Question 1
Given a $m \times n$ ($m, n \geq 1$) grid, place a pawn at the center of each square and one at each vertex of the grid (in the example shown alongside with $n=4$ and $m=3$, 32 pawns are placed).
a. Find all tables that have exactly 500 checkers.
b. Prove that there exist infinitely many positive integers $k$ such that there are no grids with exactly $k$ checkers.
a. Find all tables that have exactly 500 checkers.
b. Prove that there exist infinitely many positive integers $k$ such that there are no grids with exactly $k$ checkers.