Welcome to WordPress.com. This is your first post. Edit or delete it and start blogging!
WHAT SOME RETIRED PEOPLE DO:
For my home page visit http://bdtara.bhargavas.net
(You can also have a look at maths4fun.wordpress.com)
I will first talk of MAGIC SQUARES AND THEN MAGIC CUBES.
INTRODUCTION To MAGIC SQUARES:
In an arrangement of n horizontal rows and n vertical columns there will be nxn cells which can be filled by nxn numbers in such a way that they satisfy a set of specified conditions. For example we may require that the sum of all the rows are equal to each other or the sum of each vertical column is not only equal to each other, but also equal to the sum of each row or we may specify that only odd numbers are used to give a specified total, etc..
If we go a step further and specify that not only the sum of each row, each column and the main diagonals are equal but also that, only nxn consecutive numbers are used, the resultant square pattern has been called, from ancient times, a MAGIC SQUARE. As such a magic square of size n, denoted by Sn, is an nxn square which can be said to have the following properties:-
i) It has nxn consecutive integers, each integer occurring only once.
ii) The total for each row, each column and the main diagonals of the square is the same. This is the equisum property. The total, so obtained, is called the magic sum of the square.
(a) If these integers are 1,2,3,…n^2, the sum is n(n^2 +1)/2.
(b) In general, if the first integer is m+1, the sum is mxn + nx(n^2+1)/2. Without loss of generality we may assume that the first (lowest) integer of Sn is 1. To get a square in which the sequence of integers begins with some other number m+1, we add m to all the cells in Sn.
(c) If n is larger than 4, there are a large number of unique solutions for Sn.
An example of S4 with magic sum of 34 is given below:-
01 13 12 08
06 10 03 15
11 07 14 02
16 04 05 09
(Please note to ensure proper alignment, to suit the website requirements, I will normally be writing 1 as 01, 2 as 02, etc.)
MAGIC SQUARES IN ANCIENT TIMES
The construction of magic squares is an amusement of great antiquity, we hear of magic squares in China and India before the Christian era, while they appear to have been introduced to Europe by Moscowpulus in Constantinople in early 15th century. However, what was at first merely a practice of magicians and talisman makers, has now, for a long time, become a serious study for mathematicians. Not that they imagined that it would lead them to anything of solid advantage, but because the theory was seen to be fraught with difficulty and it was considered that some new properties of numbers might be discovered which mathematicians could turn to account. This has, in fact, proved to be the case. For, from a certain point of view, the subject has been found to be algebraic rather than arithmetical and to be intimately connected with great departments of science, such as, the infinitesimal calculus, the calculus of operations and the theory of groups.
Till the advent of modern computers, it is understood, that, no living person knew, in how many ways, it is possible to form a square Sn of order exceeding 4. Even now it is not known how many possible squares are there of order 6 and above.
MAGIC SQUARES IN CHINA
The Chinese appear to have invented and used the Magic Squares since long time back; they are mentioned in a Chinese book written four to five thousands years before our era. The Worlds oldest square, a Chinese creation, is reproduced below:-
It is known as the LO SHU.
4 9 2
3 5 7
8 1 6
According to legend, the pattern was first revealed on the shell of a turtle that crawled out of the LO river in the 23rd century B.C. in the days of the legendary Emperor Yii, reputed to be a Hydraulic Engineer.
In Chinese history the pattern is considered as a mystical symbol of enormous significance. The even numbers are identified with YIN, the female principle, and the odd numbers with YANG, the male principle. The central 5 represented the earth around which, in evenly balanced Yin and Yang, were the four elements 4 & 9 symbolizing metal, 2 & 7 fire, 1 & 6 water and 3 & 8 wood.
In Chinese cosmology, from which Chinese Astrology is derived, the Positive (YANG) and Negative (YIN) aspects of reality are not seen as opposing one another, but, on the contrary, as supporting each other, like the left and the right sides of an arch. Just as by definition one arch cannot exist without having two sides, so reality cannot exist without both its Positive and Negative aspects. In the ancient Positive and Negative Taoist philosophy, everything is classified as pertaining to one of these two aspects of inter-dependent duality. Thus the Night is negative, the Day is positive; Female sexuality is negative, Maleness is positive;
Sugar is negative, Salt is positive; and so on
And now have a look at how these numbers behave;-
492^2 + 357^2 + 816^2 = 438^2 + 951^2 + 276^2.
MAGIC SQUARES IN INDIA:
The order 4 magic square, with magic sum of 34, shown below is over 2000 years old and comes from India:
07 12 01 14
02 13 08 11
16 03 10 05
09 06 15 04
Given below now is the Jaina square of Khujaraho, also from India :
01 14 15 04
12 07 06 09
08 11 10 05
13 02 03 16
The Indian square has some interesting properties:-
1. All the four 2×2 corners and the central 2×2 square also total 34.
2. In each row one pair adds to 19 and the other to 15. In each column also one pair adds to 13 and the other to 21.
3. The squares of the numbers in the first and fourth rows are equal, as also in the second and third rows.
1^2 + 14^2 + 15^2 +4^2 = 13^2 + 2^2 + 3^2 + 16^2 = 438
12^2 + 7^2 + 6^2 + 9^2 = 8^2 + 11^2 + 10^2 + 5^2 = 310
4. Similarly the squares of the numbers in the first and fourth columns are equal, as also, in the second and third columns.
1^2 + 12^2 + 8^2 + 13^2 = 4^2 + 9^2 + 5^2 +16^2 =378
14^2 +7^2 + 11^2 + 2^2 = 15^2 + 6^2 +10^2 + 3^2 =370
Now consider the diagonals of the 2×2 squares
12 +14 +3 + 5 = 15 + 9 +8 +2 = 34 further
12^2 +14^2 + 3^2 + 5^2 =15^2 + 9^2 + 8^2 +2^2 = 374 and
12^3 +14^3 +3^3 + 5^3 = 15^3 + 9^3 + 8^3 +2^3 =4624
Similar properties are found for the Jaina square:-
1 All the four 2×2 corners and the central 2×2 square total 34.
2 In each row one pair adds to 19 and the other to 15. In each column one pair adds to 9 and the other to 25.
3 7^2+12^2+1^2+14^2 = 16^2+3^2+10^2+5^2 =390
2^2+13^2+8^2+11^2 = 9^2+ 6^2+15^2+4^2 =358
4 7^2+2^2+16^2+9^2 = 1^2+8^2+10^2+15^2 =390
12^2+13^2+3^2+6^2 = 14^2+11^2+5^2+4^2 =358
Magic Squares were constructed in India before the Christian era.
MAGIC SQUARES AND ASTROLOGY
The famous Cornelius Agrippa(1486-1535) constructed Magic Squares of the orders 3, 4, 5, 6, 7, 8 & 9 which were associated by him, with the 7 astrological “Planets”, namely Saturn, Jupiter, Mars, Sun, Venus, Mercury and the Moon.
A magic square of order 4 appears in one of Albercht Durer’s best engraving “Melancholia 1″ (1514). His square is like the one from India, slightly altered (by rotation) and is shown below:
16 03 02 13
05 10 11 08
09 06 07 12
04 15 14 01
(please note 0’s have been placed by me, they are not there in the original magic square.)
This square is also known as the Magic Square of Jupiter and is supposed to be intimately connected with all the attributes which, traditionally, asreologers have associated with Jupiter namely – Happiness, Prosperity, Good Fortune, A Long Life and so on. This square appears in the background in the Durer’s engraving of the spirit of Melancholia. Albercht Durer, a 16th Century Artist, who was considered to be the greatest of his time, held the belief, still held by some present day occulists, that numerological techniques could be used to induce mood changes, the lifting, for example, of acute depression, the psychological disorder, which our ancestors referred to as melancholia.
The Magic Square of Saturn, reproduced below, is said to have the reverse effect.
4 9 2
3 5 7
8 1 6
The Magic Square of Saturn– the harbinger of Love is reproduced below:
22 47 16 41 10 35 04
05 23 48 17 42 11 29
30 06 24 49 18 36 12
13 31 07 25 43 19 37
38 14 32 01 26 44 20
21 39 08 33 02 27 45
46 15 40 09 34 03 28
MAGIC SQUARES AND POPE URBAN V111:
Pope Urban V111 was a dedeicated student of astrology of his time and had come to the conclusion that the influence of planetary positions of Mars and Saturn in his horoscope, combined with an impending eclipse of the Sun, would have to be countered in some way or the other. He consulted a Dominician Friar named Campanella, one of the most learned man of his time, who advised that the beneficial planetary effects of Jupiter and Venus shold be used to neutralize the harmful effects of Mars and Saturn. The two squares were hung on his wall. The supposed influences of the squares were supplemented by the Pope listening to music, which he believed to be of a Jupiterian or Venusian type, scenting his apartment with roses and drinking distilled liqueurs flavoured with aromatic plants, which the astrologers of his time believed to be under the domain of Jupiter and Venus. Perhaps all these worked, certainly the Sun’s eclipse took place, without the death of Pope Urban V111, who lived on for another 16 years.
