|Ultramagic Squares of Order 7|
|In the summary I mention a lower bound for these squares.|
Notice that I mean all pan-magic 7x7-squares not only the regular ones.
I planned to make more accurate statements about this number and therefore tried to split each self-complementary pan-magic square into two auxiliary squares. I used the numbers n from 0 to 48 and filled one auxiliary square with the numbers (n mod 7) and the other with (n div 7). (div means integer-division without remainder) Surprisingly the amount of auxiliary squares is only marginally smaller than the amount of all considered squares. It should be possible to consider all combinitions of auxiliary squares in order to find a large amount of pan-magic 7x7-squares. I haven't got the time to do this right now.
|In August 2001 I started to estimate the number of special magic squares by building
Monte-Carlo-Methods in my calculation-programs.
Some cells are filled with random numbers. This reduces the found squares.
Thus you have to multipy the result with a certain factor to get an estimate of the total number of squares.|
In October 2001 I found some special equations for panmagic order-7 squares, which could be used for estimation. Much computer time was necessary to get a result with acceptable accuracy.
The Monte-Carlo-Method is more useful to find the numbers of normal, symmetrical and semimagic squares, because the probability to find such squares with random numbers is not too low. See the results of my first attempts below.
Already in 1998 K. Pinn and C. Wieczerkowski used another method (Parallel Tempering Monte Carlo) to estimate the amount of regular magic 6x6-squares: (1.7745 ±0.0016)·1019 .
They also reported a less accurate estimate of the number of 7x7-squares: (3.760 ±0.052)·1034 .
|Magic Squares of Order 7
(estimate and standard error)
|Number of Magic Squares From Parallel Tempering Monte Carlo by K. Pinn and C. Wieczerkowski (Uni Münster, Germany).|