How to Produce Perfect Magic Squares

7/31/2019 How to Produce Perfect Magic Squares

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How to produce

perfect

magic squares

Author: Arie Breedijk

(8 july 2011)

2 1 3 0

1 2 0 3

0 3 1 2

3 0 2 1


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Index

1. Introduction 3

2. Most magic square per order 4

3. 3x3 magic square 10

4. 3x3 magic square, explanation 11

5. Sudoku method (1) 13



8. 4x4 panmagic square 30

9. 4x4 panmagic square, explanation 32

10. 4x4 panmagic square, binary 35

11 Drer and Franklin transformation 37

12 Transformation method 40

13. Transformation method, analysis 46


15. 5x5 panmagic square, explanation 51


17. Ultra magic 8x8 squares 54

18. Most perfect magic square, explanation 57

19. 8x8 most perfect magic square, binary 61

20. Khajuraho method 66

21. Khajuraho method, explanation 68

22. Basic pattern method (1a) 70

23. Basic pattern method (1b) 77

24. Basic pattern method (2) 79



27. Basic pattern method (3c) 85





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33. Analysis 8x8 Franklin panmagic 104

34. Analysis 8x8 Franklin panmagic (2) 110

35. Basic key method 1 115

36. Basic key method 2 118

37. Quadrant method 122

38. 9 x 9 panmagic square 166

39. 3x extra magic 9x9 square 191


41. composite 12x12 magic square 198



44. 3x extra magic 15x15 square 214

45. The perfect magic square 219

46. 3x extra magic 18x18 magic square 226

47. Ultra panmagic 25x25 square 231



50. extra magic 35x35 square 250

51. Bordered squares 254

52. Inlaid square (1) 259

53. Inlaid square (2) 263

54. Each magic sum 272

55. Water retention challenge 273

56. Most magic 4x4x4 cube 274

57. Perfect magic 8x8x8 cube 277

58. Trick with bymagic square 284

59. My favourite links 287


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[2] Most magic square per order

2.0 What is the most magic square per order?

Does a method exist to produce magic squares of all orders? I give the answer in paragraph 2.1.

Odd, double odd and multiple of four.

It is also important to know that not all magic squares are equally magic. For each order, withexception of the 3x3 magic square, exist magic squares with extra (= more than the minimum)magic features. In this e-book you find for each order a magic square with the maximum numberof (extra) magic features.

In paragraph 2.2. Magic features, you find an explanation of the (extra) magic features. Inparagraph 2.3. Most magic squares per order, you find per order a magic square with themaximum number of (extra) magic features.

2.1. Odd, double odd and multiple offour

Does a method exist to produce magic squares of all orders? According to Wikipedia you needthree different methods to produce magic squares of all orders. You need a method to produceodd (choose one of the three classic methods to produce an odd [7x7] magic square in chapter[42] 14x14 magic square), a method to produce double odd (see the medig method in chapter [16]6x6 magic square) and a method to produce multiples of four (see the classic method to produce amultiple of four [i.e. the 16x16 inlay] in chapter [46] 3x extra magic 18x18 square).

Does a method exist to produce magic squares of all orders? My answer is yes! Use the method

to produce concentric magic squares (see chapter [51] Bordered squares).N.B.: In the execution of this method a distinction is made between odd and even orders.

Do you want to produce the most magic squares, you need more methods.

2.2. Magic features

I give an explanation of the (extra) magic features. You need this information to understand, whatI mean by the most magic square per order.

[pure] A magic square is pure if it consists of the digits 1 up to n x n. A pure magic 3x3 squareconsists of the digits 1 up to (3 x 3 =) 9. On this website you find only pure magic squares withexception of chapter [54] Each magic sum.[minimal magic features] Minimal the addition of the digits of each row/column/diagonal mustgive (the same) magic sum.


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[magic sum] For each pure magic square you can calculate the magic sum. The magic sum is [(1+ n x n) / 2] x n. For example the sum of the 3x3 magic square is: [(1 + 3 x 3) / 2) x 3 = 15.[concentric] An odd concentric magic square consists of a centre of one cell and an evenconcentric magic square consists of a centre of 2x2 cells, and you can put borders around it again

and again. For example a concentric magic 14x14 square consists of a (each time proportional)4x4 in 6x6 in 8x8 in 10x10 in 12x12 in 14x14 magic square.

[panmagic] A magic square is panmagic, if addition of the digits of each pandiagonal gives themagic sum. Apandiagonal is a broken diagonal, which consists of two parts. The first part is aline, which starts from the outside row or outside column (but not from a corner) of the magicsquare. The second part is a line or a dot (and the dotends in one of the corners of the magicsquares). See for example the pandiagonals of the panmagic 4x4 square in chapter [ 9] Panmagic4x4 square, explanation.[symmetric] In a symmetric magic squares each time addition of two digits, which can be

connected with a straightline through the centre of the magic square and which are at the samedistance to the centre, gives the same sum. The sum is 1 + n x n (for example the sum ina symmetric 5x5 magic square is: 1 + 5 x 5 = 26). It is also possible that the magic square is notsymmetric as a whole, but the magic square is symmetric in each sub-square (see for example inchapter [36] Basic key method (2).

[centre] The centre of an odd magic square is the middle cell (n.b.: in the middle cell of anodd symmetric magic square you find allways the middle digit; for example in a symmetric 5x5magic square you find the digit 13 in the middle cell). The centre of an even magic square is thecrosspoint of the middle 2x2 cells.

[compact] If a magic square is a multiple of 2, 3, 5, 7, than compact means, that each randomchosen 2x2, 3x3, 5x5, 7x7, sub- square gives the same (proportional part of the) magic sum. Amagic square can be double compact. For example the ultra magic 15x15 square on this websitegives a proportional part of the magic sum for each 3x3 sub-square and for each 5x5 sub-square.[ultramagic] For an odd order (with execption of the 3x3 magic square) is ultra magic the mostmagic square. An odd ultra magic square is always panmagic and symmetric and (if the order ofthe square is not a prime number) compact. If possible in the ultra magic square also a part ofeach row, column and/or diagonal gives a proportional part of the magic sum. For example in theultra panmagic 27x27 square on this website gives each 1/9 row, 1/9 column and 1/3 diagonal aproportional part of the magic sum.

[most perfect] For order is multiple of four the most perfect magic square is the most magicsquare. Willem Barink teached us, that a little part of the most perfect magic squares has the extramagic feature X. See for detailed explanation (including the similarities and differences with theFranklin panmagic square), chapter [18] Most perfect magic squares, explanation.[Scope] Scope means, the (absolute or relative) number of different magic squares you canproduce by using a method of construction. A scope of 100% means, that you can produce all


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possible magic squares by using a method of construction. See a method of construction with100% scope in chapter [19] 8x8 most perfect magic squares, binary.

[prime number] A prime number can only be devided by one or by the number itself.

3. Most magic square per order

See below the most magic square (with the maximum number of magic features) per order.[3x3] The 3x3 magic square is the only magic square, which has no extra magic features. Seechapter [4] 3x3 magic square, explanation.[4x4] The most magic 4x4 square is the panmagic 4x4 square. The panmagic 4x4 square is alsothe smallest most perfect magic square. The square is panmagic and (2x2) compact. See how youcan produce all (100% scope) panmagic 4x4 squares in chapter [8] 4x4 panmagic square.

[5x5] The most magic 5x5 square is the ultra magic 5x5 square. The ultra magic 5x5 square ispanmagic and symmetric. See the key to produce one (so the scope is 1) ultra magic 5x5 square,in chapter [14] 5x5 [ultra] panmagic square.[6x6] The most magic 6x6 square is a 6x6 square with a 4x4 panmagic inlay. See how you canuse each panmagic 4x4 square to produce a 4x4 in 6x6 square, in chapter [ 51] Bordered magicsquare.[7x7] The most magic 7x7 square is the ultra magic 7x7 square. The ultra magic 7x7 square ispanmagic and symmetric. See the key to produce one (so the scope is 1) ultra magic 7x7 square,in chapter [14] 5x5 [ultra] panmagic square.