ARE THEY REALLY MAGICAL?
WELL do these squares have really some magical properties? I
believe they do have, for example, the total number of 5×5 squares is over 60 millions, using numbers from 1 to 25 and to give a total of 65, but, if you are asked to construct one, I am sure, you will either not succeed or take a long time! And what if you are asked to produce a 16×16 square, using numbers from 1 to 256, yielding a total of 2056? (Let me tell you the number of solutions will be in trillions.) Well don’t despair, by the time you digest this paper, you will, probably, only take that much time as is required to draw the square and write down the numbers in the 256 cells of the given square — so aren’t they really magical?
Considering the stupendous results achieved by the engineering and the other applied sciences with the assistance of mathematics, it must be accepted that the forms of thought are wonderful indeed! As such, it is not at all surprising, that the primitive thinkers of mankind, when the importance of the laws of formal thought, in some way or another, first dawned in their minds, attributed to them supernatural powers. Somewhere herein lies lies the origin of the word MAGIC SQUARES!
However Prof. De Morgan has been reported as having said:
“Though the question of Magic Squares be in itself of no use, yet it belongs to a class of problems which call into action a beneficial species…. Without laying down any rules for their construction we shall content ourselves with destroying their magic quality and showing that the non-existence of such squares would be much more surprising than their existence”.
Truth, however, is not such a simple thing and, perhaps lies in between. When we discuss methods of formationj of these squares we will find that the squares do need different solutions and the laws are indeed difficult to formulate. We have already seen how the numbers of the 3×3 square and the 4×4 square have some intersting properties. More as we go on to the other squares!
MAGIC SQUARES – HOW MANY SOLUTIONS:
3×3 Squares.
There is a unique solution for a 3×3 square. Let us examine why?
Let us write all combinations of 3 numbers from 1 to 9 that add up to 15 They are 1,5,9: 1,6,8: 2,4,9: 2,5,8: 2,6,7: 3,4,8: 3,5,7: and 4,5,6. Now the central number of the square can only be that number which occurs in at least 4 combinations. The only number that satisfies this condition is 5. So only 5 can occupy the center space. Next examine 1, it combines with only 9 and 8 and as such cannot occupy corner space, so must come in the middle of a row or column. So we can place 1 and 9, and 8 immediately. Rest all the numbers then automatically fall in place.
4×4 Squares.
There are only 880 solutions for a 4×4 square, first given by Bernard Frenicle de Bassy in 1693, if we ignore reflections and rotations.
5×5 Squares.
There are a staggering number of solutions for a 5×5 square—-68,826,306 not counting reflections and rotations. This counting was arrived at by Richard Schroeppel, a mathematician and computer programmer at Information International, who used a standard backtracking procedure consisting of about 3500 lines and took 100 hours on a PDP-10. A final report was written by Michael Beeper and was issued in October 1975. For interested readers there is one classification by central numbers 1 through 13 (same numbers for 25 to 14) given below:-
1 – 1,091,448. 2 – 1,366,179. 3 – 1,914,984. 4 – 1,958,837.
5 – 2,431,806. 6 – 2,600,879. 7 – 3,016,881. 8 – 3,112,161.
9 – 3,472,540. 10 – 3,344,034. 11 – 3 ,933,818.
12 –3,784,618. 13 – 4,769,936
Higher Order Squares.
While no firm estimate is available for a 6×6 square, suffice it to say that, the number would run into billions. To imagine the number of solutions, simply increase all numbers in the 4×4 square by 10 to give a new 4×4 square consisting of numbers 11 to 26 with total of 34 + 40 = 74, this 4×4 square forms the inner core of the 6×6 square. Now we have to use numbers from 1 to 10 and 27 to 36 to form its boundary. Arrange these in pairs to total 37. Let us only consider one such combination of numbers arranged to give the required total of 111, see figure below formed with the Jaina square for the core 4×4 square:
01 34 05 07 35 29
33 17 22 11 24 04
31 12 23 18 21 06
28 26 13 20 15 09
10 19 16 25 14 27
08 03 32 30 02 06
Now any pair of numbers in the first and last row or the extreme left and extreme right column, excluding the corner numbers since they total 37, can be interchanged. This interchange can create 4!x4! = 576 new squares. Since there are 880 squares of order 4 which can be used for the inner core, and, further these squares can be rotated, we have, for one set of border 576×4x880 = 2,027,520 solutions. Further, in this arrangement of border numbers, if we interchange 34 & 7 from top row and corresponding pair numbers 3 & 30 from the bottom row with 31 & 10 from the left column and the corresponding pair numbers 6 & 27 from the right column, we get another set of 2,027,520 squares. But this is not all. Besides the fact that other border squares can be formed, there are other methods of construction. For example, if we reduce all odd numbers by 1 and increase all even numbers by 1 in the core 4×4 square, we get a new square, which has numbers from 10 to 27, excluding 11 and 26, still with a total of 74. (This can be done for 712 squares of order 4, as this process cannot be adopted for those squares, which have only odd and even numbers in their diagonals). Thus for any border combination for these squares we shall have 576×4x712 = 1,640,448 solutions.
This method of generating squares from lower order squares, described later in detail, can be applied to generate squares of any order, and, as such, the number of squares that can be constructed can only be imagined. Besides this method, there are other methods which, again, can result in a large number of solutions.
METHODS OF CONSTRUCTION:
We will now discuss, in greater detail, methods of construction of squares of different orders.
Border Square Method:
There is no universal method of construction of magic squares ab-initio. However, once a 3×3 square (which has only one solution) and a 4×4 square, have been constructed, the border square method, illustrated earlier for the 6×6 square, can be considered as universally applicable. In general, an order ‘n’ square can be constructed from an order ‘n-2′ square by the following method:-
To every number in the order (n-2) square add 2n-2; the numbers not yet used will be 1,2,3,….2n-2 and n^2, n^2-1, n^2-2,…..n^2-2n+3.
(these numbers are complementary numbers, in the sense that pairs of them total n^2+1). These numbers are to be placed in the 4(n-1) border cells such that the complementary numbers occur at the end of the 2 diagonals, columns and the rows of the inner square; however, the choice of the numbers for the top row and one of the columns is to be so made that the total comes to n(n^2+1)/2, the other row and column will automatically give the required total. This selection of numbers is best done by trial and error method. So long as precaution is taken not to use complementary numbers in the top row and the selected column this should not pose any problem. This method is attributed to Frenicle.
Let us try this for the 5×5 square from the 3×3 square. The order 3 square is:
4 3 8
9 5 1
2 7 6
In this square we have to increase the numbers in each cell by 2n-1 i.e. 2(5-1) = 8.we thus get:
12 11 16
17 13 09
10 15 14
The border square is to be formed with numbers from 1 to 18 and 18 to 25. Take 18, 20, 21, 4, 2 for the top row and 18, 1, 3, 19, 24, for the left-hand column.
So the final square is:
18 20 21 04 02
01 12 11 16 25
03 17 13 09 23
19 10 15 14 07
24 06 05 22 08
We have already seen construction of 6×6 square by this method earlier.
Incidentally for this square we have:-
18^2+1^2+3^2+19^2+24^2 = 2^2+25^2+23^2+7^2+8^2
and 18^2+20^2+21^2+4^2+2^2 = 24^2+6^2+5^2+22^2+8^2.
THE HINDU RULE:
For odd order squares we have a special rule, called the HINDU RULE. The Hindu rule may be enunciated as follows:-
To start with write the first number 1 in the center of the topmost row, next write 2 in the lowest space of the vertical column next adjacent to the right, and then so inscribe the remaining numbers in their natural order in the squares diagonally upwards towards the right that, on reaching the right-hand margin, the inscription shall be continued from the left-hand margin in the row just above, and again, on reaching the upper margin, shall be continued from the lower margin in the column next adjacent to the right, noting that whenever we are arrested in our progress by a square already occupied we are to fill out the square next beneath the one we have filled. In this manner, for example, the 7×7 square given below has been formed:
30 39 48 01 10 19 28
38 47 07 09 18 27 29
46 06 08 17 26 35 37
05 14 16 25 34 36 45
13 15 24 33 42 44 04
21 23 32 41 43 03 12
22 31 40 49 02 11 20
This method is neat and quick. De La Loubere, Envoy of Louis X1V to Siam learnt of this method here.
