placing coins on a table with a twist.

consider the classical puzzle, on a circular table, you and a friend take turns placing coins on the table and the first person who cannot do this loses.

You can guarantee winning by placing the first coin in the centre and then maintain symmetry.

Notice this strategy is independent of the table size. (i.e. the radius of the circle)

So I asked myself the question:

Does there exists a table shape such that the 2nd player has a winning strategy provided the table is large enough?

The answer is yes, and an annulus is an example.

My new question is does there exists a shape, topologically the same as a disc, for which, provided the table is large enough, the 2nd player is always guaranteed to win?

(consider two circles connected by a really thin rod, so you cannot place a coin on the rod, for this there is a strategy for the 2nd player to win by maintaining symmetry. but if you enlarge the table proportionally, the thin rod will becomes thick enough and the first player would be able to place a coin at the middle of the thin rod, and maintain symmetry afterward to guarantee a win. I wonder if there exists a shape, with no holes, for which, the second player is guaranteed to win, for all 'sufficiently large' table)

-
The grammar in your last sentence is poor and makes it hard to understand what you're asking. Can you fix that? – Dan Rust Nov 13 '13 at 13:14
@DanielRust okay. sorry – Lost1 Nov 13 '13 at 14:52
Ahha, so you want the table to be a simply connected subspace of the plane? (Topologically the same as a circle means something slightly different and isn't what I think you meant - you maybe meant topologically the same as the disk?) – Dan Rust Nov 13 '13 at 14:55
Although it doesn't match your criteria (as it's not homeomorphic to a disk), it's interesting to note that on a sphere, player two wins by symmetry. – Dan Rust Nov 13 '13 at 15:04
@DanielRust I mean a disc, yes. I wanted to avoid using 'topologically the same' for the reason I don't really understand topology that well. And yeah, nice example with the sphere. – Lost1 Nov 13 '13 at 15:22