There are two bimagic 8x8-squares which retain 620 units of water. No other SQ retain more water.
When all entries are squared the water retention of both squares is 37908, the maximum value.
The embenkments of the two squares are equal.
The water retention WR2 of the squares with the squared numbers is not only dependent on the water retention WR1 of the square itself.
WR2 also depends on the water area. In general the same WR1 with smaller area causes greater WR2.
Here is a sample with WR1 = 595 and WR2 = 37903. There is only one lake with an area of 18 units.
This squares has also got the maximum water density (water retention / water area).
The minimum water retention for bimagic 8x8-squares is 17. The squared numbers retain 229, which is also the minimum.
There are 13 SQ with the minimum water area of 3 units. Here are two samples with retention 25 and 70.
There are 4 SQ with the maximum water area of 26 units. Two of them with WR1=525 and WR2=30973 have got an island.
The other two squares retain WR1=516 and WR2=30128.
The maximum retention of centrally symmetric (= associative) bimagic 8x8-squares is 576.
The following squares have got a nice symmetric water pattern.
There are only 8 different super-symmetric water pattern possible (symmetric with respect to four axis).
See all 3660 unique squares with super-symmetric water pattern.
The graphics were created by Craig Knecht (who had the idea of water retention) with a program of Harry White based on an algorithm of Claudio Rocchini.