[8x8] The most magic 8x8 square is a most perfect 8x8 magic square (with the extra magicfeature X). See how you can produce all (is 100% scope) most perfect magic 8x8 squares (withthe extra magic feature X), in chapter [19] 8x8 most perfect magic squares, binary.[9x9] The most magic 9x9 square is an ultra magic 9x9 square. The ultra magic 9x9 square ispanmagic, symmetric, (3x3) compact and each 1/3 row and 1/3 column gives 1/3 of the magicsum. See how to produce one (scope of 1) ultra magic 9x9 square, in chapter [ 38] 9x9 [ultra]panmagic square.[10x10] The most magic 10x10 square is a concentric magic 10x10 square. See how to produce

one (scope is 1) con centric magic 10x10 square, in chapter [51] Bordered magic square.[11x11] The most magic 11x11 square is the ultra magic 11x11 square. The ultra magic11x11 square is panmagic and symmetric. See the key to produce one (so the scope is 1) ultramagic 11x11 square, in chapter [14] 5x5 [ultra] panmagic square.[12x12] The most magic 12x12 square is a most perfect magic 12x12 square (with the extramagic feature X; see chapter [24] Bassic pattern method (2)), or an ultra panmagic 12x12 square


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(see chapter [36] Basic key method (2)). The ultra panmagic 12x12 square is panmagic,symmetric inside each 4x4 sub/square, (2x2) compact and each 1/2 row, 1/2 column andeach 1/3 diagonal gives a proportional part of the magic sum.[13x13] The most magic 13x13 square is the ultra magic 13x13 square. The ultra magic13x13 square is panmagic and symmetric. See the key to produce one (so the scope is 1) ultra

magic 13x13 square, in chapter [14] 5x5 [ultra] panmagic square.

[14x14] The most magic 14x14 square is a 14x14 inlaid square. See how you can use each mostperfect magic 8x8 square or each magic 4x4 in 6x6 square to produce a 14x14 inlaid square inchapter [52] Inlaid square (1).[15x15] The most magic 15x15 square is an ultra magic 15x15 square. The ultra magic15x15 square is panmagic, symmetric and double (3x3 and 5x5) compact. See how to produceone (scope is 1) ultra magic 15x15 square in chapter [43] 15x15 [ultra] panmagic 15x15 square.[16x16] The most magic 16x16 square is a most perfect 16x16 magic square (with

the extra magic feauture X). See how to use each panmagic 4x4 square to produce a most perfectmagic 16x16 square (with the extra magic feature X), in chapter [27] Basic pattern method (3c).[17x17] The most magic 17x17 square is the ultra magic 17x17 square. The ultra magic17x17 square is panmagic and symmetric. See the key to produce one (so the scope is 1) ultramagic 17x17 square, in chapter [14] 5x5 [ultra] panmagic square.

[18x18] The most magic 18x18 square consists of 3x3 proportional 4x4 in 6x6 squares (seethe third method in chapter [46] 3x extra magic 18x18 square. Each 1/3 row/column/diagonal ofthe magic 18x18 square gives 1/3 of the magic sum.

[19x19] The most magic 17x17 square is the ultra magic 17x17 square. The ultra magic17x17 square is panmagic andsymmetric. See the key to produce one (so the scope is 1) ultramagic 17x17 square, in chapter [14] 5x5 [ultra] panmagic square.

[20x20] The most magic 20x20 square is a most perfect magic 20x20 square (with the extramagic feature X; see chapter [28] Basic pattern method (4)), or an ultra panmagic 20x20 square(see chapter [36] Basic key method (2)). The ultra panmagic 20x20 square is panmagic,symmetric inside each 4x4 sub-square, (2x2) compact and each 1/2 row, each 1/2 columnand each 1/5 diagonal gives a proportional part of the magic sum.[21x21] The most magic 21x21 square is an ultra magic 21x21 square. The ultra magic

21x21 square is panmagic, symmetric and double (3x3 and 7x7) compact. See how to produceultra magic 21x21 squares in chapter [43] 15x15 [ultra] panmagic 15x15 square.[22x22] The most magic 22x22 square is an inlaid square with four panmagic 7x7 inlays and fivepanmagic 4x4 inlays. See chapter [53] Inlaid square (2).[23x23] The most magic 23x23 square is the ultra magic 23x23 square. The ultra magic23x23 square is panmagic and symmetric. See the key to produce one (so the scope is 1) ultra


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magic 23x23 square, in chapter [14] 5x5 [ultra] panmagic square.[24x24] The most magic 24x24 square is a most perfect 24x24 magic square (withthe extra magic feauture X). See how to use each panmagic 4x4 square to produce a most perfectmagic 24x24 square (with the extra magic feature X), in chapter [29] Basic pattern method (5).

[25x25] The most magic 25x25 square is an ultra magic 25x25 square. The ultra magic25x25 square is panmagic, symmetric, (5x5) compact and each 1/5 row/column/diagonal gives1/5 of the magic sum. See how to use each ultra magic 5x5 square to produce an ultra magic25x25 square, in chapter [47] 25x25 ultra panmagic square.[26x26] The most magic 26x26 square is a concentric magic 26x26 square. See how to produceone (scope is 1) concentric magic 26x26 square, in chapter [51] Bordered magic square.[27x27] The most magic 27x27 square is an ultra magic 27x27 square. The ultra magic27x27 square is panmagic, symmetric, (3x3) compact and each 1/9 row, 1/9 column and 1/3diagonal gives a proportional part of the magic sum. See how to use each ultra magic 9x9 square

to produce an ultra magic 27x27 square, in chapter [48] 27x27 [ultra] panmagic square.[28x28] The most magic 28x28 square is a most perfect magic 28x28 square (with the extramagic feature X; see chapter [30] Basic pattern method (6)), or an ultra panmagic 28x28 square(see chapter [36] Basic key method (2)). The ultra panmagic28x28 square is panmagic,symmetric inside each 4x4 sub-square, (2x2) compact and each 1/2 row, each 1/2 columnand each 1/7 diagonal gives a proportional part of the magic sum.

[29x29] The most magic 29x29 square is the ultra magic 29x29 square. The ultra magic29x29 square is panmagic and symmetric. See the key to produce one (so the scope is 1) ultramagic 29x29 square, in chapter [14] 5x5 [ultra] panmagic square.

[30x30] The most magic 30x30 square is a concentric magic 30x30 square. See how to produceone (scope is 1) concentric magic 30x30 square, chapter [51] Bordered magic square.[31x31] The most magic 29x29 square is the ultra magic 31x31 square. The ultra magic31x31 square is panmagic andsymmetric. See the key to produce one (so the scope is 1) ultramagic 31x31 square, in chapter [14] 5x5 [ultra] panmagic square.[32x32] The most magic 32x32 square is a most perfect 32x32 magic square (withthe extra magic feauture X). See how to use each panmagic 4x4 square to produce a most perfectmagic 32x32 square (with the extra magic feature X), in chapter [32] Basic pattern method (7b).

[33x33] The most magic 33x33 square is an ultra magic 33x33 square. The ultra magic33x33 square is panmagic, symmetric and double (3x3 and 11x11) compact. See how to produceultra magic 33x33 squares in chapter [43] 15x15 [ultra] panmagic 15x15 square.[34x34] The most magic 34x34 square is a concentric magic 34x34 square. See how to produceone (scope is 1) concentric magic 34x34 square, chapter [51] Bordered magic square.


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[3] 3x3 magic square

The story of LowiHello I am Lowi, a five thousand year old turtle. I live in the river Lo (the yellow river) in China.

Heres my story!

When I was young (in the year 2800 before Christ) I was a servant of the river god. When theriver god was angry the river would overflow. The people of a village near by the river wouldplace a gift by the bank of the river. They hoped the river god would accept the gift, and the riverwould not overflow again. Each time as the villagers placed a gift by the bank of the river Iwould come out of the river and walk around the gift.One day there was a little boy near by the river. He looked at my shell and saw that my shell-pattern consisted of nine cells. The nine cells contain 1, 2, 3, 4, 5, 6, 7, 8 or 9 dots. He repeatedlycounted the dots of 3 cells (horizontal, vertical or diagonal) in a row. Each time the little boycounted 15 dots.

The little boy went to the headman of the village and told him about the spots on my shell. Theheadman organised a meeting. The villagers deceided to place 15 gifts by the bank of the river.I came out of the river and I walked 15 times around the presents. Thats a long time for a turtle!Then the river god appeared. He accepted the gifts and indicated to the villagers that the riverwould not overflow again.

So, that was a pretty exciting story, wasnt it! Would you like to find out how to make the magicsquare from my story. There are eight different ways to make the 3x3 magic square. You willneed the following 3 instructions for this:

[instruction 1] Always put the 5 in the middle of the square.

[instruction 2] Always put the 2, 4, 6 and 8 always in one of the corners of the square.[instruction 3] Always put the 2 and 8 and the 4 and 6 always in the same diagonal (so notin the same row or the same column).

5 5 5 5

5 5 5 5

Print this page and get stuck in.

Have fun!!!


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[4] 3x3 magic square, explanation

What is a 3x3 magic square?The 3x3 magic square is square, because it has as many rows (from left to right = horizontal) as

columns (from top to bottom = vertical).

The 3x3 magic square consists of 3 rows which multiplied by 3 columns is 9 cells.The 3x3 magic square must contain 9 different digits. A pure magic 3x3 square contains thedigits 1, 2, 3, 4, 5, 6, 7, 8 and 9.

The magic square is magic, because the sum of the digits of each row, each column and bothdiagonals always give the same result. The sum can be calculated as follows, the (odd) size of thesquare multiplied by the middle digit: 3 x 5 = 15.

