Baltic Way 2018 · Problem 6

Answer: there are no such $n$. We colour all the points in $\mathbb{Z}^3$ white and black: The point $(x, y, z)$ is colored white if $x+y+z \equiv_2 0$ and black if $x + y + z \equiv_2 1$. After the first move Elfie is at a point $(a, b, c)$ where $a^2 + b^2 + c^2 = n$. Thus, $a + b + c \equiv_2 n$ Now, if $n$ is even then $(a, b, c)$ is white. Thus, in that case Elfie only jumps between white points. On the other hand, if $n$ is odd, then $(a, b, c)$ is certainly black. And one can easily see that Elfie alternates between black and white squares after each move. But since Elfie is normal after even number of moves, and is then on a white point, she can never reach any black point being normal. Thus, there no $n$ such that Elfie can travel to any given point and be normal when she gets there.