Recent literature considers the variant of the classical Tower of Hanoi problem with n (³ 1) discs, where r (1 £ r < n) discs are evildoers, each of which can be placed directly on top of a smaller disc any number of times. Letting E(n, r) be the minimum number of moves required to solve the new variant, an explicit form of E(n, r) is available which depends on a positive integer constant N. This study investigates the properties of N.