Baltic Way shortlist

Clearly, the grasshopper can arrive at the integers $+1$ and $-1$, in one jump. And also, he can arrive at the integer $14$ using three jumps to the right ($1+4+9=14$). Note the following: if the grasshopper can arrive at a number $a\in \mathbb{Z}$ using $n$ jumps, then he can also arrive at the numbers $a-4$ and $a+4$ using $n+4$ jumps, by jumping left, right, right, left (for $a-4$), and right, left, left, right (for $a+4$), because of $$a-(n+1{)}^2+(n+2{)}^2+(n+3{)}^2-(n+4{)}^2=a-4,$$ $$a+(n+1{)}^2-(n+2{)}^2-(n+3{)}^2+(n+4{)}^2=a+4.$$ Since the numbers $0$, $+1$, $-1$ and $14$ all have distinct remainders modulo four, the grasshopper can arrive at each integer in a finite number of jumps.