Will 12 inch subwoofer be enough for an 18x20 room (with 10 ft cielings)

This is an open floor plan that opens up into a very small kitchen area then narrow hallway. Will this be enough? Room layout dictates I can't have more than one sub.
My room is similar in dimensions to yours. "Enough"? That depends on how much thundering bass you want. A room of that size must be able to accommodate at least a second sub. Don't predetermine where a second sub must go. Sub placement is one of the the most critical parts of the deal. Surely, your room size and geometrical dimensions can allow for the addition of at least a second box.

Example....In my room, equal in size to yours, I use a pair of bookshelf speakers on stands and a pair of 10" subs. Fills my bill perfectly. My listening is primarily music rather than HT. If you're unsure whether one sub is enough start with one. See how it works. A second sub can be added later. But, don't obsess about the size of the sub speaker. Focus on the placement of it/them. If music, movies, or both are your preference that will guide you in one direction or another.

I have benefited from millercarbon's suggestions about subs and other things. I would ask him directly what results you want to obtain.

Just bear in mind that not all 12 inch subs are created equal.  I also recommend SVS, they are built like tanks.  I use a SB-2000 (12" woofer) in a room similar to the OP's and have to keep the sub's volume around half way to stop my woman's ornaments from rattling off shelves.  I also use a SB-1000 nearfield under my computer desk where it doubles as a footstool and it serves both purposes quite nicely.
You should really use 2.
Not ported. 
If your on hardwood isolate them with upgraded feet.
Have them come in about 20 Hz above where your speaker's give out.
I've used 2 x 18" Eminence Kilomax drivers is 14 cu ft sealed cabinets in a room of similar volume.  EQ'ed the subs and placed them in the corners behind the main speakers.  Works OK. 

It all depends on the listening position in the room.  The optimum listening position and main speaker position may not be the optimum listening position for the subs.   
The answer to your question is NO.  Get four subs...it will be for the best.