Although this is an old solution, I’m publishing here because it has been appeared in several books from Cliff Pickover.
Below it’s my first solution and then a better one found by Rodolfo Kurchan and I proved it with an exhaustive search and finding more pandigital magic squares:
I found a smaller solution to your magic square with pandigital numbers, which appeared at Journal of Recreational Mathematics issue 28
1034786295 1035687294 1024796385 1025697384
1024697385 1025796384 1034687295 1035786294
1035697284 1034796285 1025687394 1024786395
1025786394 1024687395 1035796284 1034697285As you can see the sum is 4120967358
Later Rodolfo Kurchan found:
Pandigital magic sum = 4120736958. He says that German Gonzalez-Morris told him that this was now the smallest (just for the 4×4 case, as you will learn in short).
German Gonzalez-Morris added (May 2006) that he made a computer program and found an smaller pandigital sum (4120967358) then Rodolfo (by hand) found the smallest sum (4120736958), finally German found (and prove by exhaustive search) all smallest sums beginning from: 4120736958, 4120953678, 4120967358, 4127360958, 4129536078, …
1034728695 1035628794 1024739685 1025639784
1024639785 1025739684 1034628795 1035728694
1035629784 1034729685 1025638794 1024738695
1025738694 1024638795 1035729684 1034629785
more information can be found at: http://www.primepuzzles.net/puzzles/puzz_249.htm
http://www.mathforum.org/kb/thread.jspa?forumID=265&threadID=611491&messageID=1787982#1787982
http://mathforum.org/kb/message.jspa?messageID=1787984&tstart=0
