Perfect magic squares - Pan magic 5x5 square

Pan magic 5x5 square

	Perfect magic squares

Pan magic 5x5 square

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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 ) and 16 of the 3.600 pan magic 5x5 squares are
ultra pan magic; see http://mathsforeurope.digibel.be/magic.htm.

A 5x5 panmagic square can be produced using a method of construction which is comparable with the Sudoku
method. Every row, column and (pan)diagonal must contain all the digits 0, 1, 2, 3 and 4.

Fill in the first row of the 1^st 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.

1^st square

								0	1	2	3	4	0	1	2	3	4	0	1	2	3	4
						0	1	2	3	4	0	1	2	3	4	0	1	2	3	4
				0	1	2	3	4	0	1	2	3	4	0	1	2	3	4
		0	1	2	3	4	0	1	2	3	4	0	1	2	3	4
0	1	2	3	4	0	1	2	3	4	0	1	2	3	4

Adopt as first row of the 2^nd square, the first row of the 1^st square. Fill in row two up to five by moving the first
row each time 2 places to the right.

2^nd square

0	1	2	3	4	0	1	2	3	4	0	1	2	3	4
		0	1	2	3	4	0	1	2	3	4	0	1	2	3	4
				0	1	2	3	4	0	1	2	3	4	0	1	2	3	4
						0	1	2	3	4	0	1	2	3	4	0	1	2	3	4
								0	1	2	3	4	0	1	2	3	4	0	1	2	3	4

Take a digit from the 1^st square multiplied by 5 and add (1x) the digit from the same cell of the 2^nd square; add 1 to
each cell.

5x digit + 1x digit = +1 = panmagic 5x5

0	1	2	3	4	0	1	2	3	4	0	6	12	18	24	1	7	13	19	25
2	3	4	0	1	3	4	0	1	2	13	19	20	1	7	14	20	21	2	8
4	0	1	2	3	1	2	3	4	0	21	2	8	14	15	22	3	9	15	16
1	2	3	4	0	4	0	1	2	3	9	10	16	22	3	10	11	17	23	4
3	4	0	1	2	2	3	4	0	1	17	23	4	5	11	18	24	5	6	12

Notify that to produce the second square you can use a different combination of digits (see above) than you have
used to produce the first square. If you use the combinations 01234, 01243, 01324, 01342, 01423 or 01432 as first
row to produce the second square, than you can produce all 24 x 6 (combinations of digits) x 25 (by shifting over
the carpet) x 8 (by rotation and/or mirroring) 28.800 possible 5x5 panmagic squares.

On website www.grogono.com/magic/5x5.php you will find the ‘mother method’. With this method can be produced
(24 x 6 =)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).

Notify that the above produced panmagic 5x5 square is basis panmagic square number 2 on website www.grogono.
com/5x5pan144.php

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-a-b-c-d-e-f (instead of a up to f, you must use six
different digits out of 1 up to 6; so there are 6x5x4x3x2 = 720 possibilities!!!) as first row and multiply a digit from
the 1st square by 7. If you move the digits 3 (instead of 2) places to the left/right, than you get even more solutions
(see www.grogono.com/magic/7x7.php). There are the following 6 combinations to produce the first square / second
square:

shift 2 left / shift 2 right
shift 2 left / shift 3 right
shift 2 left / shift 3 left
shift 3 left / shift 2 right
shift 3 left / shift 3 right
shift 3 left / shift 2 left

It is possible to produce all 6 (combinations first square / second square) x 720 (combinations of digits first square)
x 720 (combinations of digits second square) x 49 (possibilties on the 2x2 carpet) x 8 (by rotation and/or mirroring)
/ 4 (correction for duplicate solutions) is 304.819.200 panmagic 7x7 squares.

Notify that you can produce by shift 2/3/4/5 & left/right 89.227.651.645.440.000, that is 89 billiard (89 million x milliard)
different panmagic 11x11 squares!!!

See ultra (pan) magic 7x7 squares on website: www.trump.de/magic-squares/

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.