How to produce 5x5 pan magic squares?
Excluding rotation and/or mirroring there are 275.305.224 (x 8 = including 2.202.441.792) different pure
magic 5x5 squares. Of the 275.305.224 squares 3.600 (x 8 = including 28.800) are panmagic (see for
example www.gaspalou.fr/magic-squares/order-5.htm ).
A 5x5 panmagic square can be produced using a method of construction which is comparable with the Sudoku
method (see also on this website). Every row, column and (pan)diagonal must contain all the digits 0, 1, 2, 3
and 4.
Fill in the first row of the 1st square. There are 24 combinations of digits which lead to unique basic solutions
(01234, 01243, 01324, 01342, 01423, 01432, 02134, 02143, 02314, 02341, 02413, 02431, 03124, 03142,
03214, 03241, 03412, 03421, 04123, 04132, 04213, 04232, 04312, 04321).
Fill in row two up to five by moving the first row each time 2 places to the left.
1st square
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Adopt as first row of the 2nd square, the first row of the 1st square. Fill in row two up to five by moving the first
row each time 2 places to the right.
2nd square
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Take a digit from the 1st square multiplied by 5 and add (1x) the digit from the same cell of the 2nd square; add 1 to
each cell.
5x digit + 1x digit = +1 = panmagic 5x5
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On website www.grogono.com/magic/5x5.php you will find the ‘mother method’. With this method can be produced
144 basic panmagic 5x5 squares. On the 2x2 carpet of one of the 144 basic panmagic 5x5 squares you can find 25
different 5x5 sub-squares (144x25=3.600).
You can use this method for odd squares that are no multiples of 3 (= 5x5, 7x7, 11x11, 13x13, 17x17, ...). For
example to produce a 7x7 panmagic square, take the digits 0, 1, 2, 3, 4, 5 and 6 as first row an multiply a
digit from the 1st square by 7.
More information on page "Pan magic 5x5 square, explanation"
Information for whiz kids
On website www.magic-squares.net/pandiag5.htm 36 essential different 5x5 panmagic squares are presented.
If you change the sequence of the rows and the columns into 1-3-5-2-4 and/or swap pandiagonals with rows (see
below) and/or shift a square on the 2x2 carpet, the 36 essential different squares can be transformed to the above
mentioned complete set of 3.600 5x5 panmagic squares.
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