| The 5x5 Magic Square Index | Main 5x5 Page Analysis All 5x5 Squares |
This page provides a listing of the 144 possible order-five pan-magic squares. Attempts to provide complete listings of magic squares go back to at least 1693 when Frenicle in France concluded there were 880 squares of order-4. (This, of course, included all varieties, not just the pan-magic ones.) Henry Dudeney, in his book first published in 1917 claims that these results have been ". . .verified over and over again", but adds a sentence later that: "the enumeration of the examples of any higher order is a completely unsolved problem." (Amusements in Mathematics, Dudeney HE. Dover Publication 1970, Page 119). He wrote this before todays proliferation of powerful personal computers. Today such ennumeration is simpler. However, it seems more valuable, and interesting, to seek common underlying patterns rather than the grand total.
The squares were derived by substituting selected sets of numbers for the high and low order letters of the 5x5 pan-magic Graeco-Latin Square. Six sets were substituted for the High order letters and all 24 sets were substituted for the low order letter. This technique produced 144 distinct squares with different unique identifiers:
Each square has 200 variations through 25 translocations, 4 rotations and 2 reflections. There are, therefore, 144 x 25 x 4 x 2 = 28800 possible regular pan-magic squares of order-4.
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Scheme for creating squaresConstruction.Table 1 (Above) shows the sets of numbers which are used to substitute for the High Order (Capital) letters in the Graeco Latin Square. For each selected row of numbers from Table 1, each of the 24 rows in Table 2 (Right) is used in turn to substitute for the Low Order (Lower Case) letters. This procedure generates 144 unique regular 5x5 pan-magic squares. The squares are then assigned unique scores and ranked in ascending order by score. Orientation.The construction automatically places the zero in the top left square with the high order 1 beside it. Many squares are, therefore, already aligned to produce a minimum score. Those that were not are reflected so that the score is the minimum. Score.The score is calculated for each square to be sure there are no duplicates. Where the columns are labeled A, B, C, etc., and the rows are labeled 1, 2, 3, etc: Score = 25*(25*(25*B1+C1)+A2)+A3 This effectively multiplies the four numbers used in the score by descending powers of 25. The scores are displayed ranked by the score in ascending order. Analysis.These squares were compared to an earlier set in which every possible combination used all 24 combinations in both axes. The earlier set produced four copies of each square; the comparison confirmed that only 144 unique squares exist and that the above simpler construction is sufficient. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Above each square is the number of the square and the unique score. Reminder: all of these squares start ot 0 and finish at 24. If you wish to compare them to a conventional square starting with 1 and ending with 25, add one to each cell.
| The 5x5 Magic Square Index | Main 5x5 Page Analysis All 5x5 Squares |
| Copyright © Mar 2010 | Magic Squares Website |
Updated Mar 6, 2010 |