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Reflections on the N + k Queens Problem
... For example, the classic n queens problem asks for placements of n queens on an n × n board so that no two queens are on the same row, column, or diagonal [1]. The n + k queens problem asks for placements of k pawns and n + k queens on an n × n board so that each pair of queens on the same row, column, or diagonal has at least one pawn between them [5]. The n queens problem has a solution for n = 1 and each n 4 [1]. ...
... For k = 1, 2, 3, the n + k queens problem has a solution for each n k + 5. For k 4, the n + k queens problem is known to have solutions for n > 25k, but it is suspected that 25k is much larger than the true lower bound [5]. ...
... In [5], a connection was established between the n + k queens problem and alternating sign matrices (ASMs), which are matrices consisting of 0s, 1s, and −1s where the nonzero elements alternate in sign and the first and last nonzero element of each row and column is a 1 [3]. We note a similar connection for the n + k dragon kings problem. ...
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. ...
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.
... Proof. We prove this lemma by first dividing the board as in [1] and from Fig.7(a), which shows the board partitioned at height 2 from bottom with width 3 till the (n − 1) th column and the remaining board at height 3. Now place a king in each of the sub-boards satisfying the concept of perfect domination. ...
... In [7] algorithms that count the number of solutions to the N + k queens problem for various values of N and k were presented and compared. The first-named author of this paper recently considered symmetric solutions to the N + k queens problem in [5] and showed that all solutions to the N + k queens problem (where N > 1) are of one of the following three types: ...
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 a chessboard graph and a given graph parameter π, a π separation number is the minimum number of pawns for which some arrangement of those pawns on the board will produce a board where π has some desired value. We extend previous results on independence and domination separation. We also consider separation of other domination-related parameters.
The paper presents new ways of n-queens problem solving . Briefly, this is a problem on a nxn chessboard of a set n-queens, so that any two of them are not in check. At the beginning, currently used algorithm to find solutions is discussed. Then sequentially 4 new algorithms, along with the interpretation of changes are given. The research results, including comparison, of calculation times of all algorithms together with their interpretation are discussed. Finally, conclusions are given. The results were obtained thanks to the pre-created application. Chapters except for "By filtering ver. 2" were based on the previous studies carried out during the Bachelor course [1].
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.
: The number of n n matrices whose entries are either -1, 0, or 1, whose row- and column- sums are all 1, and such that in every row and every column the non-zero entries alternate in sign, is proved to be [1!4! ...(3n- 2)!]/[n!(n +1)!...(2n- 1)!], as conjectured by Mills, Robbins, and Rumsey. 1 original version written December 1992. The Maple package ROBBINS accompanying this paper, can be downloaded from the www address in footnote 2 below. 2 Department of Mathematics, Temple University, Philadelphia, PA 19122, USA. E-mail:zeilberg@math.temple.edu. WWW:http://www.math.temple.edu/~ zeilberg. Supported in part by the NSF. 3 See the Exodion for a#liations, attribution, and short bios. 1 ### ############14164 ## ############# # ### ####### #### # INTRODUCTION The number of permutations ("houses") that can be made using n objects ("stones"), for n # 7, is given in Sepher Yetsira (Ch. IV, v. 12), a Cabalistic text written more than 1700 years ago. The general formula, n!, ...
An alternating sign matrix is a square matrix such that (i) all entries are 1, −1, or 0, (ii) every row and column has sum 1, and (iii) in every row and column the nonzero entries alternate in sign. Striking numerical evidence of a connection between these matrices and the descending plane partitions introduced by Andrews (Invent. Math.53 (1979), 193–225) have been discovered, but attempts to prove the existence of such a connection have been unsuccessful. This evidence, however, did suggest a method of proving the Andrews conjecture on descending plane partitions, which in turn suggested a method of proving the Macdonald conjecture on cyclically symmetric plane partitions (Invent. Math.66 (1982), 73–87). In this paper is a discussion of alternating sign matrices and descending plane partitions, and several conjectures and theorems about them are presented.
