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A261798 Maximum water retention of an associative magic square of order n. 2
0, 0, 0, 15, 59, 0, 361, 704, 1247, 0 (list; graph; refs; listen; history; text; internal format)



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.


Table of n, a(n) for n=1..10.

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.


(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.


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

Sequence in context: A183942 A012691 A020187 * A022287 A288747 A223344

Adjacent sequences:  A261795 A261796 A261797 * A261799 A261800 A261801




Craig Knecht, Sep 01 2015



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Last modified October 17 06:08 EDT 2019. Contains 328106 sequences. (Running on oeis4.)