OFFSET
3,2
COMMENTS
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.
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.
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
LINKS
B. Burger, J. S. Andrade Jr. & H. J. Herrmann, A Comparison of Hydrological and Topological Water Sheds, Nature, 10586, 2018.
Harvey Heinz, Knecht Topographical squares, Summary of contest results.
Craig Knecht, Magic Square - Topographical model
Craig L. Knecht, Walter Trump, Daniel ben-Avraham, and Robert M. Ziff, Retention Capacity of Random Surfaces, Phys. Rev. Lett. 108, 045703, 2012.
Craig Knecht, Pattern comparison table.
Hugo Pfoertner, 4 X 4 Magic square retaining 15 units of water
Hugo Pfoertner, 5 X 5 Magic square retaining 69 units of water
Hugo Pfoertner, 6 X 6 Magic square retaining 192 units of water
Hugo Pfoertner, 7 X 7 Magic square retaining 418 units of water
Hugo Pfoertner, 8 X 8 Magic square retaining 797 units of water
Hugo Pfoertner, 9 X 9 Magic square retaining 1408 units of water
Wikipedia, Water retention on mathematical surfaces
EXAMPLE
See links for illustrations.
CROSSREFS
KEYWORD
nonn,hard,nice,more
AUTHOR
Hugo Pfoertner, Dec 03 2011
STATUS
approved