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 A261798 Maximum water retention of an associative magic square of order n. 2

%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.diva-portal.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 semi-magic squares), A261347 (water retention on number squares).

%K nonn,more

%O 1,4

%A _Craig Knecht_, Sep 01 2015

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