Abstract
The famous eight queens problem with non-attacking queens placement on an 8 x 8 chessboard was first posed in the year 1848. The queens separation problem is the legal placement of the fewest number of pawns with the maximum number of independent queens placed on an N x N board which results in a separated board. Here a legal placement is defined as the separation of attacking queens by pawns. Using this concept, the current study extends the queens separation problem onto the rectangular board M x N, (M<N), to result in a separated board with the maximum number of independent queens. The research work here first describes the M+k queens separation with k=1 pawn and continue to find for any k. Then it focuses on finding the symmetric solutions of the M+k queens separation with k pawns.
Highlights
In the Queens graph QM×N, the squares on the board are taken as vertices, and edges are formed between two squares if they lie on the same path of the movement
Using the studies done on separation problems on the square boards, this paper extends the work onto the rectangular boards
We know that the independence number of queens on a rectangular board is min(M, N ), where M and N denote the number of rows and columns respectively
Summary
In the Queens graph QM×N , the squares on the board are taken as vertices, and edges are formed between two squares if they lie on the same path of the movement. Suppose M = 3, since we know that the independence number on a rectangular board of order M × N is M we need at least M + 1 columns to place M queens. Number the rows and columns as mentioned in Lemma 4.3, and place the queens as follows: (i) When M is odd, first place M queens as shown in Lemma 4.3 (i).
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