What is the secret behind the 3x3 magic square?The secret behind the 3x3 magic square is easy to explane. If you must take 3 (different) digitsout of the digits 1 up to 9, which total each time to 15, than there are the following possibilities:

1+5+91+6+82+4+92+5+82+6+73+4+8

3+5+74+5+6.

There are 8 possibilities.

The minimum features of the 3x3 square are 3 row features plus 3 column features plus 2diagonal features total to 8 features. Because there are 8 possibilities and 8 features, there is onlyone solution of the 3x3 magic square.

If we count the appearance of the 1, 2, 3, 4, 5, 6, 7, 8 and 9 in the 8 possibilities, than we get thefollowing result:

1 = 2x


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2 = 3x3 = 2x4 = 3x5 = 4x6 = 3x7 = 2x

8 = 3x9 = 2x

The middle cell takes part of the middle row, the middle column and both diagonals, that is 4features. That is why you must always put the 5 in the middle cell. The corners take part of onerow, one column and one diagonal, that is 3 features. That is why you must put the 2, 4, 6 and 8(= even digits) always in the corners. Fill the digits 1, 3, 7 en 9 in the empty cells (in the middleof the sides). Because you put the 5 in the middle, the sum of the other two cells of a diagonalmust be (15 - 5 = ) 10. To get the total of 10 with the even digits in the corners there are only twopossibilities: 2+8 or 4+6. That is why you must put 2 and 8 or 4 and 6 in the same diagonal.

Like I have already told, there is only one solution of the 3x3 magic square, that is excludingrotation and/or mirrorring (see explanation chapter [8] panmagic 4x4 square). Including rotationand/or mirroring there are (1 x 8 = ) 8 solutions of the 3x3 square

How to produce a 3x3 magic square?A trick to produce the 3x3 square is the diagonal method of (the Dutch) professor van der Blij:

1

4 2 4 9 2

7 5 3 3 5 7

8 6 8 1 6

9


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[5] Sudoku method (1)

From Sudoku to (pan)magic squareA Sudoku mostly consists of 9 rows and 9 columns. Each row and each column (and each nonet)

must contain all the digits from 1 to 9. Using a 4x4 Sudoku (which consists of 4 rows and 4columns) you can produce a magic square when you follow the next four steps.

(1st) Do not fill in the digits 1, 2, 3 and 4, but fill in the digits 0, 1, 2, 3. Ensure that every row,column and diagonal contains all the digits 0, 1, 2 and 3.

(2nd) Produce a second 4x4 Sudoku by rotating the first Sudoku (a quarter turn to the right).

(3rd) Take a digit from the first Sudoku multiplied by 4 and add (1x) the digit from the same cellof the second Sudoku.

(4th

) Add 1 to each cell.

4x + 1x = +1 = magic square

0 1 2 3 2 1 3 0 2 5 11 12 3 6 12 13

3 2 1 0 3 0 2 1 15 8 6 1 16 9 7 2

1 0 3 2 0 3 1 2 4 3 13 10 5 4 14 11

2 3 0 1 1 2 0 3 9 14 0 7 10 15 1 8

This magic square also just happens to be a panmagic square!

Information for whiz kids:


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It is possible to use the Sudoku patterns of the panmagic 4x4 square to produce a Franklin

panmagic 8x8 square. You need three 8x8 Sudoku patterns.

The first 8x8 Sudoku pattern is a 2x2 carpet of the first 4x4 Sudoku pattern.

To produce the second 8x8 Sudoku pattern you need to split up the second 4x4 Sudoku patternin two sub-squares and enter digits from the same sub-square in the empty cells crosswise:

split up the (2nd) 4x4 Sudoku pattern: Enter digits in the empty cells crosswise:

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

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

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

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

Combine the 2 sub-squares and add the same 2 sub-squares to the bottom .

The third 8x8 Sudoku pattern is fixed (= column pattern of the basic pattern method).

4x digit from first pattern + 1x digit from second pattern + 16x digit from third pattern=

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

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

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

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

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

3 2 1 0 3 2 1 0 3 2 0 1 1 0 2 3 1 2 1 2 1 2 1 21 0 3 2 1 0 3 2 0 1 3 2 2 3 1 0 2 1 2 1 2 1 2 1

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

+1 = Franklin panmagic 8x8 square

0 53 11 62 2 55 9 60 1 54 12 63 3 56 10 61

15 58 4 49 13 56 6 51 16 59 5 50 14 57 7 52

52 1 63 10 54 3 61 8 53 2 64 11 55 4 62 9

59 14 48 5 57 12 50 7 60 15 49 6 58 13 51 8

16 37 27 46 18 39 25 44 17 38 28 47 19 40 26 45

31 42 20 33 29 40 22 35 32 43 21 34 30 41 23 36

36 17 47 26 38 19 45 24 37 18 48 27 39 20 46 25

43 30 32 21 41 28 34 23 44 31 33 22 42 29 35 24

Note that above mentioned method is the Sudoku version of the basic pattern method (seechapter22).


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It is also possible to use the Sudoku patterns of the panmagic8x8 square to produce a perfect

Franklin panmagic 16x16 square. You need four 16x16 Sudoku patterns.

The first and second 16x16 Sudoku pattern is a 2x2 carpet of the first respectively second 8x8Sudoku pattern.

The third and fourth 16x16 Sudoku pattern are fixed patterns (the third and fourth patterntogether = the column pattern of the basic pattern method).

4x digit from first pattern 1x digit from second pattern

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

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

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

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

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

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

1 0 3 2 1 0 3 2 1 0 3 2 1 0 3 2 0 1 3 2 2 3 1 0 0 1 3 2 2 3 1 02 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 3 2 0 1 1 0 2 3 3 2 0 1 1 0 2 3

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

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

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

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

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

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

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

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

16x digit from third (fixed) pattern 64x digit from fourth (fixed) pattern2 3 0 1 2 3 0 1 0 1 2 3 0 1 2 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Add 1 to each digit and you produce the following magic square.


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Perfect Franklin panmagic 16x16 square

33 246 12 223 35 248 10 221 1 214 44 255 3 216 42 253

16 219 37 242 14 217 39 244 48 251 5 210 46 249 7 212

245 34 224 11 247 36 222 9 213 2 256 43 215 4 254 41

220 15 241 38 218 13 243 40 252 47 209 6 250 45 211 8

49 230 28 207 51 232 26 205 17 198 60 239 19 200 58 237

32 203 53 226 30 201 55 228 64 235 21 194 62 233 23 196229 50 208 27 231 52 206 25 197 18 240 59 199 20 238 57

204 31 225 54 202 29 227 56 236 63 193 22 234 61 195 24

97 182 76 159 99 184 74 157 65 150 108 191 67 152 106 189

80 155 101 178 78 153 103 180 112 187 69 146 110 185 71 148

181 98 160 75 183 100 158 73 149 66 192 107 151 68 190 105

156 79 177 102 154 77 179 104 188 111 145 70 186 109 147 72

113 166 92 143 115 168 90 141 81 134 124 175 83 136 122 173

96 139 117 162 94 137 119 164 128 171 85 130 126 169 87 132

165 114 144 91 167 116 142 89 133 82 176 123 135 84 174 121

140 95 161 118 138 93 163 120 172 127 129 86 170 125 131 88

It is possible to double the size of the square again and again. For example to produce a perfectFranklin panmagic 32x32 square you need five 32x32 Sudoku patterns. The first and the second

32x32 pattern is a 2x2 carpet of the first respectively second 16x16 Sudoku pattern. The third,

fourth and fifth 32x32 Sudoku pattern are the following three new fixed Sudoku patterns:


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16x digit from third (fixed) pattern

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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64x digit from fourth (fixed) pattern

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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256x digit from fifth (fixed) pattern

0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0

0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0

3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3

3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3

0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0

0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3

3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3

0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0

0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0

3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3

3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3

0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0

0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0

3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3

3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3 0 3

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 22 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2


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MOST MAGIC 12x12 SQUARE:

It is also possible to produce squares that are odd multiples of 4, for example the 12x12 square.You need the following 2 Sudoku patterns:

The first pattern of the 12x12 square is a 3x3 carpet of the second (dont split it up!) patternof the 4x4 panmagic square (see chapter[4] Sudoku method (1)).

The second pattern of the 12x12 square is a fixed pattern.

