Ping is a simple computing machine. If you put the number $x$ into ping, then $6x+2$ comes out. Pong is another machine. We know that pingpong is a really stupid machine — it does not change the number you put into it.

How does the pong machine work? How does the pongping machine work?

You buy 17 ping machines and 18 pong machines. Then you connect the 35 machines into one single machine in any order that you wish. What will this new machine do?

Let click be the machine such that if the input is $x$, then the output is $4x+1$. Can you find a machine bib which has the property that bibbib is click? Which machine bub has the property that bubbub doubles the input? Is there only one such machine bub?

You have bought a division machine. In this case you must input two (strictly) positive numbers, though you still receive one output. If you put $x$ into the first slot of the machine and $y$ into the second slot of the machine, then the output will be $x/y$. Can you multiply together two positive numbers using this machine?

You have an average machine, which works as follows: if you put $x$ and $y$ into it, it will compute $\frac{x+y}{2}$. Prove that using this machine, you can compute

Is it possible to construct two machines $X$ and $Y$ such that $XY$ and $YX$ work differently?

Is it possible to have two machines $X$ and $Y$ for which you can input any number such that $XY$ is the ‘‘stupid’’ machine (the output is the same number as the input), but $YX$ is not the ‘‘stupid’’ machine?

You have a subtraction machine. If you put $x$ into the first slot of this machine and $y$ into the second slot of it, then the output will be $x-y$. You also have a rec machine. If you put $z$ into it, it will give you $1/z$. (If you put zero into rec, it will give you an error message.) Show that you can compute the square of a positive number using these machines.

(Hard!) Show that you can use these two machines to multiply together two numbers.