A balance-scale weighing has three outcomes (left-heavy, right-heavy, balanced), so it carries at most log23 bits of information. If there are N equally-likely possibilities to distinguish (e.g. which of N coins is odd, possibly with known or unknown direction), any strategy needs at least
k≥log3N
weighings, since 3k must be ≥N. This bound is tight when a balanced ternary-style splitting is possible: divide the candidate set into three groups as equal as possible, put two on the pans, and the outcome identifies which third contains the culprit.
When the direction (heavier or lighter) is also unknown, each coin has two possible “guilty” states, so the counting bound becomes 3k≥2N+1 (allowing for the all-genuine case), and an adaptive, information-balanced construction (rotating coins between pans and the sidelined pile across weighings) achieves it exactly when equality nearly holds.