1x digit from the first pattern

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

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

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

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

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

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

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

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

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

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

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

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

4x digit from the second (fixed) pattern

35 5 30 0 34 4 31 1 33 3 32 2

0 30 5 35 1 31 4 34 2 32 3 33

5 35 0 30 4 34 1 31 3 33 2 32

30 0 35 5 31 1 34 4 32 2 33 3

29 11 24 6 28 10 25 7 27 9 26 8

6 24 11 29 7 25 10 28 8 26 9 27

11 29 6 24 10 28 7 25 9 27 8 2624 6 29 11 25 7 28 10 26 8 27 9

23 17 18 12 22 16 19 13 21 15 20 14

12 18 17 23 13 19 16 22 14 20 15 21

17 23 12 18 16 22 13 19 15 21 14 20

18 12 23 17 19 13 22 16 20 14 21 15

Add 1 to each digit and you produce the following magic square:


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Most magic 12x12 square

143 22 124 1 139 18 128 5 135 14 132 9

4 121 23 142 8 125 19 138 12 129 15 134

21 144 2 123 17 140 6 127 13 136 10 131

122 3 141 24 126 7 137 20 130 11 133 16119 46 100 25 115 42 104 29 111 38 108 33

28 97 47 118 32 101 43 114 36 105 39 110

45 120 26 99 41 116 30 103 37 112 34 107

98 27 117 48 102 31 113 44 106 35 109 40

95 70 76 49 91 66 80 53 87 62 84 57

52 73 71 94 56 77 67 90 60 81 63 86

69 96 50 75 65 92 54 79 61 88 58 83

74 51 93 72 78 55 89 68 82 59 85 64

See chapter [33] Basic key method (2) option a and b of the 12x12 most magic square and note

that the 12x12 square mentioned above has different magic features. The above mentioned 12x12

square is Franklin panmagic (not on but) on 1/3 of the rows, 1/3 of the columns and 1/3 ofthe [parallel] [mirrored] [bent] diagonals.

You can use the method mentioned above to produce perfect Franklin (for odd multiples of four:Franklin) panmagic squares for each multiple of four. You need the (2x2, 3x3, 4x4, ) carpet

of the 4x4 Sudoku pattern and a fixed (8x8, 12x12, 16x16, ) pattern.

1x digit

2 1 3 0

3 0 2 1

0 3 1 2

1 2 0 3

and

4x digit from fixed 8x8 pattern

15 3 12 0 14 2 13 1

0 12 3 15 1 13 2 14

3 15 0 12 2 14 1 13

12 0 15 3 13 1 14 2

11 7 8 4 10 6 9 5

4 8 7 11 5 9 6 10

7 11 4 8 6 10 5 9

8 4 11 7 9 5 10 6

or


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35 5 30 0 34 4 31 1 33 3 32 2

0 30 5 35 1 31 4 34 2 32 3 33

5 35 0 30 4 34 1 31 3 33 2 32

30 0 35 5 31 1 34 4 32 2 33 3

29 11 24 6 28 10 25 7 27 9 26 8

6 24 11 29 7 25 10 28 8 26 9 27

11 29 6 24 10 28 7 25 9 27 8 26

24 6 29 11 25 7 28 10 26 8 27 9

23 17 18 12 22 16 19 13 21 15 20 14

12 18 17 23 13 19 16 22 14 20 15 21

17 23 12 18 16 22 13 19 15 21 14 20

18 12 23 17 19 13 22 16 20 14 21 15

or


63 7 56 0 62 6 57 1 61 5 58 2 60 4 59 3

0 56 7 63 1 57 6 62 2 58 5 61 3 59 4 60

7 63 0 56 6 62 1 57 5 61 2 58 4 60 3 59

56 0 63 7 57 1 62 6 58 2 61 5 59 3 60 4

55 15 48 8 54 14 49 9 53 13 50 10 52 12 51 11

8 48 15 55 9 49 14 54 10 50 13 53 11 51 12 52

15 55 8 48 14 54 9 49 13 53 10 50 12 52 11 51

48 8 55 15 49 9 54 14 50 10 53 13 51 11 52 12

47 23 40 16 46 22 41 17 45 21 42 18 44 20 43 19

16 40 23 47 17 41 22 46 18 42 21 45 19 43 20 44

23 47 16 40 22 46 17 41 21 45 18 42 20 44 19 43

40 16 47 23 41 17 46 22 42 18 45 21 43 19 44 20

39 31 32 24 38 30 33 25 37 29 34 26 36 28 35 27

24 32 31 39 25 33 30 38 26 34 29 37 27 35 28 36

31 39 24 32 30 38 25 33 29 37 26 34 28 36 27 35

32 24 39 31 33 25 38 30 34 26 37 29 35 27 36 28

or



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99 9 90 0 98 8 91 1 97 7 92 2 96 6 93 3 95 5 94 4

0 90 9 99 1 91 8 98 2 92 7 97 3 93 6 96 4 94 5 95

9 99 0 90 8 98 1 91 7 97 2 92 6 96 3 93 5 95 4 94

90 0 99 9 91 1 98 8 92 2 97 7 93 3 96 6 94 4 95 5

89 19 80 10 88 18 81 11 87 17 82 12 86 16 83 13 85 15 84 14

10 80 19 89 11 81 18 88 12 82 17 87 13 83 16 86 14 84 15 85

19 89 10 80 18 88 11 81 17 87 12 82 16 86 13 83 15 85 14 84

80 10 89 19 81 11 88 18 82 12 87 17 83 13 86 16 84 14 85 15

79 29 70 20 78 28 71 21 77 27 72 22 76 26 73 23 75 25 74 24

20 70 29 79 21 71 28 78 22 72 27 77 23 73 26 76 24 74 25 75

29 79 20 70 28 78 21 71 27 77 22 72 26 76 23 73 25 75 24 74

70 20 79 29 71 21 78 28 72 22 77 27 73 23 76 26 74 24 75 25

69 39 60 30 68 38 61 31 67 37 62 32 66 36 63 33 65 35 64 34

30 60 39 69 31 61 38 68 32 62 37 67 33 63 36 66 34 64 35 65

39 69 30 60 38 68 31 61 37 67 32 62 36 66 33 63 35 65 34 64

60 30 69 39 61 31 68 38 62 32 67 37 63 33 66 36 64 34 65 35

59 49 50 40 58 48 51 41 57 47 52 42 56 46 53 43 55 45 54 44

40 50 49 59 41 51 48 58 42 52 47 57 43 53 46 56 44 54 45 55

49 59 40 50 48 58 41 51 47 57 42 52 46 56 43 53 45 55 44 54

50 40 59 49 51 41 58 48 52 42 57 47 53 43 56 46 54 44 55 45

or


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How to use only one 4x4 Sudoku to produce (in 9 steps) a most magic 1024x1024 square

4x4 Sudoku2 1 3 0

1 2 0 3

0 3 1 2

3 0 2 1

STEP 1

First we use the 4x4 Sudoku to produce a 4x4 panmagic square. You need the 4x4 Sudoku plusan - on the 2x2 carpet shifted (1 tot the right and 1 down) - version of the same 4x4 Sudoku.

4x4 Sudoku shifted on the 2x2 carpet

2 1 3 0 2 1 3 0

1 2 0 3 1 2 0 3

0 3 1 2 0 3 1 2

3 0 2 1 3 0 2 1

2 1 3 0 2 1 3 0

1 2 0 3 1 2 0 3

0 3 1 2 0 3 1 2

3 0 2 1 3 0 2 1

Take 4x a digit from the 4x4 Sudoku and add 1x a digit of the same cell from the shifted 4x4Sudoku and add 1 to each digit to produce a 4x4 panmagic square.

4x digit + 1x digit = +1 = 4x4 panmagic square

2 1 3 0 2 0 3 1 10 4 15 1 11 5 16 2

1 2 0 3 3 1 2 0 7 9 2 12 8 10 3 13

0 3 1 2 0 2 1 3 0 14 5 11 1 15 6 12

3 0 2 1 1 3 0 2 13 3 8 6 14 4 9 7

STEP 2

Use the 4x4 panmagic square to produce a 8x8 Franklin panmagic square. You need the 2x2carpet of the 4x4 panmagic square plus an 8x8 Sudoku pattern.


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1x digit

43 21 64 2 59 5 48 18 43 21 64 2 59 5 48 18

24 42 3 61 8 58 19 45 24 42 3 61 8 58 19 45

1 63 22 44 17 47 6 60 1 63 22 44 17 47 6 60

62 4 41 23 46 20 57 7 62 4 41 23 46 20 57 7

11 53 32 34 27 37 16 50 11 53 32 34 27 37 16 5056 10 35 29 40 26 51 13 56 10 35 29 40 26 51 13

33 31 54 12 49 15 38 28 33 31 54 12 49 15 38 28

30 36 9 55 14 52 25 39 30 36 9 55 14 52 25 39

43 21 64 2 59 5 48 18 43 21 64 2 59 5 48 18

24 42 3 61 8 58 19 45 24 42 3 61 8 58 19 45

1 63 22 44 17 47 6 60 1 63 22 44 17 47 6 60

62 4 41 23 46 20 57 7 62 4 41 23 46 20 57 7

11 53 32 34 27 37 16 50 11 53 32 34 27 37 16 50

56 10 35 29 40 26 51 13 56 10 35 29 40 26 51 13

33 31 54 12 49 15 38 28 33 31 54 12 49 15 38 28

30 36 9 55 14 52 25 39 30 36 9 55 14 52 25 39

+

64x digit

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

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

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

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

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

3 0 2 1 2 1 3 0 2 1 3 0 3 0 2 12 1 3 0 3 0 2 1 3 0 2 1 2 1 3 0

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

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

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

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

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

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

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

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

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

=


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Most perfect 16x16 (Franklin pan)magic square

