Initially, the number 1 and two positive numbers $x$ and $y$ are written on a blackboard. In each move, a player can choose two numbers on the board, not necessarily distinct, and write their sum or their difference on the board. We can also choose a non-zero number on the board and write its inverse. After a finite number of moves, describe how we can obtain the following numbers:

a) $x^{2}$.

b) $x y$.
Solution

a) If $x=1$ there is nothing to do. Suppose then that $x \neq 1$. First, write $x+1$ and $x-1$. Using the inverse move, we can write $\frac{1}{x+1}$ and $\frac{1}{x-1}$. Next, we can write the difference of these two numbers: $\frac{2}{x^{2}-1}$. The inverse of this last number is $p=\frac{x^{2}-1}{2}$. By calculating $p+p$ we find $x^{2}-1$. Finally, just add 1 to get $x^{2}-1+1=x^{2}$.

b) First, write $x+y$. By the previous item, we can write $(x+y)^{2}, x^{2}$ and $y^{2}$. Next, with two moves, we can write $(x+y)^{2}-x^{2}$ and $(x+y)^{2}-x^{2}-y^{2} = 2xy$. We can also write the inverse of this last number: $q=\frac{1}{2xy}$. Finally, we can write $q+q=\frac{1}{xy}$ and its inverse $xy$.