We can also start with the central cell of the bottom row, central cell of the first column or the last column as we like. I will illustrate the alternate methods by starting with the last column and I will use symbols (alphabets) to generate multiple solutions:
C+f B+e A+d G+c F+b E+a D+g
B+d A+c G+b F+a E+g D+f C+e
A+b G+a F+g E+f D+e C+d B+c
G+g F+f E+e D+d C+c B+b A+a
F+e E+d D+c C+b B+a A+g G+f
E+c D+b C+a B+g A+f G+e F+d
D+a C+g B+f A+e G+d F+c E+b
Giving A’s one of the values—0,7,14,,28,35,42, and a’s one of the values—1,2,3,4,5,6,7 – with the proviso that ‘D’ has to be given the value 21, we are able to cover all numbers from 1 to 49 and get the magic sum of 175. It gives 7!x6!/4 clear solutions. We can alternatively also give A’s one of the values – 1,2,3,4,5,6,7 and to a’s – 0,7,14,28,35,42 – with ‘d’ being given the value 21. . Of course no value should be repeated, this will ensure that all the numbers are used.
And now let us have a look at the 5×5 square by the Hindu Rule:
09 03 22 16 15
02 21 20 14 08
25 19 13 07 01
18 12 06 05 24
11 10 04 23 17
and note that:-
9^2+2^2+25^2+18^2+11^2 = 15^2+8^2+1^2+24^2+17^2 = 1155.
3^2+21^2+19^2+12^2+10^2 = 16^2+14^2+7^2+5^2+23^2 =
9^2+3^2+22^2+16^2+15^2 = 11^2+10^2+4^2+23^2+17^2 = 1055, And
2^2+21^2+20^2+14^2+8^2 = 18^2+12^2+6^2+5^2+24^2 =
22^2+20^2+13^2+6^2+4^2 =1105.
RALPH STRACHEY’S METHOD:
Mr. Ralph Strachey has devised rules for the construction of a magic square of a singly- even order n where n = 2(2m+1) ,
The rule states:– Divide the square into 4 equal quarters A, B, C,& D.
A B
D C
Construct in A, by the Hindu Rule a magic square with the numbers from 1 to u^2, where u = n/2. Construct by the identical method squares B,C & D using numbers from u^2+1 to 2u^2; 2u^2+1 to 3u^2; & 3u^2+1 to 4u^2(n^2).
(Best done by adding u^2 to numbers in A – to get B, then adding u^2 in B – to get C and then adding u^2 in C – to get D). Clearly the resulting composite square is magic in columns. Now in the middle row of A take the m cells next but one to the left-hand side and in each of the other rows take the m cells nearest to the left-hand side; interchange the numbers in these cells with the corresponding numbers in cells in D. Next interchange the numbers in the cells in each of the m+2 columns next to the left-hand side of B with the numbers in the corresponding cells of C. Now the resulting square will be truly a magic square. For ex. for a 6×6 square, n = 6, m = 1, u =3.
The preliminary square is as under:
08 01 06 17 10 15
03 05 07 12 14 16
04 09 02 13 18 11
35 28 33 26 19 24
30 32 34 21 23 25
31 36 29 22 27 28
The final square is as under:
35 01 06 26 19 24
03 32 07 21 23 35
31 09 02 22 27 20
08 28 33 17 10 15
30 05 34 12 14 16
04 36 29 13 18 11
Although Mr. Strachey talks of construction of the odd order square by the Hindu rule it is not essential, provided we add u^2 to numbers in A etc., we can use any odd order square to begin with. The Hindu Rule, of course, enables us to get the odd order square quickly. A 10×10 square, using a 5×5 square constructed differently, is generated below:-
First the 10×10 order square before application of the Strachey rule:
03 16 09 22 15 28 41 34 47 40
20 08 21 14 02 45 33 46 39 27
07 25 13 01 19 32 50 38 26 44
24 12 05 18 06 49 37 30 43 31
11 04 17 10 23 36 29 42 35 48
78 91 84 97 90 53 66 59 72 65
95 83 96 89 77 70 58 71 64 52
82 100 88 76 94 57 75 63 51 69
99 87 80 93 81 74 62 55 68 56
86 79 92 85 98 61 54 67 60 73
And now the final square:
78 91 09 22 15 53 66 59 72 40
95 83 21 14 02 70 58 71 64 27
07 100 88 01 19 57 75 63 51 44
99 87 05 18 06 74 62 55 68 31
86 79 17 10 23 61 54 67 60 48
03 16 84 97 90 28 41 34 47 65
20 08 96 89 77 45 33 46 39 52
82 25 13 76 94 32 50 38 26 69
24 12 80 93 81 49 37 30 43 56
11 04 92 85 98 36 29 42 35 73
AUXILIARY SQUARE METHOD:
Magic squares of odd order and doubly even order can be easily constructed by a method, which uses 2 auxiliary squares. It, at one go, generates a large number of squares of the given order. Form two auxiliary squares, one with A, B, C, D,…..and the other with a, b, c, d,…. These are to be constructed like any other magic square in as much as each of the alphabets appears once and once only in each row, each column and each diagonal. However the two squares should not be identical. The two squares are to be merged ensuring that each A’s gets associated with each a’s once and once only. This would ensure that each “a” gets associated with each “A” etc.,. Given below are 4×4 order squares for illustration:
A D B C….a b c d
C B D A….d c b a
D A C B….b a d c
B C A D….c d a b
and the final square:
Aa Db Bc Cd
Cc Bd Da Ab
Dd Ac Cb Ba
Bb Ca Ad Dc
Aa is now to be read as A+a, Bb as B+b, etc., Now A can be given any of the four numbers 0, 4, 8, 12 and a any of the numbers 1, 2, 3, 4. (in general 0, n, 2n, 3n, and 1, 2, 3, 4, ….) B and b can be given any of the remaining 3, C and c can be given one of the remaining 2, and .D and d the last numbers. It will be noticed that automatically we have all the required numbers from 1 to 16. This can give us 4!x4! /4 = 144 squares. One such square with A = 12, B = 8, C = 0, D = 4, a = 1, b = 2, c = 3, & d = 4, is generated below as an ex:
13 06 11 04
03 12 05 14
08 15 02 09
10 01 16 07
For a 5×5 order square we have:
Aa Dc Be Eb Cd
Bb Ed Ca Ac De
Cc Ae Db Bd Ea
Dd Ba Ec Ce Ab
Ee Cb Ad Da Bc
with a,b,c,d,e having values 1,2,3,4&5, and A, B,C,D,&E having values from 0,5,10,15 &20. This method immediately yields 5!x5! /4 = 3600 squares.
Here is an 8×8 order square by the auxiliary square method:
Cc Ga Hg De Fh Bf Ad Eb
Fg Be Ac Ea Cd Gb Hh Df
Bh Ff Ed Ab Gc Ca Dg He
Gd Cb Dh Hf Bg Fe Ec Aa
Ef Ah Bb Fd Da Hc Ge Cg
Db Hd Gf Ch Ee Ag Ba Fc
Ha Dc Ce Gg Af Eh Fb Bd
Ae Eg Fa Bc Hb Dd Cf Gh
It will yield (8!x8!)/4 squares.
De La Hire’s Method:
De La Hire has suggested a modified form of this procedure for even order square of 6×6. His method is to form two auxiliary squares with 0,6,12,18,24 & 30; and 1,2,3,4,5 & 6 in such a way that the columns of one square and the rows of the other square each so contain numbers, 3 times repeated, that the required sum summation of 90 for one swquare and 21 for the other square is achieved. We may, however, form these squaeres using A’s and a’s using the conditions stipulated by him. Our condition then will be that A’s and a’s be so combined in pairs that they total 30 and 7 respectively. We can take combinations of 0 and 30; 6 and 24; 12 and 18; 1 and 6; 2 and 5; 3 and 4. This will enable us to use permutations of numbers and thus generate multiple squares. The auxiliary squaress will take this shape:
A F F A F A ………… a e d c b f
E B E E B B ………… f b d c e a
D C C C D D ……….. f e c d b a
C D D D C C ……….. a e c d b f
B E B B E E ……….. f b c d e a
F A A F A F ………… a b d c e f
If we take A = 0, B = 6, C = 12, D =18, E = 24, F = 30, a =1,
b = 2, c = 3, d = 4, e = 5 and f = 6, our 6×6 squarec will be:
01 35 34 03 32 06
30 08 28 27 11 07
24 17 15 16 20 19
13 23 21 22 14 18
12 26 09 10 29 25
31 02 04 33 05 36
A further modification of this method, a bit more complicated, permits us use of A’s and a’s in such a way that the arithmetical sum of each row, column, and each diagonal, with appropriate values of A’s and a’s total nxn(n-1)/2 for A’s and nx(n+1)/2 for a’s. This means that there can be 2 or more A’s and a’s in diagonals too.
