King And Knight

Firstly, note that Dimitri can place the Knight in anyone of the $64$ squares. As the King and the Knight 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, let's divide the possible Knight position in cases:

• If the Knight is placed on a corner, it will be attacking just $2$ squares; then, the King can be placed on the other $61$ squares and won't be in check. As the chessboard has $4$ corners, the total possible positions are $61\cdot 4$ .
• If the Knight is placed on a square contiguous to a corner (e.g. $h2$ or $g1$ , but not $g2$ ), it will be attacking $3$ squares; thus, the King can be placed on the other $60$ squares. As each corner has $2$ contiguous squares, the total possibilities are $60\cdot 8$ .
• If the Knight is placed on an edge (but not on a square already considered), it will be attacking $4$ squares; therefore, the King can be placed on the other $59$ squares. As each edge has $4$ squares, the total possibilities are $16\cdot 59$ .

Consider the square perimeter formed by $B$ and $G$ columns, and $2$ and $7$ rows. The Knight can be placed on a corner of this square, or on an edge.

• In the latter case, it will be attacking $6$ squares; consequently, the King can be placed on the remaining $57$ squares. As each square edge consists of $4$ squares, the possibilities are $16\cdot 57$ .

• In the former case, it will be attacking $4$ squares; hence, the King can be placed on the remaining $59$ squares. As a square has $4$ corners, the possibilities are $4\cdot 59$ .

• Finally, if the Knight is placed on another square (on the expanded center), it will be attacking $8$ squares; so, the King can be placed on the other $55$ squares. As the expanded center consists of $16$ squares, the total possibilities are $16\cdot 55$ .

Adding up the results and dividing by the total possible positions, the probability is

$\frac { 61\cdot 4+60\cdot 8+16\cdot 59+16\cdot 57+4\cdot 59+16\cdot 55 }{ 63\cdot 64 } =\frac { 11 }{ 12 }$

Therefore, $m+n=23$