(Special thanks to greenturtle3141 for providing an intriguing narrative backstory to my puzzle.)
To aesthetically appeal to his customers, Hilbert has recently redesigned his hotel. You see, all rooms in the new building lay out on a plane infinitely extending in all directions, in the pattern of a hexagonal grid, the bestest geometry of all:
... * * * * * * * * * * * * * * * * * * * * * * * ... * * * # * * * ... * * * * * * * * * * * * * * * * * * * * * * * ... --------------------- (The rooms are labeled by * and # signs.) (The # sign labels Room 0, one of the rooms of all time.)
Proving Hilbert a success, guests soon filled out each room available, until one day, a one-headed hydra appeared, invited itself into Room 0 and started antagonizing its guests. Hilbert has hired you, an expert on hydras, to try and move the hydra to another room (hopefully with less guests).
If you lop a hydra head in some room, then the hydra will grow three new heads in its neighboring rooms, depending on which hand you lop the head with. You have two choices. You may choose to regrow the hydra heads as such:
. . . @ . @ . => @ . . . . . @ -------------- (The @ sign indicates a hydra head.)
Or as such:
. . @ . . @ . => . . @ . . @ . -------------- (The @ sign indicates a hydra head.)
The hydra, being benevolent to you, allows you to also reverse the process. That is, whenever you choose to simultaneously lop off three hydra heads in either of the formations above, a single new head will grow in the middle.
- Note: A room may contain multiple hydra heads, but the number of hydra heads in a room at any time should always be a non-negative integer.
For which of Hilbert's rooms X, if any, can we move the hydra to room X, in the sense that after some finite sequence of hydra head-huntings, there could exist hydra head(s) only in room X?