Other Methods:
We conclude this part with a few methods which are more of academic interest as they yield one or just a few squares.
Cosider the picture below (dashes have been used to fill up empty spaces):
- – - – 01 – - – -
- – - 06 – 02 – - -
- – 11 – - 07 – 03 – -
- 16 – 12 – 08 – 04 -
21 – 17 – 13 – 09 – 05
- 22 – 18 – 14 – 10 -
- – 23 – 19 – 15 – -
- – - 24 – 20 – - -
- – - – 25 – - – -
Now we have to simply move the numbers, which are outside the main square, 5 steps to the empty spaces to get the following 5×5 square.
11 24 07 20 03
04 12 25 08 16
17 05 13 21 09
10 18 01 14 22
23 06 19 02 15
This method is attributed to Bachet de Meziriac.
Same principle holds good for constuction of higher order odd squares.
We now have a look at a 6×6 square.
Write numbers from 1 to 36 in their natural order as shown below:
01 02 03 04 05 06
07 08 09 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
25 26 27 28 29 30
Replace 2,3,4,5 by 35,34,3,32 respectively; 7 & 25 by 30 & 12;
13 & 19 by 24 & 13; 9 & 10 by 28 & 27; 12 & 30 by 7 & 25; 18 & 24 by 19 & 18; 17 & 23 by 14 & 20; 27 & 28 by 9 & 10; 32, 3, 34, 35 by 2, 4, 33, 5 respectively, giving a 6×6 square as under:
01 35 34 03 32 06
30 08 28 27 11 07
24 23 15 16 14 19
13 17 21 22 20 18
12 26 09 10 29 25
31 02 04 33 05 36
Note that diagonal numbers were not changed.
Since 6/2 is odd a 6×6 square needs quite a few changes making it cumbersome, however, it is easier to construct a 8×8 square this way.
Write the numbers 1 to 64 in their natural order as shown below:
01 02 03 04 05 06 07 08
09 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56
57 58 59 60 61 62 63 64
Now divide by 2 vertical and 2 horizontal lines so that in each corner there is a 2×2 square and in the center a 4×4 square. Within these 5 squares inter-change all apirs of numbers symmetrically opposite. Outside the 5 squares no change is required. The resulting square will thus be as under:
64 63 03 04 05 06 58 57
56 55 11 12 13 14 50 49
17 18 46 45 44 43 23 24
25 26 38 37 36 35 31 32
33 34 30 29 28 27 39 40
41 42 22 21 20 19 47 48
16 15 51 52 53 54 10 09
08 07 59 60 61 62 02 01
Well we can go the other way round too. Keep the 4 corner 2×2 squares and the central 4×4 square as they are and reverse the other 4×2 squares in rows 3,4,5&6 and columns 1&2 and 7&8 and the rows 1&2 and 7&8 in columns 3,4,5,&6 to form the new square. I will illustrate this by 12×12 cell, but before that please note that this method is valid for all squares which are multiples of 4. So if the square is 4nx4n, we take the corner cells of the order nxn and the central cell of the order 2nx2n for reversal. Alternately we retain the 4 corner nxn squares and central 2nx2n squares unchanged and reverse the squares, as seen below for the 12×12 square.
001 002 003 141 140 139 138 137 136 010 011 012
013 014 015 129 128 127 126 125 124 022 023 024
025 026 027 117 116 115 114 113 112 034 035 036
108 107 106 040 041 042 043 044 045 099 098 097
096 095 094 052 053 054 055 056 057 087 086 085
084 083 082 064 065 066 067 068 069 075 074 073
072 071 070 076 077 078 079 080 081 063 062 061
060 059 058 088 089 090 091 092 093 051 050 049
048 047 046 100 101 102 103 104 105 039 038 037
109 110 111 033 032 031 030 029 028 118 119 120
121 122 123 021 020 019 018 017 016 130 131 132
133 134 135 009 008 007 006 005 004 142 143 144
It will be noticed that here too we can generate a large number of squares very easily. It will be noticed that pairs of numbers have same total, which can be interchanged without affecting the diagonal totals. A few examples:2+95=3+94, 2+83=3+82, 84+109=85+110, etc. More work out yourself! It will also be seen that it can be done in 8×8 square too.
SUMMING UP:
We have seen that a 3×3 square only one and 4×4 square 880 solutions and can be constructed by the auxiliary method and also by other methods. A 5×5 square can be constructed either by the Hindu Rule which gives 720 squares from one set, or by the Border Square method with the 3×3 square as the base, or from two auxiliary squares which at once gives 3,600 squares from one set. Further 5×5 squares can be constructed by either constructing 3×3 squares with total of 39 with 13 in the central cell, or by first forming a 3×3 square with any numberin the central cell and thereafter filling up thremaining cells with numbers of our choice, except that the total for any one row or columnor diagonal should not be more than62. Some illustrations are given below, first 2 with 13 in the central cell:
22 18 10 09 06
07 12 03 24 19
11 25 13 01 15
05 02 23 14 21
20 08 16 17 04
20 01 23 09 12
22 08 21 10 04
02 15 13 11 24
07 16 05 18 19
14 25 03 17 06
Now with 15 in the central cell and different totals for rows and columns for the 3×3 square:
01 10 19 23 12
09 18 22 11 05
17 21 15 04 08
25 14 03 07 16
13 02 06 20 24
It will be seen that the numbers in the border square do not folow any pattern. But we can form additional squares by subtracting all numbers from 26.
As is the case with 4×4 square, for 5×5 square too, we can frame rules which will enable us to generate squares from the one already constructed. Let the square be represented by:
a b c d e
f g h i j
k l m n o
p q r s t
u v w x y
We can now generate new squares by interchanging any two columns or rows provided this affect the diagonal totals, some examples are given below:
a) If g+s = i+q, we can interchange 2nd and 4th columns
b) If h+n = m+i and m+s=r+n, 3rd and 4th columns.
c) If g+m=h+l and h+n=i+m, 2nd and 3rd rows.
d) If g+m=h+l and l+r=m+q, 2nd and 3rd columns.
e) If l+r=m+q and m+s=r+n, 3rd and 4th rowsw.
f) Also if b+c=v+w, they can be interchanged in the 1st and 5th rows, etc.
As stated earlier new squares can be generated merely by subtracting all numbers from 26. These rules, incidentally, applicability in general for all squares of anny order.
MY METHOD:
While it is very easy to construct odd order magic squares (the Hindu Rule is a very easy method for an nxn square where “n” is odd) and doubly even order magic squares, singly even order
squares are not that easy. Border square method is the one most commonly used. I have developed a method for construction of (n+4) x (n+4) squares from nxn square where n is 6,10,14,18……to easily give multiple solutions without much effort. I will illustrate my method by first constructing a 14×14 square assuming that we have a 10×10 square. The 10×10 square will occupy the center place. 14×14 square has numbers from 1 to 196 and 10×10 square numbers from 1 to 100. If we take half the difference between 196 and 100 we get 48 and if we add this to all numbers in our 10×10 square we will have a 10×10 square with numbers from 49 to 148 and total of 505 + 10×48 = 985. The 14×14`square needs to have a total of 1379, this leaves us with a difference of 394. The numbers from 1 to 48 and 149 to 196 can be used to form six 4×4 squares with numbers from i) 1 to 8 & 189 to 196, ii) 9 to 16 and 181 to 188, iii) 17 to 24 and 173 to 180, iv) 25 to 32 and 165 to 172, v) 33 to 40 and 157 to 164, and vi) 41 to 48 and 149 to 156, to give a total of 394 for all the six 4×4 squares. We will label five from these six squares from 1 to 5 and, the 10×10 square by 0, the sixth square will need to be split into two 4×2 square and has been labeled from A to P to fit our requirements of 14×14 square.
1 1 2 2 3 3 A B 3 3 2 2 1 1
1 1 2 2 3 3 C D 3 3 2 2 1 1
4 4 0 0 0 0 0 0 0 0 0 0 4 4
4 4 0 0 0 0 0 0 0 0 0 0 4 4
5 5 0 0 0 0 0 0 0 0 0 0 5 5
5 5 0 0 0 0 0 0 0 0 0 0 5 5
I K 0 0 0 0 0 0 0 0 0 0 M O
J L 0 0 0 0 0 0 0 0 0 0 N P
5 5 0 0 0 0 0 0 0 0 0 0 5 5
5 5 0 0 0 0 0 0 0 0 0 0 5 5
4 4 0 0 0 0 0 0 0 0 0 0 4 4
4 4 0 0 0 0 0 0 0 0 0 0 4 4
1 1 2 2 3 3 E F 3 3 2 2 1 1
1 1 2 2 3 3 G H 3 3 2 2 1 1
We can select any one of the six squares by turns for 1 to 5, the remaining sixth 4×4 square has to be formed slightly differently. We have to ensure that A+B = C+D = E+F = G+H = I+J = K+L = M+N = O+P = 197, and A+C+E+G = B+D+F+H = I+K+M+O = J+L+N+P = 394.
