cp's OEIS Frontend

Determining the maximum water retention of a magic square has been the subject of the spring 2010 round of "Al Zimmermann's Programming Contests". The following description was given by Al Zimmermann: The scoring function is defined in terms of the physical characteristics of water. Simply stated, pour a gazillion units of water on top of a magic square and measure the water that doesn't run off. The cells in the magic square have heights given by their values and water cannot pass between two cells joined at a vertical edge.

Lower bounds for the next terms are a(10) >= 2267, a(11) >= 3492, a(12) >= 5185, a(13) >= 7445, a(14) >= 10397, a(15) >= 14154.

This water retention model progressed from the specific case of the magic square to a more generalized system of random levels. A quite interesting counter-intuitive finding that a random two-level system will retain more water than a random three-level system when the size of the square is greater than 51 X 51 was discovered. This was reported in the Physical Review Letters in 2012 and referenced in the Nature article in 2018. - Craig Knecht, Dec 01 2018

A number square is an arrangement of numbers from 1 to n*n in an n X n matrix with each number used only once.

The number square was used in 2009 as a stepping stone in solving the problem of finding the maximum water retention for magic squares.

In June 2009, Walter Trump wrote a program that calculates the maximum water retention in number squares up to 250 X 250.

The retention patterns for orders 3, 4, 8 and 11 show perfect symmetry.

For orders 5, 7, 30 and 58, more than one pattern gives maximum retention. (For order 7, there are 3 patterns that give maximum retention.)

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.