OFFSET
1,4
COMMENTS
The number placement starts with the lowest available number and proceeds from top left to bottom right in two separate passes. The first pass fills in the ponds. The second pass fills in the barrier cells surrounding the ponds.
A number square contains each of the numbers 1 to n*n exactly once.
The water retention model provides the definition of a pond. All the ponds have an area of 1 cell in the maximum pond example.
The immediate environment of a 1-cell pond requires four larger surrounding cells. The water retention model requires the macro environment of possible surrounding cells to be lower than the border of the 1-cell-area pond.
For even-ordered squares one of the main diagonals is made up of ponds. For odd-ordered squares both diagonals are made up of ponds.
The cells in a given row hold identical amounts of water.
A listing of the C code that calculates the water retention is given. The program gives a graphic output where the area of the ponds is color coded. Additional 3D graphics and other water retention utilites are available on Harry White's web page noted below.
The water retention model functions in three dimensions as noted in the crossrefs. The physical interpretation in three dimensions is not straightforward and the term "incarceration" of numbers is introduced.
LINKS
Craig Knecht, Center of mass for the retained water
Craig Knecht, Center symmetric ponds - alternate example
Craig Knecht, Equal number of spillways and ponds
Craig Knecht, Example for the sequence
Craig Knecht, Magic squares with the maximum number of ponds
Craig L. Knecht, Walter Trump, Daniel ben-Avraham, Robert M. Ziff, Retention capacity of random surfaces, arXiv:1110.6166 [cond-mat.dis-nn], 31 Jan 2012.
Harry White, Water retention utilites.
Wikipedia, Water retention on mathematical surfaces
EXAMPLE
Order 5 square retaining 30 units of water. The positions of the remaining numbers that do not occupy the pond cells or their immediate borders are irrelevant and a zero is placed in these positions.
0 6 0 7 0
8 1 9 2 10
0 11 3 12 0
13 4 14 5 15
0 16 0 17 0
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Craig Knecht, Jan 18 2020
STATUS
approved