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I can think of a solution with 7 pieces required. 7 Queens on a diagonal with one missing in the corner. Is there a solution which requires less pieces than this? If so, does it generalise to an nxn board?

Joshua Farrell
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    Do you mean the minimum number of white pieces so that every square not occupied by a white piece is under attack, or that every square is under attack by a white piece not occupying the square? – fleablood Jun 20 '18 at 21:08

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Because queens attack the most squares, any problem where you are looking for the minimum number of nondistinct pieces is looking for the minimum number of queens attacking every square on the board. This link has a good description of what work has been done on the problem, but it does not seem to have a closed form generalized solution at the moment:

http://mathworld.wolfram.com/QueensProblem.html

SlipEternal
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    Certainly, queens attack the same squares and more compared to any of bishop, king, rook, pawn., however queens do not attack in the same fashion as knights so it is conceivable that there may be some optimal strategy for some board which involves a combination of queens and knights rather than one which involves only queens. – JMoravitz Jun 20 '18 at 21:12
  • It's conceivable there is a position with four queens and one knight that is as good as five queens. But to have one that is better is at most 4 pieces. It may or may not be possible to calculate that the maximum number of squares attacked be 3 queens and a knight is less than 64. Or maybe not. – fleablood Jun 20 '18 at 21:17
  • Each queen may attack as many as 27 squares. Each knight may attack as many as 8 squares. It is a simple numbers game. If you want the minimum number of pieces, you will still likely use only queens. – SlipEternal Jun 20 '18 at 21:18