Since the 10×10 square occupies the center place the and that leaves us only two top rows and two bottom rows, and similarly two L.H.S. and R.H.S. columns, the five 4×4 squares have been split into two 4×2 squares each and suitably placed in the
corners. The 1 square will take care of our Diagonal totals for the 14×14 square, so we don’t have to worry about diagonal totals in rest of the 4×4 squares which means having selected the 4×4 square from the available 712 squares (880 less the squares which have 3 numbers8 and 18, ii) 8 to all numbers 8, iii) 16 to all numbers 8 and iv) 24 to all numbers 8, limiting our choice to 712 4×4 squares, excluding those which may have 3 numbers 8.We can select any one of the six squares by turns for 1 to 5, the remaining sixth 4×4 square has to be formed slightly differently as already explained. We have to ensure that A + B = C + D = E + F = G + H = I + J = K + L = M + N = O + P = 197, and A + C + E + G = B + D + F + H = I + K + M +
O = J + L + N + P = 394.
(P.S. simplest way to form the 4×4 square, say with numbers from 17 to 24 and 173 to 180, would be to add in an ordinary 4×4 square 16 to all numbers from 1 to 8 and 164 to all numbers from 9to 16.)
When we go from 14×14 to 18×18 square, the base becomes 14×14 square and to all the numbers we add 1/2(18×18 –14×14)=64 and the 14×14 square total becomes 2275. The total required for 18×18 square is 2925, this is less by 650 to be filled by the empty cells. We have now numbers from 1 – 64 and
261 – 324 to form 8, 4×4 squares with each having a total of 650. Of these one will be required for extreme corners (like 1) and one for splitting into two 4×2 squares. From rest 6 we will have 3 squares for rows and 3 squares for columns, and the number of multiple squares that can be generated from this one 18×18 squares can easily be worked out.
I now come to the first square that can be formed, the 6×6 square. I will be using one of the 880, 4×4 squares, but for those not fully conversant, I give the basic square, using alphabets, and where Aa stands for A+a.
Aa Bb Cc Dd
Dc Cd Ba Ab
Bd Ac Db Ca
Cb Da Ad bc
Here A’s can be given one of the values from 1,2,3,4 and a’s can be given one of the values from 0,4,8,12. But each value can be given only once, i.e. if you give 1 to A you can give B only from
2,3,4 and so on. Alternately you can give to A’s from 0,4,8,12 and a’s from 1,2,3,4. After constructing the 4×4 square simply add 10 to all numbers. 10 is 1/2(36-16), this will take the sum
in each row, column and diagonal to 74. Numbers now available to us are 1 to 10 and 27 to 36.
Next I construct a 2×2 square such that its diagonals total 37. 2 & 35 and 1 & 36.This enables me to take care of the diagonal totals as I am going to break up the 4×4 square into four 2×2 squares and place them in the corners. For my 4×4 square I give value 1 to a, 2 to b, 3 to c, and 4 to d, 0 to A, 4 to B, 8 to C and 12 to D. So my 6×6 square now looks like this:
11 16 00 00 21 26
25 22 00 00 15 12
00 00 02 36 00 00
00 00 01 35 00 00
18 23 00 00 24 19
20 23 00 00 14 17
(Here 00 stand for values yet to be filled in.)
Next I insert numbers, from those available, in columns 3 and 4 in rows 1,2,5 and 6 such that sum of numbers in each row is 37. Next I place numbers in rows 3 and 4 in columns1, 2, 5 and 6 such that magic sum of that each column total is 37. This ensures that the magic sum of 111 is available for each row, column and 2 main diagonals. My final 6×6 square is:
11 16 34 03 21 26
25 22 33 04 15 12
09 29 02 36 05 30
28 08 01 35 32 07
18 13 10 27 24 19
20 23 31 06 14 17
Even if we ignore choices for the central 2×2 square, we can from this one square alone generate very large number of 6×6 squares. We have choice of 880 for the4×4 squares and they can be rotated to give 4 times more squares. Next the numbers in columns 3 and 4 in the rows 1, 2, 5 and 6 can be interchanged and so can the numbers in rows 3 and 4 in columns 1, 2, 5, and 6..
To make my point clear I am replacing numbers in column 3 and 4 in rows 1,2,5 and 6 by alphabets and also in rows 3 and 4 in columns 1, 2, 5, and 6.
00 00 A1 B1 00 00
00 00 A2 B2 00 00
C1 C2 00 00 C3 C4
D1 D2 00 00 D3 D4
00 00 A3 B3 00 00
00 00 A4 B4 00 00
(Here again 00 have been placed in the empty spaces to suit the format design.)
Here A1+B1=A2+B2=A3+B3=A4+B4=37,
and C1+D1= C2+D2=C3+D3=C4+D4= 37, and so are interchangeable. So we have at least 880×4x24×24x2×2x2 squares!
Let us now go to construction of 10×10 square. First we add 32 equal to 1/2(100 –36) to all numbers in one of the 6×6 square available to me. This will give me a total of 111+32×6=303 for
the 6×6 square. Next I construct four 4×4 squares from the numbers 1 to 32 and 69 to 100 available to me by adding i) 0 to all numbers 8, ii) 8 to all numbers 8, iii) 16 to all numbers 8 and iv) 24 to all numbers 8, limiting our choice to 712 4×4 squares, excluding those which may have 3 numbers 8. To go from 10×10 to 14×14, place 10×10 square in the centre increasing all numbers by 1/2(196 100),i.e.48, to give a total of 985 for the 10×10 square, the 4×4 squares will give 394 making a total of 1,379, as required, for the 14×14 square. The 4×4 squares will number 5 and there will be 2 4×2 squares and placed just as in 10×10 square when we proceeded from 6×6 square to 10×10square.
Try to form the 10×10square and if you have difficulty get in touch with me by email bdtara@yahoo.com
ORDER 4 MAGIC SQUARES – ALL 880 SQUARES:
Coming back to order 4 squares, we examine them in detail as there are only 880 solutions.
First, when can we not have them?
First point is to be noted is that the sum of all odd numbers between 1 and 16 is 64 and all even numbers total 72. Since the magic sum figure is 34 (an even number), each row and column or diagonal will have to have either 2 or 4 odd numbers. But if we have odd numbers only in one row (or column), the other row (or column) will necessarily have all odd numbers and we will fail to get the magic sum. However, one diagonal can only have odd numbers provided the other diagonal has only even numbers. Similarly we cannot have a square if the sum of the corner numbers does not add up to 34. The sum of the corner number has to be 34 only, neither more nor less.
We also cannot have only odd (or even) numbers in corners, since if we have odd (or even) numbers in the ccorners, all middle numbers will have to be even (or odd) numbers and including diagonal numbers it makes for 12 numbers, while we have only 8 even and 8 odd numbers. And if we take only odd numbers in both diagonals again we cannot get the magic sum.
RULES FOR FORMATION:
Now let us examine the rules that will enable us to generate more squares from a given square. Let us for this purpose, take the Magic Square given below:
A B C D
E F G H
I J K L
M N O P
RULE 1. For all types of squares we get a new square by simultaneously interchanging B&C, H&L, N&O, E&I, F&K, G&J.
RULE 2. Change numbers F,G,J&K to corner numbers and A,D,M&P, and corner numbers to center, but otherwise retaining the numbers of th3e rows and columns.
RULE 3. Change all even numbers to odd numbers by decreasing them by one and odd numbers to even numbers by increasing them by one.
RULE 4. Subtract all numbers from n^2+1.
RULE 5. If F+K=G+J, we can interchange the middle rows or columns. both can be interchanged, but not necessary in view of Rule 1.
RULE 6. If B+C= N+O, we can replace B+C by N+O and vice-versa. Same holds good for E+I =H+L.
RULE 7. If B+N =C+O, we can ointerchange them. Similarly if
E+H =I+L, they can be interchanged.
(It may be noted that some of these rules will not be valid for some of the squares. In particular Rule 3 will not be valid for for a square which has only odd numbers in one diagonal and even numbers in the other diagonal. Rule 4 w2ill not produce a new square if corner numbners are complimentary, i.e. each add upto 17.)
