The Electromagnetic Balance Game: A Probabilistic Perspective
This work addresses theoretical puzzles in combinatorial optimization and coding theory, offering incremental insights into probabilistic methods for balance games.
The paper tackles variants of the counterfeit coin problem using an electromagnetic field instead of gravity, deriving tighter bounds for predetermined strategies and analyzing dishonest balance games, with results linking to Shannon's coding theorem.
Finding a counterfeit coin with the different weight from a set of visually identical coin using a balance, usually a two-armed balance, known as the balance question, is an intersting and inspiring question. Its variants involve diversified toolkits including information theory, coding theory, optimization, probabilistic theory, combinatorics and a lot of quick wits. In this paper some variants of the balance game are dicussed, especially from a probabilistic perspective. Unlike the gravity field setting, we adopt an electromagnetic field, where tighter bounds for some variants of the balance game can be found. We focus on the predetermined setting, where the player has to arrange the strategy without observing the outcome of the balancing. The sufficient condition for the balance to win is obtained by adopting a coding scheme. Apart from designing a delicate encoding framework, we also propose and analyze the performance of a completely randomized strategy. The optimal behavior of a randomized player is derived. Then we rise the dishonest balance game, in which the balance can adversely cheat the player. We present some elementary results on the analysis of dishonest balance game using probabilistic method at length. Its relationship with Shannon' s coding theorem in a noisy channel is also revealed.