Problems & Puzzles: Puzzles

Puzzle 849. 9x9 Magical square and pandigitals. Peter Kogel wrote on Sep-11-16: I recently took up the challenge of finding pandigital magic squares.  I learned from C. Rivera's web site (Prime puzzles and problems connection) that such squares of 3x3 and 4x4 are already known (puzzle 249) and I wondered whether it is possible to create one of the 9th order. I have never seen one published before so I am particularly pleased to present the example I found below (8-Sep-2016)   1093485267 1093476528 1093542768 1093524786 1093527864 1093478256 1093472856 1093546728 1093482567 1093528764 1093528647 1093478562 1093478652 1093526487 1093485726 1093487625 1093476285 1093546872 1093527468 1093486752 1093524678 1093542687 1093487265 1093527486 1093485762 1093482657 1093472865 1093476852 1093526748 1093485276 1093486725 1093546782 1093526478 1093527846 1093485627 1093475286 1093487256 1093486527 1093528746 1093478526 1093482756 1093546827 1093526847 1093475268 1093524867 1093475826 1093482675 1093475682 1093542678 1093526784 1093475628 1093542876 1093542786 1093472685 1093485672 1093546287 1093486572 1093528476 1093475862 1093524687 1093482765 1093478625 1093528674 1093542867 1093526874 1093487652 1093476825 1093476258 1093486275 1093528467 1093524876 1093487526 1093527648 1093476582 1093527684 1093478265 1093487562 1093486257 1093482576 1093524768 1093546278   This example contains a series of consecutive pandigital numbers in which all rows, columns and the two main diagonals add to the pandigital sum 9841537620. By 'consecutive' I mean an ordered set of distinct pandigital numbers from 1093472685 to 1093546872 in which no other pandigital numbers exist.  I found this example surprisingly easily using a combination of manual manipulation and a simple UBASIC program, so I have no doubt there are many other examples (a pandiagonal example would be brilliant)!   The magic constant, 9841537620 I believe to be the largest possible for this type of square (i.e. consecutive-pandigital).  There are  13 other smaller series that may give rise to a magic square with a smaller magic constant but I'll leave those as a challenge for one of your other readers to find. Q1. Confirm that this 9x9 magic square composed by 81 consecutive pandigitals is the one with the largest possible pandigital sum. Q2. Obtain the 9x9 magic square composed by 81 consecutive pandigitals with the smallest possible digital sum.

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