171 85 256 2 251 5 176 82 235 21 192 66 187 69 240 18

88 170 3 253 8 250 83 173 24 234 67 189 72 186 19 237

1 255 86 172 81 175 6 252 65 191 22 236 17 239 70 188

254 4 169 87 174 84 249 7 190 68 233 23 238 20 185 71

11 245 96 162 91 165 16 242 75 181 32 226 27 229 80 178248 10 163 93 168 90 243 13 184 74 227 29 232 26 179 77

161 95 246 12 241 15 166 92 225 31 182 76 177 79 230 28

94 164 9 247 14 244 89 167 30 228 73 183 78 180 25 231

43 213 128 130 123 133 48 210 107 149 64 194 59 197 112 146

216 42 131 125 136 122 211 45 152 106 195 61 200 58 147 109

129 127 214 44 209 47 134 124 193 63 150 108 145 111 198 60

126 132 41 215 46 212 121 135 62 196 105 151 110 148 57 199

139 117 224 34 219 37 144 114 203 53 160 98 155 101 208 50

120 138 35 221 40 218 115 141 56 202 99 157 104 154 51 205

33 223 118 140 113 143 38 220 97 159 54 204 49 207 102 156

222 36 137 119 142 116 217 39 158 100 201 55 206 52 153 103

STEP 4 UP TO 9

Repeat step 3, six times . Produce successively a perfect Franklin panmagic 32x32, 64x64,128x128, 256x256, 512x512 and 1024x1024 square (= most magic 1024x1024 square).

Notify that all used squares and Sudoku patterns are products from the same 4x4 Sudoku.

The 4x4 Sudoku is a duplicater. I found the following 32 duplicaters:

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

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

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

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

5 3 0 2 1 6 0 2 1 3 7 2 1 3 0 8 1 3 0 2

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

1 2 0 3 2 0 3 1 0 3 1 2 3 1 2 00 3 1 2 3 1 2 0 1 2 0 3 2 0 3 1

9 2 1 3 0 10 1 3 0 2 11 3 0 2 1 12 0 2 1 3

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

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

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


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13 1 2 0 3 14 2 0 3 1 15 0 3 1 2 16 3 1 2 0

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

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

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

17 2 1 0 3 18 1 0 3 2 19 0 3 2 1 20 3 2 1 0

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

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

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

21 0 3 2 1 22 3 2 1 0 23 2 1 0 3 24 1 0 3 2

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

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

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

25 3 0 1 2 26 0 1 2 3 27 1 2 3 0 28 2 3 0 1

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

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

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

29 1 2 3 0 30 2 3 0 1 31 3 0 1 2 32 0 1 2 3

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

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

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

There are 384 panmagic 4x4 squares. So with the 32 duplicaters there are 384 x 32 is 12288possibilities to produce a most perfect (Franklin pan)magic 8x8 square.

It is also possible to swap row 1&3 and/or row 2&4 and/or row 5&7 and/or row 6&8 and/orcolumn 1&2 and/or column 2&4 and/or column 5&7 and/or column 6&8 (that gives 2 x 2 x 2 x 2x 2 x 2 x 2 x 2 = 256 options).But Swapping rows and/or columnsgives a lot (unknown number)of double solutions.

See a complete classification ofall 368640 most perfect 8x8 squares in 6 groups (includingSudoku method 3): [34] Analysis Franklin panmagic 8x8 (2)


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[8] 4x4 panmagic squares

How to produce 4x4 panmagic squaresExcluding rotation and/or mirroring there are 880 different pure magic 4x4 squares. Of the 880

squares 48 are panmagic. These panmagic squares have (as well as the bigger [Franklin]panmagic squares) the following pattern:

1 8 10 15

12 13 3 6

7 2 16 9

14 11 5 4

The sum of two digits of the same colour is each time (the highest digit of the magic square + 1,in this case 16+1=) 17. For the patterns of all 880 pure magic 4x4 squares, got to: www.magic-squares.net/transform.htm

You only need to know 3 panmagic squares to produce all (excluding rotation and/or mirroring)48 panmagic squares:

1 8 13 12 1 8 11 14 1 8 10 15

15 10 3 6 15 10 5 4 14 11 5 4

4 5 16 9 6 3 16 9 7 2 16 9

14 11 2 7 12 13 2 7 12 13 3 6

On the 2x2 carpet of one of the three 4x4 squares you can find 16 different 4x4 sub-squares. See

the following example of the third square:

1 8 10 15 1 8 10 15

12 13 3 6 12 13 3 6

7 2 16 9 7 2 16 9

14 11 5 4 14 11 5 4

1 8 10 15 1 8 10 15

12 13 3 6 12 13 3 6

7 2 16 9 7 2 16 9

14 11 5 4 14 11 5 4

Select a random 4x4 sub-square on the carpet (stay out of the gray area, because of doublesolutions). The (for example yellow marked) selected 4x4 sub-square can be rotated and/ormirrored (see below for what happens to the digits):

Selected 4x4 square 4 14 11 5 Mirroring 5 11 14 4


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[9] 4x4 panmagic squares, explanation

What is a 4x4 magic square?The 4x4 magic square is square, because it has as many rows (from left to right = horizontal) as

columns (from top to bottom = vertical).

The 4x4 magic square consists of 4 rows which multiplied by 4 columns is 16 cells.The 4x4 magic square must contain 16 different digits. A pure magic 4x4 square contains thedigits 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16.

The magic square is magic, because the sum of the digits of each row, each column and bothdiagonals always give the same result. The sum can be calculated as follows, the (even) size ofthe square divided by 2 multiplied by the lowest plus the highest digit: 4 / 2 x (1 + 16) = 34.

What is a 4x4 panmagic square?A 4x4 panmagic square has the above mentioned row-, column- and diagonal features (=minimum features) plus extra magic features.

The extra magic features of the panmagic 4x4 square are:- Each 2x2 sub-square has the same (magic) sum (= 34);- Each 4x4 sub-square on a carpet of 2x2 (see below) is panmagic (in the carpet you can

find 16 different 4x4 panmagic squares).

Each 4x4 subsquare on the 2x2 carpet is panmagic, because of the parallel running diagonals on

the carpet. The sum of the digits of four sequencing cells on a diagonal is always 34 (= the magicsum).


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1 8 13 12

15 10 3 6

4 5 16 9

14 11 2 7

The sum of two digits of the same colour is each time (the highest digit of the magic square + 1,in this case 16+1=) 17, that is half of the magic sum. Each time with two colours you can produceone of the (pan)diagonals, which total to (2 x 17 = ) the magic sum of 34:

1 8 13 12 1 8 13 12 1 8 13 12 1 8 13 12

15 10 3 6 15 10 3 6 15 10 3 6 15 10 3 6

4 5 16 9 4 5 16 9 4 5 16 9 4 5 16 9

14 11 2 7 14 11 2 7 14 11 2 7 14 11 2 7

1 8 13 12 1 8 13 12 1 8 13 12 1 8 13 12

15 10 3 6 15 10 3 6 15 10 3 6 15 10 3 6

4 5 16 9 4 5 16 9 4 5 16 9 4 5 16 9

14 11 2 7 14 11 2 7 14 11 2 7 14 11 2 7

How to produce 4x4 panmagic squares?The trick is to learn the 3 basic panmagic 4x4 squares by heart; see in chapter [8] Panmagic4x4 square.


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0 1 0 1 1 0 1 0

0 1 0 1 1 0 1 0

1 0 1 0 0 1 0 1

1 0 1 0 0 1 0 1

V2a V2b0 1 0 1 1 0 1 0

1 0 1 0 0 1 0 1

1 0 1 0 0 1 0 1

0 1 0 1 1 0 1 0

Take the following 3 steps:

[1] Choose H1a or H1b and H2a or H2b and V1a or V1b and V2a or V2b (in the example

below has been chosen for H1b, H2b, V1b and V2a). There are 2x2x2x2 = 16 possibilities.

[2] Choose the sequence H1H2V1V2 or H1H2V2V1 or H1V1H2V2 or H1V1V2H2 orH1V2H2V1 or H1V2V1H2 or H2H1V1V2 or H2H1V2V1 or H2V1H1V2 or H2V1V2H1 orH2V2H1V1 or H2V2V1H1 or V1H1H2V2 or V1H1V2H2 or V1H2H1V2 or V1H2V2H1 orV1V2H1H2 or V1V2H2H1 or V2H1H2V1 or V2H1V1H2 or V2H2H1V1 or V2H2V1H1 orV2V1H1H2 or V2V1H2H1 (in the example below has been chosen for sequence H1H2V2V1).There are 24 possibilities.

