Question 157
Let $n$ be a positive integer larger than or equal to $2$. On an $n \times n$ chessboard, $r$ chess pieces have been placed such that
(i) there is at least one chess piece in each row;
(ii) there is at least one chess piece in each column; and
(iii) there is at least one chess piece in each diagonal (here a 'diagonal' refers to the collection of cells whose centres lie on a straight line which makes an angle of $45^\circ$ with a side of the chessboard). (For instance, in the $3 \times 3$ chessboard, the $1 \times 1$ cell forms a diagonal; the $1 \times 2$ cell and the $2 \times 1$ cell together form a diagonal; the $1 \times 1$ cell, $2 \times 2$ cell and the $3 \times 3$ cell also form a diagonal (a main diagonal), etc.)
Find, in terms of $n$, the smallest possible value of $r$.
(i) there is at least one chess piece in each row;
(ii) there is at least one chess piece in each column; and
(iii) there is at least one chess piece in each diagonal (here a 'diagonal' refers to the collection of cells whose centres lie on a straight line which makes an angle of $45^\circ$ with a side of the chessboard). (For instance, in the $3 \times 3$ chessboard, the $1 \times 1$ cell forms a diagonal; the $1 \times 2$ cell and the $2 \times 1$ cell together form a diagonal; the $1 \times 1$ cell, $2 \times 2$ cell and the $3 \times 3$ cell also form a diagonal (a main diagonal), etc.)
Find, in terms of $n$, the smallest possible value of $r$.