RULE 8. Applicable only to Regular Squares which number 528 in all. Change diagonal a,f,k,p to first row and m,j,g,d to 4th row and rewrite the numbers between a to m and d to p. Similarly change diagonals to columns and vice -versa to get a new square.
Now let us see how these Rules work for a particular square. We start with the square given below:
01 12 14 07
06 15 09 04
11 02 08 13
16 05 03 10
Rule 7 will give 3 more squares as shown below:
01 12 14 07 – 01 14 12 07 – 01 14 12 07
04 15 09 06 – 06 15 09 04 – 04 15 09 06
13 02 08 11 – 11 02 08 13 – 13 02 08 11
16 05 03 10 – 16 03 05 10 – 16 03 05 10
If we now apply Rule 1 to all these squares we get 4 more squares:
01 14 12 07 – 01 14 12 07 – 01 12 14 07 – 01 12 14 07
11 08 02 13 – 13 08 02 11 – 11 08 02 13 – 13 08 02 11
06 09 15 04 – 04 09 15 06 – 06 09 15 04 – 04 09 15 06
16 03 05 10 – 16 03 05 10 – 16 05 03 10 – 16 05 03 10
Now we apply Rule 8 to the main square to get 2 more squares:
01 15 08 10 – 01 12 14 07
06 12 03 13 – 15 06 04 09
11 05 14 04 – 08 13 11 02
16 02 09 07 – 10 03 05 16
Rule 7 applied to the left-hand side square along with Rule 1, will give 8 new squares. Rule 5 applied to the right-hand side square alongwith Rule 1, will give 4 new squares. In all, as such, we get 20 additional squares. Now applyinng Rule 3 to the mail sqaure, we get:
02 11 13 08
05 16 10 03
12 01 07 14
15 06 04 09
When we apply Rules 1,5,7 & 8 to this square we get 20 new squares.
Also let us apply Rule 2 to the left-hand side square and then Rule 3 to the newly generated square and see what happens. We get the following 2 squares:
06 15 09 04 – 05 16 10 03
12 01 07 14 – 11 02 08 13
03 10 16 05 – 04 09 15 06
13 08 02 11 – 14 07 01 12
These 2 will generate 20 more squares each, i.e. in all 40 squares.
So from 1 main square and the various Rules we have been successful in getting, in all, 80 squares!
REGULAR SQUARES:
For Rule 8 we talked of Regular Squares – so what is a Regular Square? If we reduce all numbers from 1 to 16 by 1, we get numbers from 1 to 15, it will be noticed that all these nmbers are a combination of 1,2,4 & 8. if we placve these numbers in a magic square its total will be 30. Now if we split the numbers of this modified square into its components of 1,2,4 & 8 and place them in seperate 4×4 squares and the squares, so formed, are such that none of the numbers 1,2,4 & 8 occur more than twice in any row, column or diagonal, in their respective squares, we get a Regular Square. Let us try this for the main square discussed earlier. Reducing all the numbers by 1 we get:
00 11 13 06
05 14 08 03
10 01 07 12
15 04 02 09
And the split numbers get arranged as under:
0 1 1 0 – 0 2 0 2 – 0 0 4 4 – 0 8 8 0
1 0 0 1 – 0 2 0 2 – 4 4 0 0 – 0 8 8 0
0 1 1 0 – 2 0 2 0 – 0 0 4 4 – 8 0 0 8
1 0 0 1 – 2 0 2 0 – 4 4 0 0 – 8 0 0 8
It will be seen that the numbers fall in the pattern requaired for a Regular Square. If we, however, examine the square reproduced below it will be found that it is not a Regular Square.
01 10 15 08 – 00 09 14 07 It will be seen that one diagonal has 2 times
05 14 11 04 – 04 13 10 03 while the other does not have it at all.
16 03 06 09 – 15 02 05 08
12 07 02 13 – 11 06 01 12
TYPES OF SQUARES — BREAK – UP:
The split of 880 squares by type as defined by me is:
Type A – 528 – Regular squares.
Type B – 120 – 3 numbers less than 9 in one diagonal and 3 numbers greater than 8 in the other diagonal.
Type C – 064 – 2 numbers less than 9, and 2 numbers greater than 8, as in regular squares, but not a Regular Square.
Type D – 120 – All odd numbers in one diagonal and all even numbers in the other, but otherwise Type C.
Type E – 048 – All odd numbers in one diagonal and all even numbers in the other, but otherwise Type B.
MAGIC SQUARES FROM MAGIC SQUARES:
We have seen earlier how we can construct a large number of squares of order 5 and above. i will now discuss another special and interesting method for construction of ‘p’ order squares, where p = mxn, with both m & n being greater than or equal to 3. the metod reuqiresthat we replace the integers by the squares of other order. I illustrate this method by constructing the 12×12 order square, 12 being equal to 3×4.
The 12×12 square will have 144 cells, numbers to be used being 1 to 144, and the magic total will be 145×12/2 = 870. Imagine these 144 cells being divided into 9 compartments, each having
16 adjacent numbers as in a 4×4 square. If we allot to the compartments numbers from 1 to 9 and form a 3×3 magic square, all we need to do is to replace these numbers 1 to 9 by 4×4 squares. This is what we have to do.
1 is to be replaced by 4×4 square with numbers from 1 to 16.
2 is to be replaced by 4×4 square with numbers from 16 to 32.
3 is to be replaced by 4×4 square with numbers from 33 to 48.
and so on, 9 being replaced by a 4×4 square with numbers from 129 to 144. It will be easily seen that the total of each, column and 2 mail diagonals is 870. And we can use any of the 880 squares and except one all can be rotated, so we have a large number of solutions – 880^9×4^8, and each will be different.
And we have the alternatve of dividing the 144 cells into 16 compartments and numbered to form a 4×4 square of 3×3 each as in 3×3 square and filling up square number 1 by numbers from 1 to 9, square with number 2 with numbers from 10 to 18, and so on, as was done earlier, and we have 880×4^15 solutions, again all different. The two sets of squares, in view of their method of construction,will be independent of each other, but with such large numbers, duplicates cannot be ruled out.
As stated earlier this method will be valid for any higher order square so long as p=mxn where both ‘m’ and ‘n’ are equal to or greater than 3. This would leave out ‘z’ order squares where ‘z’ is either a prime number or singly even number. If we consider numbers greater than 9 and lesss than 25, the squares not coveres are of order 10, 11, 13, 14, 17, 19, 22, 23. For odd numbers we can always use the Hindu Rule, while for singly even the Border Square method can be used, as already discussed.
MORE ABOUT 8X8 ORDER SQUARES:
Before we proceed further I may point out that there are exceptions. The 8×8 square can be constructed with help of 4×4 squares. This is possible because the total required is 260 and so if we construct 4 squares each having a total of 130, we can get our 8×8 square. if we name the four squares A,B,C & D and arrange them as shown:
A B
C D
we can replace A by a 4×4 square which we construct by adding 0 to all numbers less than 9 and 48 to all numbers greater than 8 to any of the
712 – 4×4 squares(excluding type B & E), B is again to be constructed from any of the 712 squares by adding 8 to all numbers less than 9 and 40 to numbers greater than 8. C is to be similarly constructed by adding 16 to all numbers less than 9 and 32 to all numbers greater than 8. For D we can use all the 880 squares and adding 24 to all numbers. A,B,C & D can be arranged in the 2×2 pattern as we like. This method yields 6×880x712×4x4×4 solutions.
A modification of this method would require addition of 0 to numbers less than 5, 12 to numbers less than 9, 36 to numbers less than 12 and 48 to numbers greater than 12. Next we add 4, 16, 32 and 44; next 8, 20, 28 & 40; and finally 12, 24, 24 & 36. So we again get a large number of new squares.
MORE ABOUT 12X12 SQUARES:
We can similarly construct 12×12 squares from 6×6 squares, provided we construct the 6×6 square by De La Hire’s method. The 4 squares required are formed by adding;
a) 0 to numbers from 1 to18, and 108 to numbers from 19 to 36.
b) 18 to numbers from 1 to 18 and 90 to numbers from 19 to 36.
c) 36 to numbers from 1 to 18 and 72 to numbers from 19 to 36. And finally
d) 54 to all numbers. (For this any 6×6 square can be used.)
This method is really general and can be extended to all doubly even order squares.
