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:

Cost Function

A mathematical formula that describes the total cost of production as a function of the quantity of output produced.

Output Of Apples

The total amount of apples produced in a given time period, often used in agricultural and economic analysis.

Cost Function

A mathematical formula or representation showing how costs vary with changes in the level of an activity, such as production volume.

Output Of Apples

The total quantity of apples produced within a given timeframe, often used to measure agricultural productivity.