[3] Produce the panmagic 4x4 square:

1x digit (H1b) + 2x digit (H2b) + 4x digit (V2a) + 8x digit (V1b) +1 = panm. 4x4 s.1 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 12 6 9 7

0 0 1 1 0 1 1 0 1 0 1 0 1 0 1 0 13 3 16 2

1 1 0 0 1 0 0 1 1 0 1 0 0 1 0 1 8 10 5 11

0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 15 4 14

Notify tha you can produce all 16 (see step 1) x 24 (see step 2) = 384 panmagic 4x4 squares(including rotating and/or mirroring) !!!


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1 16 64 49 33 48 32 17

63 50 2 15 31 18 34 47

8 9 57 56 40 41 25 24

58 55 7 10 26 23 39 42

3 14 62 51 35 46 30 19

61 52 4 13 29 20 36 456 11 59 54 38 43 27 22

60 53 5 12 28 21 37 44

Franklin panmagic (= also most perfect magic) 8x8 square:

130 130 130 130 130 130 130 130

130 130 130 130 130 130 130 130

130 130

130 130 8 9 64 49 40 41 32 17130 130 58 55 2 15 26 23 34 47 260 260

130 130 1 16 57 56 33 48 25 24 260 260

130 130 63 50 7 10 31 18 39 42 260 260

130 130 6 11 62 51 38 43 30 19 260 260

130 130 60 53 4 13 28 21 36 45 260 260

130 130 3 14 59 54 35 46 27 22 260 260

130 130 61 52 5 12 29 20 37 44 260 260

130 130

130 130 130 130 130 130 130

130 130 130 130 130 130 130

130 130 130 130 130 130 130

130 130 130 130 130 130 130130 130 130 130 130 130 130

130 130 130 130 130 130 130

130 130 130 130 130 130 130


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12 21 252 229 76 85 188 165 140 149 124 101 204 213 60 37

251 235 11 27 187 171 75 91 123 107 139 155 59 43 203 219

9 24 249 232 73 88 185 168 137 152 121 104 201 216 57 40

250 234 10 26 186 170 74 90 122 106 138 154 58 42 202 218

8 25 248 233 72 89 184 169 136 153 120 105 200 217 56 41

247 231 7 23 183 167 71 87 119 103 135 151 55 39 199 215

13 20 253 228 77 84 189 164 141 148 125 100 205 212 61 36254 238 14 30 190 174 78 94 126 110 142 158 62 46 206 222

4 29 244 237 68 93 180 173 132 157 116 109 196 221 52 45

243 227 3 19 179 163 67 83 115 99 131 147 51 35 195 211

1 32 241 240 65 96 177 176 129 160 113 112 193 224 49 48

255 226 15 18 191 162 79 82 127 98 143 146 63 34 207 210

16 17 256 225 80 81 192 161 144 145 128 97 208 209 64 33

242 239 2 31 178 175 66 95 114 111 130 159 50 47 194 223

5 28 245 236 69 92 181 172 133 156 117 108 197 220 53 44

251 230 11 22 187 166 75 86 123 102 139 150 59 38 203 214

12 21 252 229 76 85 188 165 140 149 124 101 204 213 60 37

246 235 6 27 182 171 70 91 118 107 134 155 54 43 198 219

9 24 249 232 73 88 185 168 137 152 121 104 201 216 57 40

247 234 7 26 183 170 71 90 119 106 135 154 55 42 199 218

8 25 248 233 72 89 184 169 136 153 120 105 200 217 56 41

250 231 10 23 186 167 74 87 122 103 138 151 58 39 202 215

13 20 253 228 77 84 189 164 141 148 125 100 205 212 61 36

243 238 3 30 179 174 67 94 115 110 131 158 51 46 195 222

4 29 244 237 68 93 180 173 132 157 116 109 196 221 52 45

254 227 14 19 190 163 78 83 126 99 142 147 62 35 206 211


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11 5412 5313 5214 5115 5016 49

17 4818 4719 4620 4521 4422 4323 4224 4125 4026 3927 3828 3729 3630 3531 3432 33

Second boundary condition is:a1 -/- a2 = e3 -/- e4 [in the example: 1 -/- 19 = -/- 18 versus 5 -/- 23 = -/- 18]a2 -/- a3 = e2 -/- e3 [in the example: 19 -/- 17 = 2 versus 7 -/- 5 = 2]a3 -/- a4 = e1 -/- e2 [in the example: 17 -/- 21 = -/- 4 versus 3 -/- 7 = -/- 4]

Third boundary condition is:a1 -/- b1 = a2 -/- b2 = a3 -/- b3 = a4 -/- b4 = e1 -/- f1 = e2 -/- f2 = e3 -/- f3 = e4 -/- f4 [= 32]b1 -/- c1 = b2 -/- c2 = b3 -/- c3 = b4 -/- c4 = f1 -/- g1 = f2 -/- g2 = f3 -/- g3 = f4 -/- g4 [= 8]c1 -/- d1 = c2 -/- d2 = c3 -/- d3 = c4 -/- d4 = g1 -/- h1 = g2 -/- h2 = g3 -/- h3 = g4 -/- h4 [= -/- 7]

Try it (clue: dont select the digits from column I or II to difficult)!!! Maybe an excellent programmer can let the computerdetermine all (368640 x 8) solutions of the most perfect 8x8 square from the (possible) starting positions of the trans-formation method.

Translation of the swap possibilities of the most perfect magic 8x8 square into swap of the starting position:

[1a] swap row 1&3 and/or 2&4 and/or 5&7 and/or 6&8 of magic square = swap row 1&8 and/or 2&7 and/or 4&5

and/or 3&6 of starting position[1b] swap column 1&3 and/or 2&4 and/or 5&7 and/or 6&8 of magic square = swap column 1&8 and/or 2&7 and/or 4&5

and/or 3&6 of starting position[2a] swap row 1&2 and 3&4 and 5&6 and 7&8 of magic square = swap row 1&2 and 3&4 and 5&6 and 7&8

(& vertical mirroring of starting position)[2b] swap column 1&2 and 3&4 and 5&6 and 7&8 of magic square = swap column 1&2 and 3&4 and 5&6 and 7&8

(& horizontal mirroring of starting position)[3a] swap upper with down half of magic square = swap upper with down half of starting position

[3b] swap left with right half of magic square = swap left with right half of starting position


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Notify that to produce the second square you can use a differebt combination of digits (seeabove) 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 produceall 24 x 6 (combinations of digits) x 25 (by shifting over the carpet) x 8 (by rotation and/ormirroring) 28.800 possinle 5x5 panmagic squares.

On website www.grogono.com/magic/5x5.php you will find the mother method. With thismethod can be produced 144 basic panmagic 5x5 squares. On the 2x2 carpet of one of the 144basic panmagic 5x5 squares you can find 25 different 5x5 sub-squares (144x25=3.600).

Notify that the above produced panmagic 5x5 square is basic panmagic square number 2 onwebsite www.grogono.com/magic/5x5pan144.php.

You can use this method for odd squares that are no multiples of 3 (= 5x5, 7x7, 11x11, 13x13,17x17, ...). For example for a 7x7 magic square, take the digits 0-a-b-c-d-e-f (in stead of a up to f,you must use six different digits out of 1 up to 6; so there are 6x5x4x3x2x1 = 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 ofdigits first square) x 720 (combinations of digits second square) x 49 (possibilities on the 2x2carpet) x 8 (by rotation and/or mirroring) / 4 (correction for duplicate solutions) is 304.819.200panmagic 7x7 squares.

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

Information for whiz kids

On website www.magic-squares.net/pandiag5.htm 36 essential different 5x5 panmagic squaresare presented. If you change the sequence of the rows and the columns into 1-3-5-2-4 and/orswap 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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[16] 6x6 magic squares

How to produce 6x6 squares?Panmagic (or extra magic) 6x6 squares do not exist. The structure of 6x6 squares is irregular. To

produce pure 6x6 squares, you need to puzzle. The fastest (and funniest) method is the medjigmethod of Willem Barink (source: Wikipedia, Dutch language version).

Take as 1st square a pure magic 3x3 square, but blow it up by presenting the digits 2x2. Producea 2nd square existing of 9 (2x2) medjig tiles.

Note that an 2x2 medjig tile consists of all the digits 0, 1, 2 en 3, but not always in the samesequence; You must choose the tiles, taking care that the sum of the digits of eachrow/column/diagonal is (6 x 1,5 =) 9. Take (1x) a digit from the 1st square and add 9x a digitfrom the same cell of the 2nd square.

