# Three queens and two rooks covering the chess board… again!

Three queens and two rooks can be placed on a chess board so that all empty squares are under attack, as has been shown here: 3 queens and 2 rooks covering a 8x8 chess board.

What if we require that all squares are under attack, even those occupied by any of the five pieces?

Source

I used integer linear programming to minimize the number of unattacked squares. Here is one optimal solution (unique up to symmetry), with

0 unattacked squares: $$\begin{matrix}&.&.&.&Q&.&.&.&.\\&.&.&.&.&.&.&.&.\\&.&.&.&R&.&.&.&.\\&.&.&.&.&.&.&.&.\\&.&.&.&.&.&.&.&R\\&.&.&.&.&.&.&.&.\\&.&Q&.&.&.&Q&.&.\\&.&.&.&.&.&.&.&.\\\end{matrix}$$

• Could you tell us how you came to the solution. Is it unique? – Bernardo Recamán Santos Feb 8 at 19:10
• Very cool! I didn't know it was possible. – Dmitry Kamenetsky Feb 9 at 6:48

I can manage to get

exactly one unattacked square:

(namely, the one occupied by the bottom right queen)

Methodology:

In hexomino's solution to the previous question, I noticed that the problem was essentially reduced to making three queens cover a $$6\times6$$ chessboard and then using two rooks to cover the remaining two rows and two columns. My only innovation was to shift the position of that $$6\times6$$ chessboard within the $$8\times8$$ one.

• Welp, a few minutes after I posted this, Rob Pratt edited his answer from 5 to 0 unattacked squares. +1 to him, but I'll leave this answer in place as it has a nice non-computery method. – Rand al'Thor Feb 8 at 19:22