Question

Renae has been playing Tower of Hanoi and has noticed that the minimum number of moves it takes to defeat the game is related to the number of disks she must move. She has recorded her observations below. Write an equation that describes this pattern. Show your work.

Number of Disks in the Tower ║Minimum Number of Moves

1 ║1

2║ 3

3 ║7

4 ║15

5 ║31

Answer 1

The Tower of Hanoi puzzle follows an exponential pattern for the minimum number of moves based on the number of disks, described by the equation M(n) = 2^n - 1.

Step-by-step explanation:

The Tower of Hanoi is a mathematical puzzle that follows a specific pattern for the minimum number of moves needed to solve it based on the number of disks. The pattern can be described as an exponential growth where each time you add a disk, the minimum number of moves doubles from the previous amount and then you add one more move. The equation that represents this pattern is:

M(n) = 2^n - 1

where M(n) represents the minimum number of moves and n represents the number of disks. So for 1 disk, the equation is 21 - 1 which equals 1 move. For 2 disks, it is 22 - 1 which equals 3 moves, and so on. This progression doubles the previous minimum moves and adds one for each additional disk.

Answer 2

`y = 2^x - 1`

Step-by-step explanation:

Let the number of disks be x and the minimum number of moves be y.

`y = 2^x - 1`

When x = 1: y = 1

`y = 1 = 2 - 1 = 2^1 - 1`

When x = 2: y = 3

`y = 3 = 4 - 1 = 2^2 - 1`

When x = 3: y = 7

`y = 7 = 8 - 1 = 2^3 - 1`

When x=4: y = 15

`y = 15 = 16 - 1 = 2^4 - 1`

When x = 5: y = 31

`y = 31 = 32 - 1 = 2^5 - 1`

Notice that for each derived expression of y, the exponent of 2 is being replaced by the value of x.

The general equation is:
`y = 2^x - 1`