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.

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Definitions:

Mechanoreceptors

Sensory receptors that respond to mechanical pressure or distortion.

EEG

Electroencephalogram, a test that measures and records the electrical activity of the brain.

Meissner Corpuscles

Specific nerve terminations in the skin designed to respond to the sensation of light touch.

Mechanoreceptors

Sensory receptors in the skin and other body parts that respond to mechanical pressure or distortion, playing a key role in touch, pressure, and sound sensation.