A very interesting and educationally rich puzzle. I have been using it for many years for many purposes, including explaining and illustrating the difference between formulating a problem and solving a problem, and the difference paradigms for decision making under uncertainty (eg Laplace vs minimax).
It has many versions, some are more difficult than others. In my teaching I use the simple version, namely the case where there is exactly one counterfeit coin and it is known in advanced whether it is heavier or ighter than the good coins.
It is interesting to note that the official literature on this puzzle focuses on the worst case (minmax) paradigm. This paradigm is suitable for many purposes, including the illustration that this approach completely removes the uncertainty from the problem formulation.
The more elaborate paradigm, where the objective is to minimize the expected number of weighings, is very suitable for illustrating the DP solution strategy for sequential decision making under uncertainty.
In any case, in a recent paper, I discussed these two versions of the problem from a DP perspective.
It turns out the the results for the Laplace's version of the puzzle are a bit counter intuitive: the expected number of trials is not monotonic with the size of the problem!!! For example, it takes longer to solve a problem of size 6 than a problem of size 7.
I highly recommed this paper to lecturers teaching topics related to decision making under uncertainty.
