King and Queen

Firstly, note that Dimitri can place the Queen in anyone of the $64$ squares. As the King and the Queen can't be placed in the same square, Dimitri has $63$ possible squares to place the Black King. Therefore, there are a total of $64\cdot 63$ possible positions.

Secondly, note that wherever the Queen is placed, will be attacking its whole row and column. This means that the King can't be placed on the same column or row of the Queen. As each row and column is formed by $8$ squares, and we are counting twice the square on which the Queen, there are a total of $63-14=49$ squares on which Dimitri can place the King.

Now, the we need to consider that the Queen also attacks its diagonal; so, let's divide the Queen possible position in cases:

• Consider the square delimited by $A$ and $H$ columns, and $1$ and $8$ rows. This square consists of $28$ perimeter squares. If the Queen is placed on whichever of these squares, it will be attacking $7$ squares that are diagonal to the Queen. Thus, the King can be placed on the other $49-7=42$ squares in order to avoid the Queen check. Hence, the possible positions are $28\cdot 42$ .
• Consider the square delimited by $B$ and $G$ columns, and $2$ and $7$ rows. This square consists of $20$ permiter squares. If the Queen is placed on anyone of these squares, it will be attacking $9$ diagonal squares. Therefore, the King can be placed on the remaining $49-9=40$ squares. Consequently, the total possible positions are $20\cdot 40$
• Consider the square delimited by $C$ and $F$ columns, and $3$ and $6$ rows. This square consists of $12$ perimeter squares. If the Queen is placed on any of these squares, it will be attacking $11$ diagonal squares; so, the King can be placed on the other $49-11=38$ squares. Thus, the possibilities are $12\cdot 38$ .
• Consider the $4$ squares $d4, d5, e4, e5$ . If the queen is placed on anyone of those squares, it will be attacking $13$ diagonal squares. Hence, the King can placed on the remaining $49-13=36$ squares. Therefore, the possible positions are $4\cdot 36$ .

Finally, the probability that Dimitri places the King and the Queen on the chessboard such that the King is not in check is given by the expression

$\frac { 42\cdot 28+40\cdot 20+38\cdot 12+36\cdot 4 }{ 64\cdot 63 } =\frac { 23 }{ 36 }$

Thus, $m=23$ and $n=36$ , so $n-m=13$ .

Assuming that Dimitri equally likely chooses any empty squares.

If the queen is at any field, the king could go at 63 other square. Of these 63 squares, 7 are on the same row as the queen, 7 others are on the same file, and some are on the diagonals, a number which depends on the position of the queen. It turns out that the distance of the queen to the edge of the board is enough to tell the number of squares it sees diagonally. If the queen is at the outer edge of the board (28 squares), the number of squares it sees diagonally is 7. If the queen is at distance 1 to the edge (b2-g2-g7-b7-b2) (20 squares) that number is 9. At distance 2: 12 squares for the queen, seeing 11 squares, at center: 4 squares, seeing 13 squares diagonally.

To calculate the number of squares where the king is NOT in check take 63-7-7-the number of diagonally seen squares (as described above). For the edge that number is 63-7-7-7=42 (out of 63 positions for the king). Multiply with the number of fields where the queen can be (out of 64 positions for the queen)

28/64× 42/63+20/64×40/63+12/64×38/63+4/64×36/64 =( 28×42+20×40+12×38+4×36)/(64×63) =( 7×21+5×20+3×19+1×18)/(4×63) =322/504=23÷36 =0.6388888

The asked answer is 36-23=13.