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%I #70 Jul 02 2020 04:54:32
%S 0,15,69,192,418,797,1408
%N Maximum water retention of a magic square of order n.
%C Determining the maximum water retention of a magic square has been the subject of the spring 2010 round of "Al Zimmermann's Programming Contests". The following description was given by Al Zimmermann: The scoring function is defined in terms of the physical characteristics of water. Simply stated, pour a gazillion units of water on top of a magic square and measure the water that doesn't run off. The cells in the magic square have heights given by their values and water cannot pass between two cells joined at a vertical edge.
%C Lower bounds for the next terms are a(10) >= 2267, a(11) >= 3492, a(12) >= 5185, a(13) >= 7445, a(14) >= 10397, a(15) >= 14154.
%C This water retention model progressed from the specific case of the magic square to a more generalized system of random levels. A quite interesting counter-intuitive finding that a random two-level system will retain more water than a random three-level system when the size of the square is greater than 51 X 51 was discovered. This was reported in the Physical Review Letters in 2012 and referenced in the Nature article in 2018. - _Craig Knecht_, Dec 01 2018
%H B. Burger, J. S. Andrade Jr. & H. J. Herrmann, <a href="https://www.nature.com/articles/s41598-018-28470-2">A Comparison of Hydrological and Topological Water Sheds</a>, Nature, 10586, 2018.
%H Harvey Heinz, <a href="http://www.magic-squares.net/square-update-2.htm#Knecht%20Topographical%20squares">Knecht Topographical squares,</a> Summary of contest results.
%H Craig Knecht, <a href="http://www.knechtmagicsquare.paulscomputing.com/topographical.html">Magic Square - Topographical model</a>
%H Craig L. Knecht, Walter Trump, Daniel ben-Avraham, and Robert M. Ziff, <a href="http://dx.doi.org/10.1103/PhysRevLett.108.045703">Retention Capacity of Random Surfaces</a>, Phys. Rev. Lett. 108, 045703, 2012.
%H Craig Knecht, <a href="/A261347/a261347_8.jpg">Pattern comparison table.</a>
%H Hugo Pfoertner, <a href="/A201126/a201126.png">4 X 4 Magic square retaining 15 units of water</a>
%H Hugo Pfoertner, <a href="/A201126/a201126_1.png">5 X 5 Magic square retaining 69 units of water</a>
%H Hugo Pfoertner, <a href="/A201126/a201126_2.png">6 X 6 Magic square retaining 192 units of water</a>
%H Hugo Pfoertner, <a href="/A201126/a201126_3.png">7 X 7 Magic square retaining 418 units of water</a>
%H Hugo Pfoertner, <a href="/A201126/a201126_4.png">8 X 8 Magic square retaining 797 units of water</a>
%H Hugo Pfoertner, <a href="/A201126/a201126_5.png">9 X 9 Magic square retaining 1408 units of water</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Water_retention_on_mathematical_surfaces">Water retention on mathematical surfaces</a>
%e See links for illustrations.
%Y Cf. A201127 (water retention of semi-magic squares), A261347 (water retention of number squares), A261798 (water retention of an associative magic square).
%K nonn,hard,nice,more
%O 3,2
%A _Hugo Pfoertner_, Dec 03 2011
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