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CENTROSYMMETRIC SOLUTIONS TO CHESSBOARD SEPARATION PROBLEMS
Abstract
Chessboard separation problems are modifications to classic chessboard problems, such as the N queens problem, in which obstacles are placed on the chessboard. The N + k queens problem requires placements of k pawns and N + k mutually non-attacking queens on an N -by-N chessboard. Here we examine centrosymmetric (half-turn symmetric) and doubly centrosymmetric (quarter-turn symmetric) solutions to the N + k queens problem. We also consider solutions in which the queens have a different type of symmetry than the pawns have.
... For k = 0, take the 6 × 6 board with dragon kings on squares (0, 3), (1, 5), (2, 2), (3, 0), (4, 4) and (5, 1). For k = 1, take the 8 × 8 board with a pawn on square (5, 5) and dragon kings on squares (0, 5), (1, 1), (2,4), (3,6), (4, 2), (5, 0), (5, 7), (6,3), and (7, 5). We use Lemma 12 for another induction on k. ...
... For k = 0, take the 6 × 6 board with dragon kings on squares (0, 3), (1, 5), (2, 2), (3, 0), (4, 4) and (5, 1). For k = 1, take the 8 × 8 board with a pawn on square (5, 5) and dragon kings on squares (0, 5), (1, 1), (2,4), (3,6), (4, 2), (5, 0), (5, 7), (6,3), and (7, 5). We use Lemma 12 for another induction on k. ...
... The bound in Proposition 13 is not tight, as we can see in the monodiagonally symmetric 9 + 3 dragon kings problem solution with pawns on squares (4, 4), (5,6), and (6, 5) and dragon kings on squares (0, 4), (1, 7), (2,5), (3,3), (4, 0), (4, 6), (5, 2), (5,8), (6,4), (6, 6), (7, 1), and (8, 5). ...
A dragon king is a shogi piece that moves any number of squares vertically or horizontally or one square diagonally but does not move through or jump over other pieces. We construct infinite families of solutions to the n + k dragon kings problem of placing k pawns and n + k mutually nonattacking dragon kings on an n × n board, including solutions symmetric with respect to quarter-turn or half-turn rotations, solutions symmetric with respect to one or two diagonal reections, and solutions not symmetric with respect to any nontrivial rotation or reection. We show that an n + k dragon kings solution exists whenever n > k + 5 and that, given some extra conditions, symmetric solutions exist for n > 2k + 5 .
... Chatham et al., in [2,5] defined the symmetric solutions such as ordinary, centrosymmetric, and doubly centrosymmetric solutions on a square board, and proved that all the solutions of N + k queens solutions belong to one of these symmetries. Using the studies done on separation problems on the square boards, this paper extends the work onto the rectangular boards. ...
... Proof. If both M and N are even, the board must have an equal number of pieces on the left half and the right half of the board as mentioned in [5]. Thus, k must be always even. ...
The famous eight queens problem with non-attacking queens placement on an 8 × 8 chessboard was 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 × 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 × N (M < N) to result in a separated board with the maximum number of independent queens. The research work here first shows that M + k queens are separated with 1 pawn and continues to prove that k pawns are required to separate M + k queens. Then it focuses on finding the symmetric solutions to the M + k Queens separation problem.
We define a legal placement of Queens to be any placement in which any two attacking Queens can be separated by a Pawn. The Queens separation number is defined to be equal to the minimum number of Pawns which can separate some legal placement of m Queens on an order n chess board. We prove that n + 1 Queens can be separated by 1 Pawn and conjecture that n + k Queens can be separated by k Pawns for large enough n. We also provide some results on the separation number of other chess pieces.
Abstract Chessboard separation problems are modiflcations to classic chess- board problems, such as the N Queens Problem, in which obstacles are placed on the chessboard. This paper focuses on a variation known as the N + k Queens Problem, in which k Pawns and N + k mutually non-attacking Queens are to be placed on an N-by-N chess- board. Results are presented from performance studies examining the e‐ciency of sequential and parallel programs that count the number of solutions to the N + k Queens Problem using traditional back- tracking and dancing links. The use of Stochastic Local Search for determining existence of solutions is also presented. In addition, pre- liminary results are given for a similar problem, the N +k Amazons.
A legal placement of Queens is any placement of Queens on an order N chessboard in which any two attacking Queens can be separated by a Pawn. The Queens independence separation number is the minimum number of Pawns which can be placed on an N × N board to result in a separated board on which a maximum of m independent Queens can be placed. We prove that N + k Queens can be separated by k Pawns for large enough N and provide some results on the number of fundamental solutions to this problem. We also introduce separation relative to other domination-related parameters for Queens, Rooks, and Bishops.
