Question

Need help with this and question two2. Use a proof by mathematical induction to show that your equation from question 1 applied to the minimum number of moves required to defeat the tower of Hanoi game, based on the number of disks you must move. Think about the process of the game; and describe how your equation applies to it

Answer 1

SOLUTION

We want to write an equation that describes the pattern on the table

Number of disks is taken as x-values and minimum number of moves is taken as y-values

So since it's a square function, let's assume it takes the form

$y=ax^2-bx+c$

plotting the table in a graphing calculator, we have

From the image above, we have that

$a=1.5,b=2.9,c=2.5$

Substituting the values we have

$\begin{gathered} y=ax^2-bx+c \\ y=1.5x^2-2.9x+2.5 \end{gathered}$

So, when x is 1, we have

$\begin{gathered} y=1.5x^2-2.9x+2.5 \\ y=1.5(1)^2-2.9(1)+2.5 \\ y=1.5-2.9+2.5 \\ y=1.1\cong1 \\ \end{gathered}$

So, it works for 1.

Assume it works for x = 2, let us try 3, we have

$y(3)=y(2+1)$

So,

$\begin{gathered} y(k+1)=1.5(k+1)^2-2.9(k+1)+2.5 \\ y(k+1)=1.5(k^2+2k+1)-2.9(k+1)+2.5_{} \\ y(k+1)=1.5k^2+3k+1.5-2.9k-2.9+2.5 \\ y(k+1)=1.5k^2+3k+1.5-2.9k-0.4 \end{gathered}$

So, from

$\begin{gathered} y(3)=y(2+1) \\ k=2 \end{gathered}$

substituting k for 2, we have

$\begin{gathered} y(k+1)=y(3)=1.5(2)^2+3(2)+1.5-2.9(2)-0.4 \\ =6+6+1.5-5.8-0.4 \\ =7.3\cong7 \end{gathered}$

So, 7.3 is approximately 7,

Hence the equation

$y=1.5x^2-2.9x+2.5$

works to describe this pattern.

So, the equation is

$y=1.5x^2-2.9x+2.5$