Question 34
Let $a$ and $b$ be two positive integers satisfying $\gcd(a, b)=1$. Consider a pawn standing on lattice point $(x, y)$. A step of type A consists of moving the pawn to one of the following points: $(x+a, y+a), (x+a, y-a), (x-a, y+a)$ or $(x-a, y-a)$. A step of type B consists of moving the pawn to $(x+b, y+b), (x+b, y-b), (x-b, y+b)$ or $(x-b, y-b)$.
Now place a pawn on $(0,0)$. You may perform a (finite) number of steps, namely alternately steps of type A and type B, starting with a step of type A. You may perform an even or an odd number of steps, so the last step may be of type A as well as of type B. Determine the set of all lattice points $(x, y)$ that you can reach with such a series of steps.
Now place a pawn on $(0,0)$. You may perform a (finite) number of steps, namely alternately steps of type A and type B, starting with a step of type A. You may perform an even or an odd number of steps, so the last step may be of type A as well as of type B. Determine the set of all lattice points $(x, y)$ that you can reach with such a series of steps.