%I
%S 0,0,0,15,59,0,361,704,1247,0
%N Maximum water retention of an associative magic square of order n.
%C Two of the most famous magic squares are associative magic squares  the Lo Shu magic square and Dürer's magic square. Al Zimmermann's programing contest in 2010 produced the presently known maximum retention values for magic squares order 4 to 28 A201126. No concerted effort has been made to find the maximum retention for associative magic squares.
%C There are 4211744 different water retention patterns for a 7 x 7 square A054247 and 1.12*10^18 different order 7 associative magic squares. There is no proof that the presently stated maximum retention values greater than order 5 are actually the maximum possible retention.
%C a(11) >= 3226, a(12) >=4840, a(13) >= 6972.
%C The Goo Wikipedia associative magic square link below shows the first attempt to classify a set of data by its water retention. Here the 48 associative order 4 magic squares are thus classified. Perhaps there might be some correlation between this surface evaluation and Mohs hardness scale.
%H Goo Wikipedia, <a href="http://wpedia.goo.ne.jp/enwiki/Associative_magic_square">Associative magic square</a>.
%H Craig Knecht, <a href="/A261798/a261798_1.png">Order 5 associative magic square.</a>
%H Craig Knecht, <a href="/A261798/a261798_3.jpg">Order 7 associative magic square.</a>
%H Craig Knecht, <a href="/A261798/a261798_1.jpg">Order 8 associative magic square.</a>
%H Craig Knecht, <a href="/A261798/a261798_4.jpg">Order 9 associative magic square.</a>
%H Craig Knecht, <a href="/A261798/a261798_5.jpg">Order 12 associative magic square.</a>
%H Johan Ofverstedt, <a href="http://uu.divaportal.org/smash/record.jsf?pid=diva2%3A534020">Water Retention on Magic Squares with Constraint Based Local Search</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/User:Knecht03/sandbox#/media/File:Magic_Square_Construction.gif">Magic square construction</a> and <a href="http://en.wikipedia.org/wiki/Water_retention_on_mathematical_surfaces">Water retention on mathematical surfaces</a>.
%e (16 3 2 13)
%e (5 10 11 8)
%e (9 6 7 12)
%e (4 15 14 1)
%e This is Albrecht Dürer's famous magic square in Melencolia I. Dürer put the date of its creation (1514) in the numbers in the bottom row. This square holds 5 units of water.
%Y Cf. A201126 (water retention on magic squares), A201127(water retention on semimagic squares), A261347 (water retention on number squares).
%K nonn,more
%O 1,4
%A _Craig Knecht_, Sep 01 2015
