Numerous but Rare: An Exploration of Magic Squares
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{"title"=>"Numerous but rare: An exploration of magic squares", "type"=>"journal", "authors"=>[{"first_name"=>"Akimasa", "last_name"=>"Kitajima", "scopus_author_id"=>"35875065200"}, {"first_name"=>"Macoto", "last_name"=>"Kikuchi", "scopus_author_id"=>"36043162900"}], "year"=>2015, "source"=>"PLoS ONE", "identifiers"=>{"scopus"=>"2-s2.0-84929340274", "sgr"=>"84929340274", "issn"=>"19326203", "pui"=>"604377741", "doi"=>"10.1371/journal.pone.0125062", "pmid"=>"25973764"}, "id"=>"35ebb6a3-697a-30bd-b5ba-5395d9f16091", "abstract"=>"How rare are magic squares? So far, the exact number of magic squares of order n is only known for n ≤ 5. For larger squares, we need statistical approaches for estimating the number. For this purpose, we formulated the problem as a combinatorial optimization problem and applied the Multicanonical Monte Carlo method (MMC), which has been developed in the field of computational statistical physics. Among all the possible arrangements of the numbers 1; 2, …, n(2) in an n × n square, the probability of finding a magic square decreases faster than the exponential of n. We estimated the number of magic squares for n ≤ 30. The number of magic squares for n = 30 was estimated to be 6.56(29) × 10(2056) and the corresponding probability is as small as 10(-212). Thus the MMC is effective for counting very rare configurations.", "link"=>"http://www.mendeley.com/research/numerous-rare-exploration-magic-squares", "reader_count"=>3, "reader_count_by_academic_status"=>{"Professor > Associate Professor"=>2, "Student > Ph. D. Student"=>1}, "reader_count_by_user_role"=>{"Professor > Associate Professor"=>2, "Student > Ph. D. Student"=>1}, "reader_count_by_subject_area"=>{"Biochemistry, Genetics and Molecular Biology"=>1, "Agricultural and Biological Sciences"=>1, "Physics and Astronomy"=>1}, "reader_count_by_subdiscipline"=>{"Physics and Astronomy"=>{"Physics and Astronomy"=>1}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>1}, "Biochemistry, Genetics and Molecular Biology"=>{"Biochemistry, Genetics and Molecular Biology"=>1}}, "reader_count_by_country"=>{"Japan"=>1}, "group_count"=>0}

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  • {"month"=>"5", "year"=>"2015", "pdf_views"=>"77", "xml_views"=>"5", "html_views"=>"2032"}
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  • {"files"=>["https://ndownloader.figshare.com/files/2067953"], "description"=>"<p>Numbers in the parentheses indicate the statistical errors (3 times the standard error) in the last digits.</p><p>Estimated number and qppearance probability of magic squares.</p>", "links"=>[], "tags"=>["combinatorial optimization problem", "magic square decreases", "Magic squares", "Multicanonical Monte Carlo method", "mmc"], "article_id"=>1414952, "categories"=>["Uncategorised"], "users"=>["Akimasa Kitajima", "Macoto Kikuchi"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0125062.t001", "stats"=>{"downloads"=>1, "page_views"=>13, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Estimated_number_and_qppearance_probability_of_magic_squares_/1414952", "title"=>"Estimated number and qppearance probability of magic squares.", "pos_in_sequence"=>0, "defined_type"=>3, "published_date"=>"2015-05-14 03:03:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/2067950"], "description"=>"<p><i>P</i><sub><i>n</i></sub> decreases faster than exponentially with the size <i>n</i>. Two fitted functions are also shown: exp((<i>An</i> + <i>B</i>)ln(<i>n</i>) + <i>Cn</i> + <i>D</i>) (solid line) and exp((<i>En</i> + <i>F</i>)ln(<i>n</i> + <i>G</i>) + <i>H</i>) (dotted line) with <i>A</i> = −4.99 and <i>E</i> = −4.88. We used <i>P</i><sub><i>n</i></sub> of <i>n</i> ≥ 10 for the fitting. Enlarged plot for <i>n</i> < 6 is shown in the inset, in which difference of two functions are visible.</p>", "links"=>[], "tags"=>["combinatorial optimization problem", "magic square decreases", "Magic squares", "Multicanonical Monte Carlo method", "mmc"], "article_id"=>1414949, "categories"=>["Uncategorised"], "users"=>["Akimasa Kitajima", "Macoto Kikuchi"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0125062.g001", "stats"=>{"downloads"=>1, "page_views"=>12, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Semi_log_plot_of_the_appearance_probability_P_n_of_magic_squares_/1414949", "title"=>"Semi-log plot of the appearance probability <i>P</i><sub><i>n</i></sub> of magic squares (•).", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-05-14 03:03:43"}
  • {"files"=>["https://ndownloader.figshare.com/files/2067952"], "description"=>"<p><i>P</i><sub><i>n</i></sub>/exp{(<i>An</i> + <i>B</i>)ln(<i>n</i>) + <i>Cn</i> + <i>D</i>}} (•) and <i>P</i><sub><i>n</i></sub>/exp{(<i>En</i> + <i>F</i>)ln(<i>n</i> + <i>G</i>) + <i>H</i>} (×) with <i>A</i> = −4.99 and <i>E</i> = −4.88. Both functions seem to express <i>P</i><sub><i>n</i></sub> equally well.</p>", "links"=>[], "tags"=>["combinatorial optimization problem", "magic square decreases", "Magic squares", "Multicanonical Monte Carlo method", "mmc"], "article_id"=>1414951, "categories"=>["Uncategorised"], "users"=>["Akimasa Kitajima", "Macoto Kikuchi"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0125062.g002", "stats"=>{"downloads"=>2, "page_views"=>11, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_The_ratio_of_the_appearance_probability_P_n_to_two_fitted_functions_/1414951", "title"=>"The ratio of the appearance probability <i>P</i><sub><i>n</i></sub> to two fitted functions.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2015-05-14 03:03:43"}

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{"start_date"=>"2015-01-01T00:00:00Z", "end_date"=>"2015-12-31T00:00:00Z", "subject_areas"=>[]}
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