In a Double Tower of Hanoi with Adjacency Requirement There

question 34

Essay

In a Double Tower of Hanoi with Adjacency Requirement there are three poles in a row and
2n disks, two of each of n different sizes, where n is any positive integer. Initially pole A (at
one end of the row) contains all the disks, placed on top of each other in pairs of decreasing
size. Disks may only be transferred one-by-one from one pole to an adjacent pole and at no
time may a larger disk be placed on top of a smaller one. However a disk may be placed on
top of another one of the same size. Let C be the pole at the other end of the row and let $$s _ { n } = \left[ \begin{array} { l } \text { the minimum number of moves } \\\text { needed to transfer a tower of } 2 n \\\text { disks from pole } A \text { to pole } C\end{array} \right]$$
(a) Find $s _ { 1 }$ and $s _ { 2 }$ .
(b) Find a recurrence relation expressing $s _ { k }$ in terms of $s _ { k - 1 }$ for all integers $k \geq 2$ . Justify your answer carefully.

---

Definitions:

Ammonia Molecule

A compound consisting of one nitrogen atom and three hydrogen atoms (NH3), often used in cleaning products and as a fertilizer.

Trigonal Pyramidal

A molecular geometry where a central atom is surrounded by three atoms placed at the corners of a pyramid, with a vertex missing.

Central Atom

In a molecular structure, the atom that is bonded to several other atoms.

Equivalent Atoms

Equivalent atoms are atoms in a molecule that are indistinguishable from each other in their chemical environment, resulting in identical physical and chemical properties.