- Table of Contents
- Introduction
- Part One
- Part Two
- The Luo Shu - 3x3 Magic Square
- The Enneagram and the Lo Shu
- 5x5 Magic Square
- 7x7 Magic Square
- 9x9 Magic Square
- 11x11 Magic Square
- 13x13 Magic Square
- Discussion - 13x13 Magic Square
- 15x15 Magic Square
- 17x17 Magic Square
- 19x19 Magic Square
- 21x21 Magic Square
- 23x23 Magic Square
- 25x25 Magic Square
- 27x27 Magic Square
- The Significance of the 27x27 Luo Shu Magic Square
- Part Three
- Bibliography
Outside Contributions
FIRST SUBMISSION
post date 9/9/09
all submissions should be sent to CONTACT
WATER RETENTION SQUARES
by Craig Knecht
The topographical model for the magic square is served adequately for orders < 6 by the "list all pathways" method; the code and flow sheet is shown above. Gareth McCaughan provided a working algorithm in Python that will determine the water capacity of any order of square.
Walter Trump thought the most interesting direction for work on the topographical model would be to determine the patterns for maximum retention. Trump looked at all 500 billion semimagic 5x5 Magic Squares and found 2 examples with 78 units retained in a pattern of two ponds. Trump examined 7x7 Magic Squares and found a pond and a lake hold more water than a single lake. He sorted through his data set of > 20 million order 7 pandiagonal associated Magic Squares and found that 284 was the maximum retention with 6 examples retaining no water. Trump then used normal number squares to provide a example of patterns for maximum retention as the order of the square increases. These patterns may translate to Magic Squares.
Ultimately, Trump found that large populations of Magic Squares for orders < 250 could be readily generated by a random number generator and then evaluated for examples with maximum retention with a ultrafast version of the retention algorithm.
It is fascinating to be able to create examples of higher order Magic Squares .. say order 150 .. that have a large lake capacity.
The above shows the program sampled 200,000 16x16 Magic Squares. The program calculated the water capacity of each square and shows the square with the maximum retention. It also shows the average retention for the group at 9143. The program took approximately 4 hours on my computer to accomplish this.
Blue = cells retaining water. Red = barrier cells. Green = all other cells.
It should be noted that the example above used the "lake" option. Trump defines a lake as a body of water that has a width of N-2 and a height of N-2 where N = order of the square. All other water retaining bodies are referred to as "ponds"
The 16x16 Magic Square has 16408 units retained. The numbers below in blue show the number of units of water retained for each cell. The program writes the square to an excel spread sheet file that has outoput as shown above.
If you find a example of a 16x16Magic Square with > 16,408 units retained or if you have an interest in having a copy of the water retention program, McCaughan’s algorithm or spread sheet data on the retention patterns for the known sets of 5x5 and 7x7 Magic Squares, send me (Craig Knecht) a e-mail.
Illustration provided by Walter Trump