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Baltic Way
2018 Baltic Way
6
6
Part of
2018 Baltic Way
Problems
(1)
A weird and a normal elf
Source: Baltic Way 2018, Problem 6
11/6/2018
Let
n
n
n
be a positive integer. Elfie the Elf travels in
R
3
\mathbb{R}^3
R
3
. She starts at the origin:
(
0
,
0
,
0
)
(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}
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
n
n
n
can Elfie travel to any point with integer coordinates and be normal when she gets there?
combinatorics