A pencil game, strategy based on symmetry

[rayman123]

Homework Statement

Two players A and B place in alternated way pens on the quadratic table.
http://img37.imageshack.us/img37/3051/grazn.jpg (my table is not quite quadratic as it should be) The only condition is that the pens cannot come into a contact with one another. The player who can not add any more pens loses. Is there any strategy so that the player A can win this game? The same question regarding the player B.
Try to solve the problem with the help if symmetry as the strategy.

Does anyone has the slightest clue how to do it?
Thanks

Homework Equations

The Attempt at a Solution

 

[hgfalling]

The hint you have is a pretty strong one.

You really need to show us what you've come up with; then we can help some more.