Chess game

We numbered the chessboard's rows and columns as is shown on the figure below.

We write the number to each field of the chessboard, which is equal to the sum of its row's number and its column's number. For example to the left top corner we write $2$ , because $1+1=2$ .

Let $S$ be the sum of the numbers of fields, which contain a chess piece. Since in each row and in each column there are exactly two pieces, $S=2*(1+2+3+4+5+6+7+8)+2*(1+2+3+4+5+6+7+8)=72+72=144$ , which is even. (First we added the numbers from the rows, and then the numbers from the columns.) We can notice that in each white cell there is an even number and in each black cell there is an odd number. (It is true, because $\text{even}+\text{even}=\text{even}=\text{odd}+\text{odd}$ , and $\text{even}+\text{odd}=\text{odd}$ . So suppose there are $n$ number of pieces on black squares, and $k$ number of pieces on white cells. Then n would have to be odd. Then $\underbrace{o_1+o_2+\dots+o_n}_{n\space \text{number of}\space o_x\text{-s}}+\underbrace{e_1+e_2+\dots+e_k}_{k\space \text{number of}\space e_x\text{-s}}=S$

where $o_x$ is odd and $e_x$ is even, but then $\underbrace{o_1+o_2+\dots+o_n}_{n\space \text{number of}\space o_x\text{-s}}$ is odd and $\underbrace{e_1+e_2+\dots+e_k}_{k\space \text{number of}\space e_x\text{-s}}$ is even, but since $S$ is even:

$\text{odd}+\text{even}=\text{even}$

which is not true.

So it is not possible.