In this work I study a modified Tower of Hanoi puzzle, which I term Magnetic Tower of Hanoi (MToH). The original Tower of Hanoi puzzle, invented by the French mathematician Edouard Lucas in 1883, spans "base 2". That is - the number of moves of disk number k is 2^(k-1), and the total number of moves required to solve the puzzle with N disks is 2^N - 1. In the MToH puzzle, each disk has two distinct-color sides, and disks must be flipped and placed so that no sides of the same color meet. I show here that the MToH puzzle spans "base 3" - the number of moves required to solve an N+1 disk puzzle is essentially three times larger than he number of moves required to solve an N disk puzzle. The MToH comes in 3 flavors which differ in the rules for placing a disk on a free post and therefore differ in the possible evolutions of the Tower states towards a puzzle solution. I analyze here algorithms for minimizing the number of steps required to solve the MToH puzzle in its different versions. Thus, while the colorful Magnetic Tower of Hanoi puzzle is rather challenging, its inherent freedom nurtures mathematics with remarkable elegance.