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Chessboard squares

Amanda G. Chetwynd, Susan J. Rhodes
<span title="">1995</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
An array is said to be avoidable if an n x n latin square on the same symbols can be found which differs from the given array in every cell.  ...  We describe a family of arrays, known as chessboard arrays, and classify these arrays as avoidable or non-avoidable. * Corresponding author. 0012-365X/95/$09.50 © 1995--Elsevier Science B.V.  ...  We may assume, without loss of generality, that P has the form of the square of Fig. 6 .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(94)e0206-w">doi:10.1016/0012-365x(94)e0206-w</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/huyine6ofzd4bljentglptxjsm">fatcat:huyine6ofzd4bljentglptxjsm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170926104907/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/3c9/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wMDEyMzY1eDk0ZTAyMDZ3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/48/cb/48cb71562cfd63bf0d47a617e2c2a2fdb53356e4.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(94)e0206-w"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Perfect domination separation on square chessboard

Sowndarya K S P., Lakshmi Naidu Y.
<span title="">2020</span> <i title="Malaya Journal of Matematik"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/d2s26e277ra2vlsq7jnndzx34i" style="color: black;">Malaya Journal of Matematik</a> </i> &nbsp;
The separation number for the rooks, bishops and kings on a square chessboard were obtained by decreasing the perfect domination number mentioned in [6] using the pawns.  ...  In this paper, we find the perfect domination separation on square chessboard by placing the pawns in the place of non-dominating cells to decrease γ p f values obtained in [6] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.26637/mjm0804/0027">doi:10.26637/mjm0804/0027</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/orp2zh4pg5ahzpujzlqxkz3zwu">fatcat:orp2zh4pg5ahzpujzlqxkz3zwu</a> </span>
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Knight's Tours on Rectangular Chessboards Using External Squares

Grady Bullington, Linda Eroh, Steven J. Winters, Garry L. Johns
<span title="">2014</span> <i title="Hindawi Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ukdw2rc3qreyhlaihgmrddedzu" style="color: black;">Journal of Discrete Mathematics</a> </i> &nbsp;
The classic puzzle of finding a closed knight's tour on a chessboard consists of moving a knight from square to square in such a way that it lands on every square once and returns to its starting point  ...  In this paper we define anextended closed knight's tourfor a rectangular chessboard as a closed knight's tour that includes all squares of the board and possibly additional squares beyond the boundaries  ...  Specifically for the knight's tour problem, if we label each square on an 8 × 8 chessboard with an ordered pair using its row and column position, and assign (1, 1) to the upper left square, then a knight  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1155/2014/210892">doi:10.1155/2014/210892</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ihwasibcmbbc3a32lgkmf35uqq">fatcat:ihwasibcmbbc3a32lgkmf35uqq</a> </span>
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Dominator and Total Dominator Coloring on Square Chessboard

K.S.P. Sowndarya, Y. Lakshmi Naidu
<span title="2019-08-30">2019</span> <i title="Research India Publications"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/x7vq7yygtrdvxbmccivqkjeqaq" style="color: black;">Global Journal of Pure and Applied Mathematics</a> </i> &nbsp;
In this paper, we would discuss the dominator and total dominator coloring parameters of bishops and rooks on square chessboard and give the values for dominator chromatic number and total dominator chromatic  ...  number for these chessboard graphs.  ...  ) and Rooks graph ( ) on a square chessboard.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37622/gjpam/15.4.2019.499-504">doi:10.37622/gjpam/15.4.2019.499-504</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ycf4bhytfveatislwg2zlcjogm">fatcat:ycf4bhytfveatislwg2zlcjogm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200725024903/http://ripublication.com/gjpam19/gjpamv15n4_13.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/1f/f0/1ff0d09290cafb7fe51cd175ec29077bacc36d80.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37622/gjpam/15.4.2019.499-504"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Perfect Domination for Bishops, Kings and Rooks Graphs On Square Chessboard

K.S.P. Sowndarya, Sri Sathya Sai Institute of Higher Learning, Y. Lakshmi Naidu
<span title="2018-07-01">2018</span> <i title="House of Scientific Research"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cpzkilke7zfn3oytc5gxra4kei" style="color: black;">Annals of Pure and Applied Mathematics</a> </i> &nbsp;
The chessboard domination problems were said to be origin of the study of graphs.  ...  The n 2 cells in the chessboard of order ݊ × ݊ are treated as vertices and two vertices are adjacent if one can cover the other vertex in one move directly.  ...  ሺ‫,ݑ‬ ‫ݒ‬ሻ = 1 where u and v are two squares, or each King on an ݊ × ݊ chessboard is said to attack its own square and its neighboring squares, i.e., the nine or fewer squares within one move of the King  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.22457/apam.v18n1a8">doi:10.22457/apam.v18n1a8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5si26pymojdc7l43tcwge3mc3q">fatcat:5si26pymojdc7l43tcwge3mc3q</a> </span>
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Knight's Tours on 3 x n Chessboards with a Single Square Removed

