Puzzle: Table and coins

Question: 

Suppose two of us have to play a game on a table. Each of us has an infinite number of identical coins. Each of us will take turns and put one coin on the table. The coins cannot overlap. The other player wins when there is no place to keep the coin for any of us. There is a strategy for winning this game, independent of the size of the table. Find this strategy and find conditions when this strategy works.

Note: Conditions should answer questions like: What is the shape of the table? When does this strategy not work? 

Solution: 

The solution for this problem when the shape of the table is fixed, i.e., circular, is explained in this article: Puzzle | (Round table coin game)

The strategy for winning exists if the table is radially symmetric, that is, the table should have a geometric centre. Examples of these include ellipses, rectangles, circles, etc. The strategy also dictates winning. You must play first and put your coin in the centre of the table. After that, you can simply, imitate the opponent, i.e., place the coin at the mirror image of your opponent’s position when viewed through the central point.

The above strategy ensures victory because if your opponent can place a coin, then so can you.

Another way to get at the solution: If both of us are playing the game, I win if I place a coin on the table and you were not able to. Assuming the table is radially symmetric if I play first and occupy the centre of the table I can guarantee that all other points in the table have a complementary such that any line passing through the centre and any other point, is bisected by the central point. So, I have to win after occupying the centre, I will simply imitate you. That is if you place a coin on point A, I will find a point A’ such that the line segment is AA’ is bisected at the centre of the table. The radial symmetry of the table simply guarantees that such a point exists. This strategy also ensures that I can’t lose, since there is no scope for a tie in this game if I don’t lose it means I win. 

Note: Radial symmetry is the necessary condition, but by no means is it sufficient. The strategy will not work if the centre is not present, e.g., in an annulus (doughnut-shaped) or any other radially symmetric shape with a hole in between. In this case, though the strategy for winning remains precisely the same, only you should play second.