1. Magic Squares

 

A 5×5 magic square with the maximal retention.

A 5×5 magic square with the maximal retention. https://handwiki.org/wiki/index.php?curid=1957973

Magic squares have been studied for over 2000 years. In 2007, the idea of studying the water retention on a magic square was proposed.^[1] In 2010, a competition at Al Zimmermann's Programing Contests^[2] produced the presently known maximum retention values for magic squares order 4 to 28.^[3] Computing tools used to investigate and illustrate this problem are found here.^[4][5][6][7]

There are 4,211,744 different retention patterns for the 7×7 square. A combination of a lake and ponds is best for attaining maximum retention. No known patterns for maximum retention have an island in a pond or lake.^[1]

The figures below show the 10x10 magic square. Is it possible to look at the patterns above and predict what the pattern for maximum retention for the 10x10 square will be? No theory has been developed that can predict the correct combination of lake and ponds for all orders, however some principles do apply. The first color-coded figure shows a design principle of how the largest available numbers are placed around the lake and ponds. The second and third figures show promising patterns that were tried but did not achieve maximum retention.^[1]

Several orders have more than one pattern for maximum retention. The figure below shows the two patterns for the 11x11 magic square with the apparent maximum retention of 3,492 units:^[3]

This figure also provides an example of a square and its complement that have the same pattern of retention. There are 137 order 4 and 3,254,798 order 5 magic squares that do not retain water.

₃ for L ≤ 51, but R₂ > R₃ for L ≥ 52:^[20]

16 x 16 associative magic square retaining 17840 units. The lake in the first image looks a little uglier than common. Jarek Wroblewski notes that good patterns for maximum retention will have equal or near equal number of retaining cells on each peripheral edge ( in this case 7 cells on each edge) ^[2] The second image is doctored, shading in the top and bottom 37 values.

Maximum-retention magic squares for orders 7-9 are shown below:^[3]

The most-perfect magic squares require all (n-1)^2 or in this case all 121 2x2 planar subsets to have the same sum. ( a few examples flagged with yellow background, red font). Areas completely surrounded by larger numbers are shown with a blue background.^[8]

Before 2010 if you wanted an example of a magic square larger than 5×5 you had to follow clever construction rules that provided very isolated examples. The 13x13 pandiagonal magic square below is such an example. Harry White's CompleteSquare Utility ^[4] allows anyone to use the magic square as a potter would use a lump of clay. The second image shows a 14x14 magic square that was molded to form ponds that write the 1514 - 2014 dates. The animation notes how the surface was sculptured to fill all ponds to capacity before the water flows off the square. This square honors the 500th anniversary of Durer's famous magic square in Melencolia I. allows anyone to use the magic square as a potter would use a lump of clay. The second image shows a 14x14 magic square that was molded to form ponds that write the 1514 - 2014 dates. The animation notes how the surface was sculptured to fill all ponds to capacity before the water flows off the square. This square honors the 500th anniversary of Durer's famous magic square in Melencolia I.

13 x 13 Pandiagonal Magic Sqauare. https://handwiki.org/wiki/index.php?curid=1475210

This figure also provides an example of a square and its complement that have the same pattern of retention. There are 137 order 4 and 3,254,798 order 5 magic squares that do not retain water.^[1]

Magic square pattern. https://handwiki.org/wiki/index.php?curid=1173902	Associative magic square 2013. https://handwiki.org/wiki/index.php?curid=1894224

The figure below is a 17x17 Luo-Shu format magic square.^[9] The Luo-Shu format construction method seems to produce a maximum number of ponds. The drainage path for the cell in green is long eventually spilling off the square at the yellow spillway cell.

The figure to the right shows what information can be derived from looking at the actual water content for each cell. Only the 144 values are highlighted to keep the square from looking too busy. Focusing on the green cell with a base value 7, the highest obstruction on the path out is its neighbor cell with the value of 151 (151-7=144 units retained). Water rained into this cell exits the square at the yellow 10 cell.

Mario Mamzeris invented his own method for constructing odd order magic squares. His Order 19 associative magic square is shown below.^[10]

In the 21 x 21 magic square below all the even numbers form dams and ponds and all the odd numbers provide exit paths.^[11]

The computer age now allows for the exploration of the physical properties of magic squares of any order. The figure below shows the largest magic square studied in the contest. For L > 20 the number of variables/ equations increases to the point where it makes the pattern for maximum retention predictable.

This is a 32x32 panmagic square. Dwane Campbell using binary construction methods produced this interesting water retention example.^[12] The GET TYPE utility applied to this square shows that it has the following properties: 1) normal magic 2) pandiagonal 3) bent diagonal two way 4) self-complement.

2. Random Surfaces

 

Water retention on a random surface of 10 levels.

Water retention on a random surface of 10 levels. https://handwiki.org/wiki/index.php?curid=1931701

Another system in which the retention question has been studied is a surface of random heights. Here one can map the random surface to site percolation, and each cell is mapped to a site on the underlying graph or lattice that represents the system. Using percolation theory, one can explain many properties of this system. It is an example of the invasion percolation model in which fluid is introduced in the system from any random site.^[13][14][15]

In hydrology, one is concerned with runoff and formation of catchments.^[16] The boundary between different drainage basin (watersheds in North America) forms a drainage divide with a fractal dimension of about 1.22.^{[17] [18] [19]}

The retention problem can be mapped to standard percolation.^[20][21][22] For a system of five equally probable levels, for example, the amount of water stored R₅ is just the sum of the water stored in two-level systems R₂(p) with varying fractions of levels p in the lowest state:

R₅ = R₂(1/5) + R₂(2/5) + R₂(3/5) + R₂(4/5)

Typical two-level systems 1,2 with p = 0.2, 0.4, 0.6, 0.8 are shown on the right (blue: wet, green: dry, yellow: spillways bordering wet sites). The net retention of a five-level system is the sum of all these. The top level traps no water because it is far above the percolation threshold for a square lattice, 0.592746.

The retention of a two-level system R₂(p) is the amount of water connected to ponds that do not touch the boundary of the system. When p is above the critical percolation threshold p _c, there will be a percolating cluster or pond that visits the entire system. The probability that a point belongs to the percolating or "infinite" cluster is written as P_∞ in percolation theory, and it is related to R₂(p) by R₂(p)/L² = p − P_∞ where L is the size of the square. Thus, the retention of a multilevel system can be related to a well-known quantity in percolation theory.

To measure the retention, one can use a flooding algorithm in which water is introduced from the boundaries and floods through the lowest spillway as the level is raised. The retention is just the difference in the water level that a site was flooded minus the height of the terrain below it.