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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 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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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
How to produce 7x7 panmagic squares
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 pro-
duce 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.
How to produce 11x11 panmagic 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!!!
The 36 essential different panmagic 5x5 squares
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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Key to produce ultra panmagic squares (for odd squares, that are no multiples of 3):
| 1x digit |
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4 |
1 |
2 |
3 |
| 2 |
3 |
0 |
4 |
1 |
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1 |
2 |
3 |
0 |
| 3 |
0 |
4 |
1 |
2 |
| 1 |
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3 |
0 |
4 |
+ 5x digit +1 |
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| 0 |
2 |
4 |
3 |
1 |
| 4 |
3 |
1 |
0 |
2 |
| 1 |
0 |
2 |
4 |
3 |
| 2 |
4 |
3 |
1 |
0 |
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1 |
0 |
2 |
4 |
= ultra panm. 5x5 |
| 1 |
15 |
22 |
18 |
9 |
| 23 |
19 |
6 |
5 |
12 |
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2 |
13 |
24 |
16 |
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21 |
20 |
7 |
3 |
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8 |
4 |
11 |
25 |
Two digits crosswise from the centre give the lowest plus the highest digit, so for a 5x5 square: 1 + 25 = 26.
The keys to produce ultra panmagic 7x7, 11x11, 13x13, 17x17, ... squares are:
0 - 6 - 1 - 2 - 3 - 4 - 5
0 - 10 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9
0 - 12 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11
0 - 16 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 - 14 - 15
...
See ultra (pan) magic 7x7 squares on website: www.trump.de/magic-squares/
More information on page: Pan magic 5x5 square, explanation
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