DOUBLY MAGIC SQUARES:
Now I come to doubly magic squares:
In these squares not only the basic square has the equisum property, but, if the numbers in each cell are replaced by their squares, the resultant square maintains the equisum property. Of course, in the new squares, the numbers will not be in sequence. I reproduce the 8×8 and 9×9 squares with these properties:
05 31 35 60 57 34 08 30
19 09 53 46 47 56 18 12
16 22 42 39 52 61 27 01
63 37 25 24 03 14 44 50
26 04 64 49 38 43 13 23
41 51 15 02 21 28 62 40
54 48 20 11 10 17 55 45
36 58 06 29 32 07 33 59
26 65 32 63 48 15 43 01 76
61 46 13 44 02 77 27 66 33
45 03 78 25 64 31 62 47 14
29 33 71 12 60 54 73 40 07
10 58 52 74 41 08 30 24 72
75 42 09 28 22 70 11 59 53
68 35 20 51 18 57 04 79 37
49 16 55 05 80 38 69 36 21
06 81 39 67 34 19 50 17 56
Here too from this one square a number of new squares can be generated. For example for the 8×8 square if we simultaneously interchange rows 1 & 2, and 7 & 8, and thereafter columns 1 & 2 and 7 & 8, we get the square:
09 19 53 46 47 56 12 18
31 05 35 60 57 34 30 08
22 16 42 39 52 61 01 27
37 63 25 24 03 14 50 44
04 26 64 49 38 43 23 13
51 41 15 02 21 28 40 62
58 36 06 29 32 07 59 33
48 54 20 11 10 17 45 55
It will be seen that combination of numbers in all rows, columns and diagonals is the same, so the resulting square will be doubly magic square. There are more such changes possible.
In the case of 9×9 square besides such changes there are additional changes possible. For example 23+58+42=40+24+59, and their squares are also equal being 529+3364+1764=5657=1600+576+3481.
Similarly 44+2+77=5+80+38, and their squares are also equal being 7869.
TREBLY MAGIC SQUARES:
It is possible to have magic squares where on replacement of numbers in each cell by the cube of the numbers the resultant square retains the equisum property, but, so far as it is known, we can have it only for the 128×128 square.
OTHER TYPES OF SQUARES;
I had stated that the Magic Squares must have, as one of the conditions, consecutive numbers. I now consider some other types of Squares which have the equisum property.
PRODUCT SQUARES:
We can, certainly, have squares where the sum of the products of the numbers in each row, column or diagonal is the same constant number. All we have to do is to write, instead of the numbers m,m+1,m+2,m+3………..: m,m^2,m^3…………
SQUARES WITH ONLY ODD OR EVEN NUMBERS:
We can have squares which have only odd or even numbers in all the cells. Foe odd numbers square, all we have to do, is to increase all even numbers by n^2-1, and for all even number square merely increase all odd numbers by n^2+1, or simply double all the numbers in the cells.
COMBINATION OF 4X4 AND 3X3 SQUARE:
Here is another illustration of what we can do with these squares. Study the square below below which has 4 numbers and 3 numbers in alternate rows.
30 — 44 — 43 — 33
– 49 — 54 — 47 –
41 — 35 — 36 — 38
– 48 — 50 — 52 –
37 — 39 — 40 — 34
– 53 — 46 — 51 –
42 — 32 — 31 — 45
The 4×4 square totals 150 and has numbers from 30 to 45.
The 3×3 square totals 150 too and has numbers from 46 to 54.
The main diagonals total 300 as expected.
UPSIDE DOWN SQUARE:
Another intersting square which totals the same read upside down and has
multiple solutions is shown below:
96 11 89 68
88 69 91 16
61 86 18 99
19 98 66 81
To get multiple solution all we have to do is for the alphabets A,B,C& D and a,b,c,&d take the numbers 1,6,8,9. (Please note here Aa is not to be taken as A+a).
STAR-SHAPED SQUARE:
Some elegant constructions on star shaped figures (pentagons, hexagons, etc.,) can also be developed. An instance of octagon is illustrated below:
– — – — – — 01 — – — – — –
– 02 — – 11 — – 16 — – 05 –
– — 12 — – — – — – 09 — – –
10 — – — – — – — – — – — 08
– — 06 — – — – — – 07 — – –
– 14 — – 03 — – 04 — – 13 –
– — – — – — 15 — – — – — –
There are magic circles, rectangles, crosses and diamonds which can be developed for fun!
CHESS BOARD MOVES:
Attempts have also been made to develop a 8×8 square based on the KNIGHT’S MOVES on the chess board. So far nobody has been successful. The square, as worked out, does not give the magic sum of 260 for the diagonals. See the square below:
47 10 23 64 49 02 59 06
22 63 48 09 60 05 50 03
11 46 61 24 01 52 07 58
62 21 12 45 08 57 04 51
19 36 25 40 13 44 53 30
26 39 20 33 56 29 14 43
35 18 37 28 41 16 31 54
38 27 34 17 32 55 42 15
SQUARES FOR ANY GIVEN NUMBER:
We can easily construct magic squares for a given calendar year or for any other number, but not always. For example for the year 1892 or 2002 we can construct a 11×11 square as both these numbers are divisible by 11. All we have to do is to start with 112 for getting 1892 and 122 for getting 2002.
We can also construct 4×4 squares for odd magic sum. To get 35 add 1 to 13,14,15,&16. To get 36 add 2 etc. Of course for this it is necessary that we select a square in which 13,14,15&16 are so distributed that they appear in all rows, all columns and the diagonals.
POLYHEDRONS:
Polyhedrons aredefined as solid objects with 6 or more surfaces. Hexahedron has 6 surfaces and is popularly called a cube. We can decide to have same order square on all sides or, if we so like, have a different order squares on each side. For example for a Octahedron (which has eight sides)
if we decide to have on all sides 4×4 squares with identical total, we will be required to use the numbers from 1 to 128 and the equisum will have to be 258. All we are required to do first is to select a 4×4 square which has 2 numbers less than 9 and two numbers greater than 8. Next we have to get the 8 squares by adding:
0 to all numbers from 1 to 8 and 112 to all numbers from 9 to 16.
8 to all numbers from 1 to 8 and 104 to all numbers from 9 to 16.
16 to all numbers from 1 to 8 and 96 to all numbers from 9 to 16.
24 to all numbers from 1 to 8 and 88 to all numbers from 9 to 16.
32 to all numbers from 1 to 8 and 80 to all numbers from 9 to 16.
40 to all numbers from 1 to 8 and 72 to all numbers from 9 to 16.
48 to all numbers from 1 to 8 and 64 to all numbers from 9 to 16.
56 to all numbers from 1 to 8 and 56 to all numbers from 9 to 16.
If we decide to have squares of, say, order 8 on all sides, we require numbers from 1 to 512 with equisum of 2052, follow aq similar procedure increasing numbers from 1 to 32 bu 0, 32, 64 …… and increasing numbers from 33 to 64 by 448,416, 384….
If, however, we decide on a 4×4 order square on one side, 5×5 order square on second side, 6×6 order square on the third side….. there is no problem except that totals on each side will depend on the numbers selecte for each square. If we want a uniform total for all sides that will be possible but we may have to deal with very large numbers.
HYPER – MAGIC SQUARES:
I have stated earlier that we can impose varying conditions for these squares. I will now discuss some of these, they will be found only in some of the squares.
DIABOLIC SQUARES:
A Magic Sqaure is called Diabolic, Perfect or Pandiagonal. It is diabolic if it has the same constant dum for its broken diagonals as it has for its rows, columns and diagonals. Examine the square:
01 08 11 14
12 13 02 07
06 03 16 09
15 10 05 04
The broken diagonals are; 1 & 10, 16, 09; 4 & 11, 13, 06; 12, 08 & 05, 09;
06, 10 & 11, 07; 15 & 08, 02, 09; 14 & 12, 03, 05.
The square from ancient India and the Jaina square also have this property. I also reproduce below a 5×5 square with tthis property.
01 08 15 17 24
20 22 04 06 13
09 11 18 25 02
23 05 07 14 16
12 19 21 03 10
Obviously there can’t be a diabolic square of order 3. Also there are no diabolic singly even squares like 6, 10, etc. An 8×8 square constructed from diabolic square of order 4 will also be diabolic.
One property of diabolic squares, worth mentioning, is that if a pandiagonal square be cut into 2 pieces along a line between any two rows or columns, the pieces be interchanged, the new square, so formed, will also be a magic square and retain the property of being diabolic. This gives us another method for generating squares from an existing square. Reproduced below is a 5×5 square from the one shown above:
08 15 17 24 01
22 04 06 13 20
11 18 25 02 09
05 07 14 16 23
19 21 03 10 12
ASSOCIATIVE SQUARES:
An Associative or Symmetrical square is one that has pair of numbers symetrically opposite the center and add upto n^2+1. A 4×4 square may be associative or diabolic, but not both, but an odd square order can be both (in thar case the center number has necessarily to be in the center). Both a 4×4 square and 5×5 square with this property are shown below:
05 04 16 09 – - – - 01 15 24 08 17
10 15 03 06 – - – - 23 07 16 05 14
11 14 02 07 – - – - 20 04 13 22 06
08 01 13 12 – - – - 12 21 10 19 03
- – - – - – - – - – — 09 18 02 11 25
REGULAR SQUARES:
Just as we have regular squares od order 4, we can also have regular squares of order 8, 16, 32, 64, etc. The procedure is the same as for 4×4 square.
