Order 5 Retention Patterns for the Topographical model
Figures above show a order 5 magic square with 7 of the 9 interior cells retaining water.
figure above shows the 102 “elementary” retention patterns for the order 5 square
Of the 512 patterns only 102 patterns are “elementary patterns, “
Notice that different patterns have different symmetry and thus different numbers of redundant patterns. There are only 8 elementary patterns for examples having 2 cells retaining water.
The above figure shows the retention patterns for the 3600 pandiagonal order 5 magic squares. The top pattern shows that 1800 of the 3600 squares in this set have two cells retaining water in only one pattern. All pandiagonal squares retain water. Of the possible 102 elementary patterns .. only 6 different patterns are found in this subset of order 5 magic squares.
The above figure shows the retention patterns for the 48,544 order 5 associated magic squares.
1266 squares in this set retain no water. The rarest pattern is three cells in a row retaining water on a outer row or column = 542 examples.
figure above shows specific examples of retention patterns . The top example has three cells retaining water in the same row. The water would drain out the outer cell with value 8. Thus the total units retained by this square = 6 units. The triangle pattern at the bottom drains out through the outer cell with value 9. Thus this square retains 7 total units of water
The above figure shows that the pandiagonal set and the associated set of order 5 magic squares do not have any two cell retention patterns in common. There is a set of pandiagonal associated magic squares (one example shown at the beginning of this section). They share a common 3 cell retention pattern.
Computer programming notes:
It is a simple task to code a program that accepts as input the 25 values of the order 5 square and gives as output the 9 values for the water retention for the 9 interior cells. Applying this program to various sets allows a complete depiction of retention patterns and maximum possible retention…etc.
Note 1 example with 59 units retained …. Maximum retention for associated magic square set.