Analysis of a Collatz Game and Other Variants of the  \(3n+1\) Problem

Introduced and discussed are new variants of the \(3n+1\) problem, in particular a 2-player game with moves \(3n+1\) and \(3n-1\) and subsequent halving, where the player is winner who first reaches the 1. Variants of the classical \(3n+1\) problem are discussed with respect to a general convergence conjecture. Finally, we set up prizes for solutions of the Collatz problem and simpler variants.