Question 69
Let $n$ be a positive integer. Elfie the Elf lives in a three dimensional space $\mathbb{Z}^3$. She starts at the origin: $(0,0,0)$. In each turn she can teleport into any point in $\mathbb{Z}^3$ which lies at the distance $\sqrt{n}$ 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 in $\mathbb{Z}^3$ and be normal when she gets there?
For which $n$ can Elfie travel to any given point in $\mathbb{Z}^3$ and be normal when she gets there?