Example 1: Either Sente or Gote 
It is not unusual to have both sente and gote options in a position, but we typically consider a position to be sente or gote according to the largest option. Here they are the same size, and the position is ambiguous. Black has the privilege of playing sente.
This play is worth three points. If White saves one stone, there is another three point play left.
This isn't exactly sente, because White cannot normally make the play at a time when Black cannot afford to play here instead (White does not have the privilege). But it isn't exactly gote, either, because White will normally make the play when Black's reply will be the largest play on the board. It's ambiguous.
So in other words, it is indeed sente for white, but not in the sense that black cannot afford the reverse sente. ~srn347
With four copies of the situation above and White starting, it could either go 4 * 6 (Black treats as sente) or 0 + 6 + 9 + 9 (Black treats as gote): 24 in each case.
Is that what's behind?
Is below also sote? (2 + 2 + 2 + 2 = 0 + 2 + 3 + 3)
- /Discussion. (Includes a question by Jesusin about which one-point play to make.)
- One Thousand and One Life-and-Death Problems / Problem 593 discussion
- Ongoing Game 1 / Move 249 after White's play.
 The original sentence was: "The Ongoing Game has a position that, with a couple of changes, is ambiguous whether it is sente or gote." If anybody knows why it starts with "ongoing game" and what was meant by "with a couple of changes" please clarify. I propose to delete this (and this footnote) - this is sufficiently expressed by Bill's footnote 2. BTW, I also set the difficulty from n.a. to "Intermediate" - please feel free to change. Also, I moved part of this to the /Discussion page. -- Sebastian
Bill: Hi, Sebastian. :-)
I intended ambiguous position to be a technical term with a more specific meaning than the new definition above. However, I did not want to give a formal definition, in line with my view that such definitions tend to be stumbling blocks for my intended audience of regular go players. I thought that examples, instead, were better for getting the idea across.