Solution
The Twelve Balls
Show the problem again
Twelve visually identical balls include exactly one counterfeit that is either heavier or lighter than the rest, but you don't know which. Using only a balance scale, what is the minimum number of weighings that always suffices to identify the counterfeit and determine whether it is heavy or light?
Worked solution
The answer is 3. Three weighings have 3³ = 27 outcomes, enough to distinguish the 24 possibilities (12 balls × heavy/light), and a careful scheme achieves it. Weigh 4 vs 4: if balanced, the fake is among the remaining 4 and two more weighings against known-good balls finish. If unbalanced, weigh a carefully chosen mix (rotate three balls from the heavy pan onto the light side, remove three others, and add known-good balls) so that each of the three outcomes isolates a distinct small set resolvable in the final weighing. Two weighings give only 9 outcomes < 24, so 3 is optimal.
Source: Classic twelve-coin balance-weighing puzzle, in wide circulation since the 1940s. Statement written for AxiomIQ.