Putnam 2013 Problem B6

gistfile1.txt

http://inside-bigdata.com/2014/07/20/putnam-mathematical-competitions-unsolved-problem/

All final boards are full of stones (LAAEFTR).
A turn can only net 0 or 1 stones (LAAEFTR).
Only the removal of an edge stone nets 0 stone (LAAEFTR).
The removal of an edge stone is always optional (LAAEFTR).
A stone adjacent to a stone cannot be removed (LAAEFTR).

Since A plays first on an odd sized board, an even number of spots are left over, meaning A will win unless B plays a turn that nets 0 stones.
If A's first move is anywhere but at the edge with subsequent moves building a chain of stones outward, B will only be able to play a net 0 stone turn on a self-laid edge stone.
However, if B places an edge stone, A can on the next turn place a stone next to it, preventing the edge stone's removal.
Therefore, as long as A's first move is in the center, A wins. ∎