A261798 Maximum water retention of an associative magic square of order n.

0, 0, 0, 15, 59, 0, 361, 704, 1247, 0

Offset: 1

Author

Craig Knecht, Sep 01 2015

Keywords

• nonn
• more

Comments

Two of the most famous magic squares are associative magic squares - the Lo Shu magic square and Dürer's magic square. Al Zimmermann's programming contest in 2010 produced the presently known maximum retention values for magic squares order 4 to 28 A201126. No concerted effort has been made to find the maximum retention for associative magic squares.

There are 4211744 different water retention patterns for a 7 x 7 square A054247 and 1.12*10^18 different order 7 associative magic squares. There is no proof that the presently stated maximum retention values greater than order 5 are actually the maximum possible retention.

a(11) >= 3226, a(12) >= 4840, a(13) >= 6972.

The Wikipedia link below shows the first attempt to classify a set of data by its water retention. Here the 48 associative order 4 magic squares are thus classified. Perhaps there might be some correlation between this surface evaluation and Mohs hardness scale.

Examples

			(16  3  2  13)
(5  10 11   8)
(9   6  7  12)
(4  15  14  1)
This is Albrecht Dürer's famous magic square in Melancholia I. Dürer put the date of its creation (1514) in the numbers in the bottom row. This square holds 5 units of water.
		

Links

• Craig Knecht, Order 5 associative magic square.
• Craig Knecht, Order 7 associative magic square.
• Craig Knecht, Order 8 associative magic square.
• Craig Knecht, Order 9 associative magic square.
• Craig Knecht, Order 12 associative magic square.
• Johan Ofverstedt, Water Retention on Magic Squares with Constraint Based Local Search.
• Wikipedia, Listing by water retention capacity. and Water retention on mathematical surfaces.

Crossrefs

Cf. A201126 (water retention on magic squares), A201127 (water retention on semi-magic squares), A261347 (water retention on number squares).