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A weird and a normal elf

Source: Baltic Way 2018, Problem 6

November 6, 2018
combinatorics

Problem Statement

Let nn be a positive integer. Elfie the Elf travels in R3\mathbb{R}^3. She starts at the origin: (0,0,0)(0,0,0). In each turn she can teleport to any point with integer coordinates which lies at distance exactly n\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 turns normal again, then strange again... etc. For which nn can Elfie travel to any point with integer coordinates and be normal when she gets there?
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