(Special because of greenturtle3141 for offering an intriguing narrative backstory to my puzzle.)

To aesthetically attraction to his clients, Hilbert has lately redesigned his resort. You see, all rooms within the new constructing lay out on a aircraft infinitely extending in all instructions, within the sample of a hexagonal grid, the bestest geometry of all:

         ...
   * * * * * * * *
    * * * * * * *
   * * * * * * * *
... * * * # * * * ...
   * * * * * * * *
    * * * * * * *
   * * * * * * * *
         ...
---------------------
(The rooms are labeled by * and # indicators.)
(The # signal labels Room 0, one of many rooms of all time.)

Proving Hilbert a hit, company quickly stuffed out every room out there, till sooner or later, a one-headed hydra appeared, invited itself into Room 0 and began antagonizing its company. Hilbert has employed you, an professional on hydras, to try to transfer the hydra to a different room (hopefully with much less company).

If you lop a hydra head in some room, then the hydra will develop three new heads in its neighboring rooms, relying on which hand you lop the top with. You have two decisions. You might select to regrow the hydra heads as such:

 . .      . @
. @ . => @ . .
 . .      . @
--------------
(The @ signal signifies a hydra head.)

Or as such:

 . .      @ .
. @ . => . . @
 . .      @ .
--------------
(The @ signal signifies a hydra head.)

The hydra, being benevolent to you, means that you can additionally reverse the method. That is, everytime you select to concurrently lop off three hydra heads in both of the formations above, a single new head will develop within the center.

• Note: A room might include a number of hydra heads, however the variety of hydra heads in a room at any time ought to at all times be a non-negative integer.

For which of Hilbert’s rooms X, if any, can we transfer the hydra to room X, within the sense that after some finite sequence of hydra head-huntings, there may exist hydra head(s) solely in room X?