This paper presents and analyzes the results of several new approaches to the problem of finding the folding nucleus in a given 3D protein structure. Firstly, we show that the participation of residues in the hydrophobic core and the secondary structure of native protein has a rather modest correlation with the experimentally found phi values characterizing the participation of residues in the folding nuclei. Then we tried to find the nuclei as the free energy saddle points on the network of the folding/unfolding pathways using the branch-and-bound technique and dynamic programming. We also attempted to estimate the phi values from solving of kinetic equations for the network of protein folding/unfolding pathways. These approaches give a better correlation with experiment, and the estimated folding time is consistent with the experimentally observed rapid folding of small proteins.