A Towers of Hanoi puzzle has all of its discs colored black or white according to parity. Looking at the starting/finished tower the discs alternate back and forth between black and white.

Take a colored set like this and separate the white discs from the black discs. The white discs are placed on one pole in order and the black discs are placed on a second pole in order.

Devise an algorithm that will transfer the discs back into the complete tower on the third pole. As a function of N, how few moves can a tower of N discs be reassembled on the third pole?

An example: the XS, S, M, L, XL discs in the linked puzzle would start with the XS, M, and XL discs colored black and be on the first pole while the S and L discs would be colored white and be on the second pole.