Consider usual $3$-dimensional space with integer points $\{(x,y,z)|x,y,z\in \mathbb{Z}\}$. The player is trying to trap the cat at the origin.
The player can block one point. The cat can move to either direction of $(±1,0,0), (0,±1,0), (0,0,±1)$, but cannot move to the point which has been blocked. Repeat these steps alternately. (Assume that the space is large enough.)
The problem is: Can the cat go arbitary far away from the origin? Or can the playrer trap the cat?
It seems difficult to trap the cat, but it also seems difficult to proof that the cat can go arbitary far away. I appreciate any information about this problem.
-- supplemental information --
This puzzle is motivated by this website. https://www.gamedesign.jp/sp/cat/
The player can trap the cat in $2$-dimensional version.
I think I have read that player can also trap the cat in $2$-dimensional version even if the cat can move $10$ steps (or arbitary finite large steps) during the player block one point, but I am not quite sure. I also appreciate information about this problem.
My very partial result: Assume that
(1) the $7$ points $\{(x,y,z)|x+y+z=5,\ 0\leq x,y,z\leq 5,\ x≡y \pmod 3\}$ $= \{(0,0,5),(1,1,3),(3,0,2),(0,3,2),(2,2,1),(4,1,0),(1,4,0)\}$ are already blocked
and that
(2) the cat can only move to either direction of (1,0,0), (0,1,0), (0,0,1),
then the cat cannot cross over the plane $x+y+z=5$. (https://twitter.com/icqk3/status/1360877256407613443/photo/1)