Chapter 7.1, Problem 65PS

Solution

To state: A formula for the sequence and the minimum number of moves required to move 6 rings, 7 rings, 8 rings if a move consists of moving exactly one ring, and no ring may be placed on top of a smaller ring.

The minimum number $a_n$ of moves required to move n rings is 1 for 1 ring, 3 for 2 rings, 7 for 3 rings, 15 for 4 rings, and 31 for 5 rings.

The resultant formula is $a_n=2^n-1$ and the minimum number of moves required to moves 6 rings, 7 rings, and 8 rings are 63, 127, and 255 respectively.

Given information:

The minimum number $a_n$ of moves required to move n rings is 1 for 1 ring, 3 for 2 rings, 7 for 3 rings, 15 for 4 rings, and 31 for 5 rings.

Explanation:

Consider the given information:

The minimum number of moves required to move n rings is 1 for 1 ring, 3 for 2 rings, 7 for 3 rings, 15 for 4 rings, and 31 for 5 rings.

The sequence can be formed for the above said information:

  $1,3,7,15,31$

The sequence can be written as:

  $2^1-1=1,2^2-1=3,2^3-1=7,2^4-1=15,2^5-1=31$

Now the formula for the nth term is:

  $a_n=2^n-1$

Now substitute n as 6 to find out moves required to move 6 rings,

  $\begin{array}{c}{a}_6=2^6-1\\ =64-1\\ =63\end{array}$

Now substitute n as 7 to find out moves required to move 7 rings,

  $\begin{array}{c}{a}_7=2^7-1\\ =128-1\\ =127\end{array}$

Now substitute n as 7 to find out moves required to move 7 rings,

  $\begin{array}{c}{a}_8=2^8-1\\ =256-1\\ =255\end{array}$

Therefore, the minimum number of moves required to moves 6 rings, 7 rings, and 8 rings are 63, 127, and 255 respectively.