can you extend this to other numbers of coins? Powers of $4$ are pretty easy, others take more work.