Amanda M. Miller, David L. Farnsworth
<span title="">2013</span> <i title="Scientific Research Publishing, Inc,"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/t3hxsybnxjg55o6z4c2pbf2sge" style="color: black;">Open Journal of Discrete Mathematics</a> </i> &nbsp;
The following theorem is proved: A knight's tour exists on all  chessboards with one square removed unless: n is even, the removed square is (i, j) with i + j odd, n = 3 when any square other than the  ...  center square is removed, n = 5, n = 7 when any square other than square (2, 2) or (2, 6) is removed, n = 9 when square (  ...  Introduction A knight's tour on a chessboard is a path, consisting of at least two moves, in which the knight chess piece visits each square on the chessboard exactly once.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4236/ojdm.2013.31012">doi:10.4236/ojdm.2013.31012</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/z6fgstauubcafihdxlfyu7n2qu">fatcat:z6fgstauubcafihdxlfyu7n2qu</a> </span>
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A Method for the Estimation of the Square Size in the Chessboard Image using Gray-level Co-occurrence Matrix

Guan Xu, Xiaotao Li, Jian Su, Rong Chen, Jianfang Liu
<span title="2012-01-01">2012</span> <i title="Walter de Gruyter GmbH"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jx4g6sjrrffvvm3ypedhd7eoyi" style="color: black;">Measurement Science Review</a> </i> &nbsp;
The experiments prove that the described method has the potential to measure square size of the chessboard.  ...  The paper proposes a new simple procedure for measuring the square size employing the gray-level co-occurrence matrix of a chessboard image.  ...  The length and width of the square over the chessboard are obtained by the boundary information in the image.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2478/v10048-012-0011-z">doi:10.2478/v10048-012-0011-z</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/meq6ltz3z5h6feh7nav2lftegi">fatcat:meq6ltz3z5h6feh7nav2lftegi</a> </span>
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Exploring mod 2 n-queens games

Tricia Muldoon Brown, Abrahim Ladha
<span title="2019-09-01">2019</span> <i title="Walter de Gruyter GmbH"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ixiatdie5zabti7ionmxxbhilm" style="color: black;">Recreational Mathematics Magazine</a> </i> &nbsp;
We introduce a two player game on an n × n chessboard where queens are placed by alternating turns on a chessboard square whose availability is determined by the parity of the number of queens already  ...  on the board which can attack that square.  ...  Before we look at some complete chessboards, let us review some chessboard terminology Each square on a n × n chessboard will be indexed by an ordered pair (i, j) where 1 ≤ i ≤ n indicates the row numbered  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2478/rmm-2019-0002">doi:10.2478/rmm-2019-0002</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pyoer45xpbfxhaguqx2q7js7ce">fatcat:pyoer45xpbfxhaguqx2q7js7ce</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200321160140/http://www.math.armstrong.edu/faculty/brown/Brown_Ladha_QueensGame.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/8e/62/8e62326d5e5ab276d3243534807b44de75bebbfd.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2478/rmm-2019-0002"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Exploring mod2 n-queens games [article]

Tricia Muldoon Brown, Abrahim Ladha
<span title="2015-10-10">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We introduce a two player game on an n x n chessboard where queens are placed by alternating turns on a chessboard square whose availability is determined by the number of queens already on the board which  ...  can attack that square modulo two.  ...  Before we look at some complete chessboards, let us review some chessboard terminology Each square on a n × n chessboard will be indexed by an ordered pair (i, j) where 1 ≤ i ≤ n indicates the row numbered  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1510.02875v1">arXiv:1510.02875v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/j2d3qvr73zg3tk237c4efq2rge">fatcat:j2d3qvr73zg3tk237c4efq2rge</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20191016043923/https://arxiv.org/pdf/1510.02875v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c6/3f/c63feed1cd99451005048b745885bcd2df768264.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1510.02875v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

The rook problem on saw-toothed chessboards

Hon-Chan Chen, Ting-Yem Ho
<span title="">2008</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/deqidnohqjdu7gsln5vdm6obre" style="color: black;">Applied Mathematics Letters</a> </i> &nbsp;
A saw-toothed chessboard, or STC for short, is a kind of chessboard whose boundary forms two staircases from left down to right without any hole inside it.  ...  A rook at square (i, j) can dominate the squares in row i and in column j. The rook problem of an STC is to determine the minimum number of rooks that can dominate all squares of the STC.  ...  Concluding remarks In this paper, we define a saw-toothed chessboard (or STC) and model it by its rook graph G r and board graph G b . We also introduce some properties of the rook problem on STCs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.aml.2007.12.003">doi:10.1016/j.aml.2007.12.003</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/meh5pxo6cbdjzm4ufgigexhau4">fatcat:meh5pxo6cbdjzm4ufgigexhau4</a> </span>
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Folding Optimal Polygons from Squares

