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%I #38 Jan 07 2020 11:47:38
%S 0,0,5,26,84,222,488,946,1664,2723,4227,6277,8993,12514,16976,22538,
%T 29364,37649,47563,59321,73149,89254,107892,129308,153764,181547,
%U 212931,248223,287747,331780
%N Maximum water retention of a number square of order n.
%C A number square is an arrangement of numbers from 1 to n*n in an n X n matrix with each number used only once.
%C The number square was used in 2009 as a stepping stone in solving the problem of finding the maximum water retention for magic squares.
%C In June 2009, _Walter Trump_ wrote a program that calculates the maximum water retention in number squares up to 250 X 250.
%C The retention patterns for orders 3, 4, 8 and 11 show perfect symmetry.
%C For orders 5, 7, 30 and 58, more than one pattern gives maximum retention. (For order 7, there are 3 patterns that give maximum retention.)
%H Craig Knecht, <a href="/A261347/a261347_1.jpg">Maximum retention 5 X 5 number square.</a>
%H Craig Knecht, <a href="/A261347/a261347_2.jpg">Maximum retention 6 X 6 number square.</a>
%H Craig Knecht, <a href="/A261347/a261347_5.jpg">Maximum retention 7 X 7 number square.</a>
%H Craig Knecht, <a href="/A261347/a261347_4.jpg">Maximum retention 8 X 8 number square.</a>
%H Craig Knecht, <a href="/A261347/a261347.jpg">Maximum retention 9 X 9 number square.</a>
%H Craig Knecht, <a href="/A261347/a261347_6.jpg">Order 7 - three patterns for maximum retention.</a>
%H Craig Knecht, <a href="/A261347/a261347_7.jpg">Order 30 - two patterns for maximum retention.</a>
%H Craig Knecht, <a href="/A261347/a261347_8.jpg">Pattern comparison table.</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Water_retention_on_mathematical_surfaces">Water retention on mathematical surfaces</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/File:Prime_number_surface_.svg">Retention with prime numbers</a>.
%e (2 6 3)
%e (7 1 8)
%e (4 9 5)
%e The values 6,7,8,9 form the dam with the value 6 being the spillway. 5 units of water are retained above the central cell. The boundaries of the system are open and allow water to flow out.
%Y Cf. A201126 (water retention on magic squares), A201127 (water retention on semi-magic squares).
%K nonn
%O 1,3
%A _Craig Knecht_, Aug 15 2015
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