How many mutually non-attacking, non-knight pieces can we place on an <i>m × n</i> chessboard?

Abstract How many black and white chess pieces – excluding knights – can we place on the squares of an m × n chess board so that none of those pieces attack any other piece? We answer this question for all rectangular boards. If n &gt; 1, the answer is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>⌊</m:mo> <m:mrow> <m:mfrac> <m:mrow> <m:mn>2</m:mn> <m:mi>m</m:mi> </m:mrow> <m:mn>3</m:mn> </m:mfrac> </m:mrow> <m:mo>⌋</m:mo> </m:mrow> </m:mrow> </m:math> \left\lfloor {{{2m} \over 3}} \right\rfloor plus either 0, 1, or 2.