The problem "The Fake Sample" typically involves identifying a counterfeit coin among a set of identical-looking coins using a balance scale. Here, we address the classic version: 12 coins, one fake (either heavier or lighter), find the fake and its relative weight in three weighings.

1. First Weighing: Divide the 12 coins into three groups of four:
   • Group A: Coins 1, 2, 3, 4
   • Group B: Coins 5, 6, 7, 8
   • Group C: Coins 9, 10, 11,