A261347 Maximum water retention of a number square of order n.

0, 0, 5, 26, 84, 222, 488, 946, 1664, 2723, 4227, 6277, 8993, 12514, 16976, 22538, 29364, 37649, 47563, 59321, 73149, 89254, 107892, 129308, 153764, 181547, 212931, 248223, 287747, 331780

Offset: 1

Author

Craig Knecht, Aug 15 2015

Keywords

• nonn

Comments

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.)

Examples

			(2 6 3)
(7 1 8)
(4 9 5)
  The values 6,7,8,9 form the dam with the value 6 being the spillway. 5 units of water are retained above the central cell. The boundaries of the system are open and allow water to flow out.
		

Links

• Craig Knecht, Maximum retention 5 X 5 number square.
• Craig Knecht, Maximum retention 6 X 6 number square.
• Craig Knecht, Maximum retention 7 X 7 number square.
• Craig Knecht, Maximum retention 8 X 8 number square.
• Craig Knecht, Maximum retention 9 X 9 number square.
• Craig Knecht, Order 7 - three patterns for maximum retention.
• Craig Knecht, Order 30 - two patterns for maximum retention.
• Craig Knecht, Pattern comparison table.
• Wikipedia, Water retention on mathematical surfaces
• Wikipedia, Retention with prime numbers.

Crossrefs

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