The insight for size of groups can be shown by constructing the weighing that maximizes entropy. With lg(9) bits of information needed to identify the odd coin

The insight for size of groups can be shown by constructing the weighing that maximizes entropy. With lg(9) bits of information needed to identify the odd coin, and a scale with only three possible outcomes (< > =), we can show the best sequence of weighings must reveal lg(3) + lg(3) bits of information by yielding a uniform distribution on the outcomes.