In a Triple Tower of Hanoi, There Are Three Poles $t _ { n}$

question 36

Essay

In a Triple Tower of Hanoi, there are three poles in a row and 3n disks, three of each of n
different sizes, where n is any positive integer. Initially, one of the poles contains all the disks
placed on top of each other in triples of decreasing size. Disks are transferred one by one from one pole to another, but at no time may a larger disk be placed on top of a smaller disk.
However, a disk may be placed on top of one of the same size. Let $t _ { n}$
be the minimum number of moves needed to transfer a tower of 3n disks from one pole to another. Find a recurrence relation for $t _ { 1 } , t _ { 2 } , t _ { 3 } , \ldots$ Justify your answer carefully.

---

Definitions:

Sensorimotor Stage

The first stage in Piaget's theory of cognitive development, where infants learn through interacting with their environment using senses and motor actions.

Piaget

Swiss psychologist known for his pioneering work in child development, particularly regarding cognitive development and the stages of learning in children.

Formal Operational

The stage in Piaget's theory of cognitive development during which individuals develop the ability to think abstractly and reason logically.

Concrete Operational

The concrete operational stage is a phase in Jean Piaget's theory of cognitive development where children aged 7 to 11 years develop logical thinking but are still mostly focused on concrete objects and situations.