BdMO National Higher Secondary 2011/10

[BdMO]

Problem 10:
Consider a square grid with $n$ rows and $n$ columns, where $n$ is odd (similar to a chessboard). Among the $n^2$ squares of the grid, $p$ are black and the others are white. The number of black squares is maximized while their arrangement is such that horizontally, vertically or diagonally neighboring black squares are separated by at least one white square between them. Show that there are infinitely many triplets of integers $(p, q, n)$ so that the number of white squares is $q^2$.

[Avik Roy]

There's a note about this problem-
This was one of the earliest problems of the question camp (might be even the first one). I like this problem very much, it is a beautiful one. One who solves it should appreciate its elegance as well.

[Mehfuj Zahir]

Let $n=2x+1$ then try to prove that $(x+1)(x+1)$ is the maximum number of black square in the $n^2$ square of the grid.Now find the value of white square.
$q^2=(2x+1)^2-(x+1)^2$
Then by solving equation show that $1+3q^2$ is must be perfect square and also $1$ modulo to $3$.
Then complete the prove by bionomial equation.


or, $q^2=3x(x+2)$ has infinite solution for the value of $x$ where $x$ is a natural number.

[Moon]

Mehfuj, could you please write your equations inside two dollar signs? Most of your equations are not complex at all, so just writing them between two dollar signs would make them readable.

I have edited the last post for you; now it is more readable.