1x digit + 9x digit = pure magic 6x6 square2 2 9 9 4 4 2 3 0 2 0 2 20 29 9 27 4 222 2 9 9 4 4 1 0 3 1 3 1 11 2 36 18 31 13

7 7 5 5 3 3 3 1 1 2 2 0 34 16 14 23 21 37 7 5 5 3 3 0 2 0 3 3 1 7 25 5 32 30 12

6 6 1 1 8 8 3 2 2 0 0 2 33 24 19 1 8 266 6 1 1 8 8 0 1 3 1 1 3 6 15 28 10 17 35

You can use this method to produce even squares that are not multiples of 4 (= 6x6, 10x10,

14x14, 18x18, ). For example to produce a 10x10 square use as 1 st square a pure (pan)magic5x5 square, use a 2nd square existing of 25 (2x2) medjig tiles, taking care that the sum of thedigits of each row/column/diagonal is (10 x 1,5 =) 15 and add 15x a digit from the 2nd square.

Instead of the mejig method you can use the method of Strachey. Make a 2x2 carpet of a magic3x3 square and add 9x the digit from a fixed pattern.

1x digit + 9x digit = magic 6x6 square

2 9 4 2 9 4 0 0 3 2 2 2 2 9 31 20 27 22

7 5 3 7 5 3 0 3 0 2 2 2 7 32 3 25 23 21

6 1 8 6 1 8 0 0 3 2 2 2 6 1 35 24 19 26

2 9 4 2 9 4 3 3 0 1 1 1 29 36 4 11 18 137 5 3 7 5 3 3 0 3 1 1 1 34 5 30 16 14 12

6 1 8 6 1 8 3 3 0 1 1 1 33 28 8 15 10 17

To use the method of Strachey to produce bigger double odd magic squares (= 10x10, 14x14,18x18, ) it is a disadvantage that you need to swap more and more digits to get a correct magicsquare. See the improved method of Strachey to produce a magic 10x10 square [chapter 37].


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[17] 8x8 Ultra magic square

The Franklin panmagic (= most perfect magic) 8x8 square has the most magic features (click on[NEXT>>). The most magic odd squares are ultra (pan)magic squares. But also (even, forexample) 8x8 ultra (pan)magic squares exist. Just like the the most perfect magic square, the ultra

magic 8x8 square consists of four 4x4 squares with the same structure. These 4x4 squares havethe following structure:

The sum per colour is the lowest plus the highest digit, that is (as we take a 8x8 square) 1 + 64 =65.

First we produce (as basic pattern) a 4x4 square using four binary grids. N.b.: Note that the fourbinary grids have the right (see above) structure!!! Secondly we use the right Sudoku grid to getall digits from 1 up to 64 in the ultra (pan)magic 8x8 square.

1x digit 2x digit 4x digit 8x digit +1 Magic 4x4 square

0 0 1 1 0 1 1 0 0 1 1 0 0 1 0 1 1 15 8 101 1 0 0 0 1 1 0 1 0 0 1 1 0 1 0 14 4 11 51 1 0 0 1 0 0 1 0 1 1 0 1 0 1 0 12 6 13 30 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 7 9 2 16

N.B.: The 4x4 square is semi panmagic (group III of the 880 possible 4x4 squares excludingrotating and/or mirroring). You can put the binary grids in random order (and that gives 4x3x2x1is 24 possibilities). It is possible for each binary grid to swap digits 0 and 1 or not (and that gives2x2x2x2 is 16 possibilities). In total there are including rotating and/or mirroring (24 x 16 =) 384different semi panmagic 4x4 squares of group III.

The problem is that the 4x4 square is (semi panmagic, but) not panmagic. The solution is to usein the upper half of the 8x8 grid two times the 4x4 square and to use in the lowest half of the 8x8grid two times the 4x4 square up site down.

1x digit from 4x 4x4 magic


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square[up site down]

1 15 8 10 1 15 8 10

14 4 11 5 14 4 11 5

12 6 13 3 12 6 13 3

7 9 2 16 7 9 2 16

16 2 9 7 16 2 9 7

3 13 6 12 3 13 6 125 11 4 14 5 11 4 14

10 8 15 1 10 8 15 1

+ 16x digit from Sudoku grid 0 3 1 2 1 2 0 3

3 0 2 1 2 1 3 0

2 1 3 0 3 0 2 1

1 2 0 3 0 3 1 2

0 3 1 2 1 2 0 3

3 0 2 1 2 1 3 0

2 1 3 0 3 0 2 1

1 2 0 3 0 3 1 2

= ultra (pan)magic 8x8 square

1 63 24 42 17 47 8 58

62 4 43 21 46 20 59 5

44 22 61 3 60 6 45 19

23 41 2 64 7 57 18 48

16 50 25 39 32 34 9 55

51 13 38 28 35 29 54 12

37 27 52 14 53 11 36 3026 40 15 49 10 56 31 33

Alternative

It is also possible to choose the following structure:

Now we produce a semi panmagic 4x4 square of group II.

1x digit 2x digit 4x digit 8x digit +1 Mag. 4x4 sq.


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0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 11 8 141 1 0 0 0 1 1 0 1 1 0 0 0 1 0 1 6 16 3 91 1 0 0 1 0 0 1 0 0 1 1 1 0 1 0 12 2 13 70 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 15 5 10 4

And now we have to puzzle to find the right Sudoku grid to produce another ultra (pan)magic 8x8square:

1x digit from 4x4 [up site down] 1 11 8 14 1 11 8 14

6 16 3 9 6 16 3 9

12 2 13 7 12 2 13 7

15 5 10 4 15 5 10 4

4 10 5 15 4 10 5 15

7 13 2 12 7 13 2 12

9 3 16 6 9 3 16 6

14 8 11 1 14 8 11 1

+ 16x digit from Sudoku grid 0 1 2 3 2 3 0 1

2 3 0 1 0 1 2 3

0 1 2 3 2 3 0 1

2 3 0 1 0 1 2 3

0 1 2 3 2 3 0 1

2 3 0 1 0 1 2 3

0 1 2 3 2 3 0 1

2 3 0 1 0 1 2 3

= ultra (pan)magic 8x8 square

1 27 40 62 33 59 8 30

38 64 3 25 6 32 35 57

12 18 45 55 44 50 13 23

47 53 10 20 15 21 42 52

4 26 37 63 36 58 5 31

39 61 2 28 7 29 34 60

9 19 48 54 41 51 16 22

46 56 11 17 14 24 43 49

N.B.: The groups IV, V and VI of the 4x4 squares are semi panmagic as well. Puzzle yourself toproduceultra (pan)magic 8x8 squares using the 4x4 magic squares of group IV, V and VI!!!


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[18] Most perfect magic squares, explanation

What do I need to know, before I read about most perfect magic squares?Do you know everything about the panmagic 4x4 square (= the smallest most perfect magicsquare)? If the answer is no, than read first chapter [9] panmagic 4x4 square, explanation.

Which size (order) have most perfect squares?Most perfect squares are multiples of 4 (4x4, 8x8, 12x12, 16x16, 20x20, magic squares).

Which special features have most perfect magic squares? Most perfect squares are panmagic and the sum of the digits of each 2x2 sub-square is 4 / n (n= the size/order of the square) x the magic sum. A most perfect magic square, which is a multiple of 8 (= 8x8, 16x16, 24x24, 32x32, 40x40, magic square), has all features of a Franklin panmagic squares. It means (simply spoken), that the

total of the digits of rows/columns/diagonals is of the magic sum. For each multiple of 8 from 16x16 and up, the most perfect magic square is even more magicthan a Franklin panmagic square. For example the 16x16 most perfect square: (not only the totalof but also) the total of rows/columns/diagonals is of the magic sum.

What is the structure of a most perfect magic square?Do you know the structure of the most perfect magic square, than you understand why the mostperfect magic square has its special features.

The most perfect magic square consists of one or more proportional panmagic 4x4 (sub-)squares.

The meaning of proportional becomes clear as a panmagic 4x4 square is compared with (one ofthe four 4x4 sub-squares of) a most perfect 8x8 magic square.

panmagic 4x4 4x4 sub-square of 8x81 8 13 12 1 54 12 63

15 10 3 6 16 59 5 50

4 5 16 9 53 2 64 11

14 11 2 7 60 15 49 6

In both squares the sum of two digits of a colour always equals to the lowest plus the highest digitof the magic square (1+16=17 respectively 1+64=65). With each time two colours you can get alleight (pan)diagonals (see page panmagic 4x4 square, explanation).

Look below at the patterns of the panmagic 4x4 square and the most perfect 8x8 square.


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See a complete classification ofall 368640 most perfect (Franklin pan)magic 8x8 squares:[34] Analysis Franklin panmagic squares (2)

See also [45] the perfect magic square. I have produced this magic square by using the basicpattern method 3. I have used as input the 1st basic panmagic 4x4 square. And I have made some

classic row- and column swaps; see Analysis of Franklin panmagic 8x8 square.


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[20] Khajuraho method (from panmagic 4x4 to [Franklin]panmagic 8x8)

From a panmagic 4x4 square, for example the famous Khajuraho square, you can produce a

bigger panmagic square (8x8, 12x12, 16x16, 20x20, ).

