**To find the equation for the inverse of the Tower Puzzle
rule - Explanation**

**By: Ratan & Don**

**We played with the Tower
Puzzle.**

**The object and rules are:
**The object is to move the tower of discs
from one peg to another. The rules are: you can't put a bigger disc on top of a
smaller one, you can only move one disc at a time, and later you will need to
move the discs in the **minimum # of moves. The table and graph show the # of
discs vs the minimum # of moves**.

The function that we found for the tower puzzle rule was 2** ^{x}** - 1 = y.
We
found the the graph of the inverse by switching the x and y-coordinates. So the
point
(2,3) on the original graph (black), will go to (3,2) on the orange graph- the
inverse-whose equation was

2^{y}**-** 1 = x

Notice also that
the y=x line is the line of symmetry of the 2 graphs. [Don noticed on
Ratan's graph that between 0 and 1 the orange graph shows up, but the black does
not, and is covered by the blue-it got unnoticed by both of us.].

So to solve for
y we first had to figure **
how to use logs,** because y was an exponent. We first
figured out that that
the log2^{3} = 3*log2 which we then found out equals the log of 8. This means that
the identity is log A^{B} equals B*log A. Then we combined this with our
skills of simplifying equations to solve the problem. So we tried to find an
equation with y in terms of x But then we realized a problem with
the equation it was solving for x instead of y. So we the following: **
Work (simplified equation)**

2^{y}
- 1 = x

Add 1 to both sides of the equation to get

2^{y}
= x + 1

Then take the log of both sides, to get

log 2^{y} = log (x+1)

Using the identity above, the log 2^{y}=
y* log2

So y* log 2 = log (x+1),

So we get the equation of the inverse,

**y = log(x+1)/log2**

[Ratan made a second mistake when he wrote to Don and said there
was a second equation for the inverse. He had 2y=x+1, (2y instead of 2^{y})
then divided both sides by 2 and got y=.5x+.5 which is a linear equation and
couldn't be right. Don wrote an email to Ratan about this error].

**To check our work, if we put 3 in
for x, we should get 2 for y **

**then y =
log(3+1)/log2= log(4)/log2 = .602/.301= 2 it
works!**

**FINE WORK RATAN !!**