The N-Queens problem is a commonly used example in computer science. There are numerous approaches proposed to solve the problem. We introduce several definitions of the problem, and review some of the algorithms. We classify the algorithms for the N-Queens problem into 3 categories. The first category comprises the algorithms generating all the solutions for a given N. The algorithms in the second category are desinged to generate only the fundamental solutions [34]. The algorithms in the last category generate only one or several solutions but not necessarily all of them.
Two stones build two houses. Three build six houses. Four build four and twenty houses. Five build hundred and twenty houses. Six build Seven hundreds and twenty houses. Seven build five thousands and forty houses. From now on, [exit and] ponder what the mouth cannot speak and the ear cannot hear. (Sepher Yetsira IV,12) Abstract: The number of n × n matrices whose entries are either −1, 0, or 1, whose row- and column- sums are all 1, and such that in every row and every column the non-zero entries alternate in sign, is proved to be [1!4!... (3n −2)!]/[n!(n+1)!... (2n −1)!], as conjectured by Mills, Robbins, and Rumsey.
Robbins conjectured, and Zeilberger recently proved, that there are 1!4!7!...(3n-2)!/n!/(n+1)!/.../(2n-1)! alternating sign matrices of order n. We give a new proof of this result using an analysis of the six-vertex state model (also called square ice) based on the Yang-Baxter equation.
An alternating sign matrix is a square matrix with entries 1, 0 and -1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the symmetry classes of alternating sign matrices and their variations, G. Kuperberg associate square ice models with appropriate boundary conditions, and give determinanat and Pfaffian formulae for the partition functions. In this paper, we utilize several determinant and Pfaffian identities to evaluate Kuperberg's determinants and Pfaffians, and express the round partition functions in terms of irreducible characters of classical groups. In particular, we settle a conjecture on the number of vertically and horizontally symmetric alternating sign matrices (VHSASMs).
In a previous article [math.CO/9712207], we derived the alternating-sign matrix (ASM) theorem from the Izergin-Korepin determinant for a partition function for square ice with domain wall boundary. Here we show that the same argument enumerates three other symmetry classes of alternating-sign matrices: VSASMs (vertically symmetric ASMs), even HTSASMs (half-turn-symmetric ASMs), and even QTSASMs (quarter-turn-symmetric ASMs). The VSASM enumeration was conjectured by Mills; the others by Robbins [math.CO/0008045]. We introduce several new types of ASMs: UASMs (ASMs with a U-turn side), UUASMs (two U-turn sides), OSASMs (off-diagonally symmetric ASMs), OOSASMs (off-diagonally, off-antidiagonally symmetric), and UOSASMs (off-diagonally symmetric with U-turn sides). UASMs generalize VSASMs, while UUASMs generalize VHSASMs (vertically and horizontally symmetric ASMs) and another new class, VHPASMs (vertically and horizontally perverse). OSASMs, OOSASMs, and UOSASMs are related to the remaining symmetry classes of ASMs, namely DSASMs (diagonally symmetric), DASASMs (diagonally, anti-diagonally symmetric), and TSASMs (totally symmetric ASMs). We enumerate several of these new classes, and we provide several 2-enumerations and 3-enumerations. Our main technical tool is a set of multi-parameter determinant and Pfaffian formulas generalizing the Izergin-Korepin determinant for ASMs and the Tsuchiya determinant for UASMs [solv-int/9804010]. We evaluate specializations of the determinants and Pfaffians using the factor exhaustion method.
It was shown by Kuperberg that the partition function of the square-ice model
related to the quarter-turn symmetric alternating-sign matrices of even order
is the product of two similar factors. We propose a square-ice model whose
states are in bijection with the quarter-turn symmetric alternating-sign
matrices of odd order, and show that the partition function of this model can
be also written in a similar way. This allows to prove, in particular, the
conjectures by Robbins related to the enumeration of the quarter-turn symmetric
alternating-sign matrices.
It was shown by Kuperberg that the partition function of the square-ice model related to half-turn symmetric alternating-sign matrices of even order is the product of two similar factors. We propose a square-ice model whose states are in bijection with half-turn symmetric alternating-sign matrices of odd order. The partition function of the model is expressed via the above mentioned factors. The contributions to the partition function of the states corresponding to the alternating-sign matrices having 1 or -1 as the central entry are found and the related enumerations are obtained. Comment: 28 pages, 14 figures
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