Rewrite the Khajuraho square as follows:

Khajuraho square Basic square7 12 1 14 7 h-4 1 h-22 13 8 11 2 h-3 8 h-5

16 3 10 5 h 3 h-6 59 6 15 4 h-7 6 h-1 4

If you want to produce an 8x8 panmagic square, you need four 4x4 squares (the basic square andthree additional squares):

7 h-4 1 h-2 +8 -8 +8 -8

2 h-3 8 h-5 +8 -8 +8 -8

h 3 h-6 5 -8 +8 -8 +8

h-7 6 h-1 4 -8 +8 -8 +8

+16 -16 +16 -16 +24 -24 +24 -24

+16 -16 +16 -16 +24 -24 +24 -24

-16 +16 -16 +16 -24 +24 -24 +24

-16 +16 -16 +16 -24 +24 -24 +24

The highest digit of the (pure) 8x8 square is 64, so h is 64. Calculate first the digits of the basicsquare. Then you take a digit from a cell of the basic square and add a digit from the same cell ofthe first, the second or the third additional square. Result:

Panmagic 8x8 square7 60 1 62 15 52 9 54

2 61 8 59 10 53 16 51

64 3 58 5 56 11 50 13

57 6 63 4 49 14 55 12

23 44 17 46 31 36 25 38

18 45 24 43 26 37 32 35

48 19 42 21 40 27 34 29

41 22 47 20 33 30 39 28


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[21] Khajuraho method, explanation

How I discovered the Khajuraho method (and the basic pattern method).

I think it will be interesting to tell how I have discovered the Khajuraho method of construction.

Strange but true, the story begins with the discovery of the method of construction to producesquares for each random chosen magic sum (see chapter54 Each magic sum). The key of thismethod is an impure 4x4 magic square with (8 different) positive and (8 different) negativedigits and the magic sum of 0. The minimum difference between digits is four because of theremainder. The random chosen magic sum must be devided by four, which gives a maximumremainder of 3 (for example 403 / 4 = 100 remainder 3). Because the minimum differencebetween the digits is four, after correction of the maximum remainder of 3, there are still 16different digits in the square.

What I have discovered is that it is possible to translate a panmagic 4x4 square, for example theKhajuraho square, to the key of the magic sum of 0. Translate the digits 1 up to 16 in -30, -26, -

22, -18, -14, -10, -6, -2, +2, +6, +10, +14, +18, +22, +26 en +30; see below.

Khajuraho square translation in key of magic sum of 07 12 1 14 -6 14 -30 222 13 8 11 -26 18 -2 1016 3 10 5 30 -22 6 -149 6 15 4 2 -10 26 -18

I had already discovered that it is possible to produce from the 4x4 key square of the magic sum

of 0 a bigger, for example a 8x8 key square of the magic sum of 0. You need three additional 4x4squares. In the first additional square you use the digits +/- 34, 38, 42, 46, 50, 54, 58 en 62 (putthe lowest, second lowest, , second highest and highest digit in exactly the same place as in the4x4 key square). In the second additional square you use the digits +/- 66, 70, 74, 78, 82, 86, 90en 94. In the third additional square you use the digits +/- 98, 102, 106, 110, 114, 118, 122 en126. Produce the following square:

-6 14 -30 22 -38 46 -62 54-26 18 -2 10 -58 50 -34 42

30 -22 6 -14 62 -54 38 -46

2 -10 26 -18 34 -42 58 -50

-70 78 -94 86 -102 110 -126 118-90 82 -66 74 -122 114 -98 10694 -86 70 -78 126 -118 102 -110

66 -74 90 -82 98 -106 122 -114


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Translate this square (back) to a pure 8x8 square. Translate the lowest digit (= most negative)digit in 1 and the highest (most positive) digit in 64; see below:

31 36 25 38 23 44 17 4626 37 32 35 18 45 24 43

40 27 34 29 48 19 42 2133 30 39 28 41 22 47 2015 52 9 54 7 60 1 6210 53 16 51 2 61 8 5956 11 50 13 64 3 58 549 14 55 12 57 6 63 4

As key square of the Khajuraho method of construction (see chapter [20] Khajuraho method) Itook the third additional 4x4 square (see at the right of the bottom of the 8x8 square).

4x4 sub-square at the right of the bottom Key square7 60 1 62 7 h-4 1 h-22 61 8 59 2 h-3 8 h-5

64 3 58 5 h 3 h-6 557 6 63 4 h-7 6 h-1 4

The square produced by the Khajuraho method of construction is almost Franklin panmagic.Only the sum of four 2x2 sub-squares in the middle two columns is not half of the magic sum

(1/2 x 260 = 130). If you swap digits in the rows, you get a (perfect) Franklin panmagic 8x8square. In stead of the Khajuraho method of construction and swapping digits in the rows you canuse directly the basic pattern method of construction (see chapter [22] basic pattern method(1a)) to produce the same (perfect) Franklin panmagic 8x8 square.


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You can find an easy method of construction to produce larger (than 8x8): chapter [35] Basickey method (1).


It is possible to produce the following perfect 16x16 Franklin panmagic square by enlarging the

8x8 square (see above):

47 243 10 214 48 244 9 213 15 211 42 246 16 212 41 245

2 222 39 251 1 221 40 252 34 254 7 219 33 253 8 220

247 43 210 14 248 44 209 13 215 11 242 46 216 12 241 45

218 6 255 35 217 5 256 36 250 38 223 3 249 37 224 4

63 227 26 198 64 228 25 197 31 195 58 230 32 196 57 229

18 206 55 235 17 205 56 236 50 238 23 203 49 237 24 204

231 59 194 30 232 60 193 29 199 27 226 62 200 28 225 61

202 22 239 51 201 21 240 52 234 54 207 19 233 53 208 20

111 179 74 150 112 180 73 149 79 147 106 182 80 148 105 181

66 158 103 187 65 157 104 188 98 190 71 155 97 189 72 156

183 107 146 78 184 108 145 77 151 75 178 110 152 76 177 109

154 70 191 99 153 69 192 100 186 102 159 67 185 101 160 68

127 163 90 134 128 164 89 133 95 131 122 166 96 132 121 165

82 142 119 171 81 141 120 172 114 174 87 139 113 173 88 140

167 123 130 94 168 124 129 93 135 91 162 126 136 92 161 125

138 86 175 115 137 85 176 116 170 118 143 83 169 117 144 84

The 16x16 square has the same basic pattern as the 8x8 square (only the 2 sub-squares have

switched places).

Basic (row) pattern 16x16 square

15 3 10 6 16 4 9 5 15 3 10 6 16 4 9 5

2 14 7 11 1 13 8 12 2 14 7 11 1 13 8 12

7 11 2 14 8 12 1 13 7 11 2 14 8 12 1 13

10 6 15 3 9 5 16 4 10 6 15 3 9 5 16 4

15 3 10 6 16 4 9 5 15 3 10 6 16 4 9 5

2 14 7 11 1 13 8 12 2 14 7 11 1 13 8 12

7 11 2 14 8 12 1 13 7 11 2 14 8 12 1 13

10 6 15 3 9 5 16 4 10 6 15 3 9 5 16 4

15 3 10 6 16 4 9 5 15 3 10 6 16 4 9 5

2 14 7 11 1 13 8 12 2 14 7 11 1 13 8 12

7 11 2 14 8 12 1 13 7 11 2 14 8 12 1 13

10 6 15 3 9 5 16 4 10 6 15 3 9 5 16 4

15 3 10 6 16 4 9 5 15 3 10 6 16 4 9 5

2 14 7 11 1 13 8 12 2 14 7 11 1 13 8 12

7 11 2 14 8 12 1 13 7 11 2 14 8 12 1 13

10 6 15 3 9 5 16 4 10 6 15 3 9 5 16 4

If you place the digits from 1 to 256 in rows of 16 under each other, then you get 16 columns (of16 digits). The column pattern of the 16x16 square is as follows:


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75

Column pattern 16x16 square

3 16 1 14 3 16 1 14 1 14 3 16 1 14 3 16

1 14 3 16 1 14 3 16 3 16 1 14 3 16 1 14

16 3 14 1 16 3 14 1 14 1 16 3 14 1 16 3

14 1 16 3 14 1 16 3 16 3 14 1 16 3 14 1

4 15 2 13 4 15 2 13 2 13 4 15 2 13 4 152 13 4 15 2 13 4 15 4 15 2 13 4 15 2 13

15 4 13 2 15 4 13 2 13 2 15 4 13 2 15 4

13 2 15 4 13 2 15 4 15 4 13 2 15 4 13 2

7 12 5 10 7 12 5 10 5 10 7 12 5 10 7 12

5 10 7 12 5 10 7 12 7 12 5 10 7 12 5 10

12 7 10 5 12 7 10 5 10 5 12 7 10 5 12 7

10 5 12 7 10 5 12 7 12 7 10 5 12 7 10 5

8 11 6 9 8 11 6 9 6 9 8 11 6 9 8 11

6 9 8 1

Documents

How to Produce Perfect Magic Squares