David Dureisseix
<span title="2006-10-01">2006</span> <i title="Informa UK Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7u5yos36cbcydmndshi7sl26i4" style="color: black;">Mathematics Magazine</a> </i> &nbsp;
When folding a chessboard from an initial square of paper, the color changes are numerous.  ...  John Montroll produced one in [3] from a 36x36 square of paper. Can a similar chessboard be folded from a smaller square of paper? What is the smallest one?  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2307/27642951">doi:10.2307/27642951</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fa2t5nthfnc6douh7typdyj2ra">fatcat:fa2t5nthfnc6douh7typdyj2ra</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200313013413/https://hal.archives-ouvertes.fr/hal-01380815/file/ChessboardColorChange.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/2d/38/2d386801be7554ebef05c7041e7b26f3562d27c1.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2307/27642951"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> jstor.org </button> </a>

Design of smart chess board that can predict the next position based on FPGA

Alaa Hamza Omran, Yaser Muhammad Abid
<span title="">2018</span> <i title="ASTES Journal"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5z5fhzfgard2vod5er2hdyuq54" style="color: black;">Advances in Science, Technology and Engineering Systems</a> </i> &nbsp;
with a high efficiency in the prediction of the position of any piece on the chessboard and the similarities to the human brain.  ...  This paper proposed an intelligent chessboard which works in a way that similar to the human brain that predicts the next positions for any piece of the chess.  ...  There are also 64 LASER diodes which are equally distributed as the push buttons, one LED for each square of the chessboard.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.25046/aj030417">doi:10.25046/aj030417</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ufhvyk7q2bccfd422vehbo4ona">fatcat:ufhvyk7q2bccfd422vehbo4ona</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200216042119/https://www.astesj.com/publications/ASTESJ_030417.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d9/b3/d9b3cdc8f07cf22c34768f6df34ce538ca5e26e1.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.25046/aj030417"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Relation-algebraic modeling and solution of chessboard independence and domination problems

Rudolf Berghammer
<span title="">2012</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/anq6kvwvqbegtipq5vfhvonzqq" style="color: black;">The Journal of Logic and Algebraic Programming</a> </i> &nbsp;
It can easily be applied to other chessboard problems.  ...  We describe a simple computing technique for solving independence and domination problems on rectangular chessboards.  ...  (b) Arrange n − m − 1 bishops along row d of the chessboard starting with square d, d + 1 and ending with square d, n − d .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jlap.2012.05.001">doi:10.1016/j.jlap.2012.05.001</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nkp4yznmyfhg3eij7xqdmjkeeu">fatcat:nkp4yznmyfhg3eij7xqdmjkeeu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190415205307/https://core.ac.uk/download/pdf/82639670.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c7/76/c7763e0f7ec82f29245b0e3fe1bf3085bf96b952.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jlap.2012.05.001"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Across the board: The mathematics of chessboard problems

Arthur T. Benjamin
<span title="">2005</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/gsfwd6dwnzc37ejtshafjrqxiy" style="color: black;">The Mathematical intelligencer</a> </i> &nbsp;
The book begins with a discussion of knight's tours whereby a chess knight visits every square on a chessboard exactly once, beginning and ending the tour at the same square.  ...  Since knight moves alternate colors, then on an 8-by-8 chessboard, we could place 32 knights on all the black squares or all of the white squares, and no two knights will attack each other.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/bf02985847">doi:10.1007/bf02985847</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/u3gmpdkqdbcf3aekb4l2y3qtnu">fatcat:u3gmpdkqdbcf3aekb4l2y3qtnu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200319191446/https://scholarship.claremont.edu/cgi/viewcontent.cgi?referer=&amp;httpsredir=1&amp;article=1124&amp;context=hmc_fac_pub" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/85/2d/852d4825be3f0970bdade736e411e80a0316aa1b.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/bf02985847"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

The Problem of Pawns [article]

Tricia Muldoon Brown
<span title="2018-11-24">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
of nonattacking pawns on a 2n × 2m chessboard is m+n n^2.  ...  Using a bijective proof, we show the number of ways to arrange a maximum number of nonattacking pawns on a 2m× 2m chessboard is 2m m^2, and more generally, the number of ways to arrange a maximum number  ...  For 1 ≤ j ≤ m + 1, let m 1,j be the arrangement where the leftmost (m + 1 − j) 2 × 2 squares of the rectangular chessboard are of Type A and the remaining rightmost (j − 1) squares are of Type B.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1811.09606v1">arXiv:1811.09606v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nialy2ytjbaxxcv4ax3sut6qnm">fatcat:nialy2ytjbaxxcv4ax3sut6qnm</a> </span>
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