MathNet 2016-part-07 — 182 problems (EN–VI) Question 36

Theseus starts at the point $(0,0)$ in the plane. If Theseus is standing at the point $(x, y)$ in the plane, he can step one unit to the north to point $(x, y+1)$, one unit to the west to point $(x-1, y)$, one unit to the south to point $(x, y-1)$, or one unit to the east to point $(x+1, y)$. After a sequence of more than two such moves, starting with a step one unit to the south (to point $(0,-1)$), Theseus finds himself back at the point $(0,0)$. He never visited any point other than $(0,0)$ more than once, and never visited the point $(0,0)$ except at the start and end of this sequence of moves.
Let $X$ be the number of times that Theseus took a step one unit to the north, and then a step one unit to the west immediately afterward. Let $Y$ be the number of times that Theseus took a step one unit to the west, and then a step one unit to the north immediately afterward. Prove that $|X-Y|=1$.