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Bishops at peace

BrainteasersEasy

On a $6$ by $6$ board, $2$ identical bishops are placed on $2$ different cells. A bishop attacks along diagonals. In how many ways can the bishops be placed so that they do not attack each other? Swapping the $2$ bishops is not a new placement.

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Hint

Count all pairs of cells, then subtract the pairs that share a diagonal.

Solution

There are $(2 36) = 630$ pairs of cells.
The diagonals in one direction have lengths $1$ , $2$ , $3$ , $4$ , $5$ , $6$ , $5$ , $4$ , $3$ , $2$ , $1$ , and a diagonal of length $k$ holds $(2 k)$ attacking pairs, which gives $0 + 1 + 3 + 6 + 10 + 15 + 10 + 6 + 3 + 1 + 0 = 55$ .
The other direction gives another $55$ pairs, and no pair shares a diagonal of both directions, so $110$ pairs attack.
The number of peaceful placements is $630 - 110 = 520$ .

Answer: $520$

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