Problem $10($Infinite Doubling$).$ In this problem we construct a Turing machine that $($infinitely$)$ prints out out powers of two. Starting with $1**$ we will see

$1**$

$1**11**$

$1**11**1111**$

$1**11**1111**11111111**$

$1**11**1111**11111111**1111111111111111**$ and $soon.$

One way we can tackle this problem is by the following:

Read a $1$ and $x$ it out.

$($while seeking a $1)$ If you encounter a $*,$ then seek the end and print a $*.$

Seek the end and print two $1s.$

Seek the swapped $x$ and return it to a $1$ and repeat step $1.$

For example, the first few moves should look roughly like the following. Here we did not include all of the individual movements, but you should get a feel for the movements. $($Sorry the spacing with the blank is a bit funky.$)$