BALTIC-WAY 2018 Problem 6

Let $n$ be a positive integer. Elfie the Elf travels in $\mathbb{R}^{3}$. She starts at the origin: $(0,0,0)$. In each turn she can teleport to any point with integer coordinates which lies at distance exactly $\sqrt{n}$ from her current location. However, teleportation is a complicated procedure. Elfie starts off normal but she turns strange with her first teleportation. Next time she teleports she becomes normal again, then strange again... etc.

For which $n$ can Elfie travel to any given point with integer coordinates and be normal when she gets there?