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

Doug Chatham

doi:10.2478/rmm-2026-0002

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

Recreational Mathematics Magazine

Volume 13 (2026): Issue 22 (June 2026)

By: Doug Chatham

Open Access

|May 2026

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 > 1, the answer is $⌊ \frac{2 m}{3} ⌋$ \left\lfloor {{{2m} \over 3}} \right\rfloor plus either 0, 1, or 2.

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