Suppose we know that exactly $1$ of $n$ gold coins is counterfeit, and weighs slightly less than the rest. The maximum number of weighings on a balance needed to identify the counterfeit coin can be shown to be $f(n)=\lceil \log_3(n) \rceil.$
Now let $n\ge 3$, and suppose we don't know whether the counterfeit is lighter or heavier than an authentic gold coin? In this case, one might ask three related questions:
- What is the maximum number of weighings, $f(n)$, needed to identify the counterfeit?
- What is the maximum number of weighings, $g(n)$, needed to identify whether the counterfeit is lighter or heavier than an authentic coin, but perhaps not identifying the actual counterfeit itself?
- What is the maximum number of weighings, $h(n)$, needed to identify both the counterfeit and its weight relative to an authentic coin?
It isn't hard to see that $h(n)=f(n)$ or $h(n)=f(n)+1$. Also, I think that $g(n)\le f(n)$. It seems for large $n$ that strategies can get fairly complex, so I'd be interested to see if there is a nice formula for $f,g,$ and $h$. (perhaps recursive?)