MY GENERAL METHOD FOR CONSTRUCTION OF ‘N+4′ ORDER SQUARE FROM
‘N’ ORDER SQUARE:
Various methods of construction of magic squares have been propounded, some giving a limited number of squares while some yield a large number of squares but there is no general method of construction which gives all possible solutions or is uniformly applicable to all types of squares. An attempt has been made here to work out a solution which gives at one go a large number of solutions and is easy to follow and use. Before we start I would like to make a mention of a general mode of construction of magic squares first enunciated by De La Hire for odd-numbered squares and perfected by Professor Scheffler. To acquaint ourselves with this general method I have made a slight modification, in that, instead of using in the auxiliary squares numbers o,n,2n, 3n…(n-1)n and 1,2,3,…n, I shall be using A,B,C,…N, and a,b,c,…n. For an nxn order square, if we arrange alphabets A,B, C,…N and a,b,c,…n in the square in such a way that each of A to N and a to n occurs only once in each row, each column and the diagonals with an additional condition that each A, B, C,.. has an association with each of a to n, we can at one stroke construct a large number of squares of that order. All we need to do is to give values 0, n, 2n,.( n-1)n to A’s and values 1,2,3,…n to a’s, with the stipulation that Aa is to be read as A+a, Bb is to be read as B+b etc., we can thus cover all the numbers from 1 to n2 and the resultant total of the rows, columns and diagonals is the magic sum n(n2+1)/2. This method when applicable will give n!xn! /8 different squares ( excluding rotations and reflections). A 4×4 order square will have as one of the combinations:
Aa Bb Cc Dd
Cd Dc Ab Ba
Db Ca Bd Ac
Bc Ad Da Cb
Link 1: If we have A=4, B=0, C=12 and D=8 ; and a=1, b=4, c=2, d=3 we get the 4×4 square:
05 04 14 11
15 10 08 01
12 13 03 06
02 07 09 16
A 5×5 order square will have as one of the combinations:
Be Ca Db Ec Ad
Dc Ed Ae Ba Cb
Aa Bb Cc Dd Ee
Cd De Ea Ab Bc
Eb Ac Bd Ce Da
Link 2: This square yields 5!x5!/4 all different squares.
If we use the Hindu rule we get the square:
Bd Ac Eb Da Ce
Ab Ea De Cd Bc
Ee Dd Cc Bb Aa
Dc Cb Ba Ae Ed
Ca Be Ad Ec Db
Link 3: However in this square C has to be given the value 10 only. A, B, D, E have the choice of 0, 5,15 and 20, of course a’s can be given any of the 5 values, 1,2,3,4,5. We thus have 5!x4!/4 squares only but these will all be different from the one produced by the earlier combination.
A 6×6 order square cannot be constructed strictly on these lines but with slight modification we get
Aa Fe Fd Ac Fb Af
Ef Bb Ed Ec Be Ba
Df De Cc Cd Cb Da
Ca Ce Dc Dd Db Cf
Bf Eb Bc Bd Ee Ea
Fa Ab Ad Fc Ae Ff
Link 4: Here we will have to ensure that A+F = B+E = C+D = 30 and similarly a+f = b+e = c+d = 7. This will greatly reduce the number of squares that can be generated. However if we use the border square method mentioned above we get a larger number of squares. All we need to do is to take any of the 880 4×4 order square and add (36-16)/2 = 10 to all the numbers, thus using numbers from 11 to 26 and getting a total of 74 as the magic sum. Now for border we have numbers from 1 to 10 and 17 to 26, and we have to ensure that the two rows and two columns total 111 and the opposite corner numbers total 37 with the additional condition that opposite pair of numbers in rows and columns total 37 too. One such combination of border numbers is again given below;
01 34 05 07 35 29
33 – - – - – - – -04
31 – - – - – - – -06
28 – - – - – - – -09
10 – - – - – - – -27
08 03 32 30 02 36
Link 5: Since all the 880 squares can be rotated and the 4 pairs 34 and 3, 5 and 32, 7 and 30, 35 and 2, can switch positions and similarly the pairs 33 and 4, 31 and 6, 28 and 9, 10 and 27, can switch positions we have in all 880×4x4!x4! independent solutions.
I now come to our method of construction of n+4 order square from an n order square, the procedures are slightly different for odd order and even order squares. I start with 3×3 order square and go to 7×7 and 11×11 order square. Also for identification of squares I will number the rows R1, R2, R3, .. and columns C1, C2, C3,.. starting from bottom left hand corner. The 7×7 order square has numbers from 1 to 49 and 3×3 order square has numbers from 1 to 9. We construct the 3×3 square by adding (72-32)/2=20 to all the numbers and then place all the corner numbers in the corners and middle numbers in the middle spaces as shown;
24 – - – - – -29 – - – - – - 22
23 – - – - – -25 – - – - – - 27
28 – - – - – -21 – - – - – - 26
In the next step we take any 5×5 order square which has number 13 in the central square and add (72-52)/2=12 to all the numbers, this will make the square have numbers from 13 to 37, and this will take the magic sum to 125 from 65. We now make an additional correction by reducing all numbers less than 25 by 4 and adding 4 to all the numbers greater than 25, (this is done to avoid repetition of numbers). Since we have not touched the central square the diagonal totals, middle row and column totals, will remain unaffected, but the total of two of the rows and two columns will get altered to 129 and the total of the other two rows and columns will now have a total of 121. We now place this square in the center omitting the central square, this makes our square to appear like this;
24 ——– 29 ——- 22
—– 17 11 38 32 31
—– 10 37 36 30 16
23 — 41 35 25 15 09 27
—– 34 20 14 13 40
—– 19 18 12 39 33
28 ——— 21 —— 26
We have not used the numbers 1 to 8 and 41 to 49. We arrange these numbers such that combination of 4 numbers gives us 100 and place them such that 4 pairs total 54 and 4 pairs total 46. One such combination is: 49, 6; 1, 44 ; 5,48; 45, 2 ; 43, 4; 46, 7 ; 3, 42; 8, 47.
The pairs totaling 54 and 46 are interchangeable and the 2 rows or two columns are also interchangeable. Besides other combinations are possible and with choice of 4,769,936 squares of order 5 which have 13 in the central square we have just billions of squares. However if we go only by the limited number of order 5 squares that can be generated by the 2 sets of order 5 squares in Link 3 and Link 4 we still have the choice of 5!x5! +5!x4! = 17280 squares.
We now place the first two combinations in the top row and bottom row and the other combination in the left hand column and the right hand column. This completes our square which now takes final shape;
24 49 06 29 01 44 22
43 17 11 38 32 31 03
04 10 37 36 30 16 42
23 41 35 25 15 09 27
46 34 20 14 13 40 08
07 19 18 12 39 33 47
28 05 48 21 45 02 26
Now we will go from this 7 order square to 11 order square we first construct a 7×7 order square which has 61 in the central square. For this we have to use numbers from 37 to 85 and the magic sum will be 36×7 +175 = 427. This square we split up in 4 parts, as we did with order 3 square earlier, and transfer to the 4 corners, retaining the central number in the central square and middle row and middle column numbers in the middle row and middle column but in the top and bottom rows and left and right columns. As a next step we again construct a 5×5 order square with 61 in the central square using numbers from 49 to 73 and magic sum 48×5 + 65 = 305. In this square we reduce all numbers less than 61 by 24 and in numbers greater than 61 we add 24, to avoid repetition of numbers, so we are now using numbers from 25 to 36, 61, and 86 to 97. This square (except the central number) we place in the center. We now have the correct magic square total of 671 for the two diagonals and the middle row and middle columns and our square looks likes this;
60 85 42 —— 65 ——- 37 80 58
79 53 47 —— 74 ——- 68 67 39
40 46 73 —— 72 ——- 66 52 78
———- 33 27 94 88 87
———- 26 93 92 86 32
59 77 71 97 91 61 31 25 51 45 63
———- 90 36 30 29 96
———- 35 34 28 95 89
82 70 56 —— 50 —— 49 76 44
43 55 54 —— 48 —— 75 69 83
64 41 84 —— 57 —— 81 38 62
Two rows and two columns of the 5×5 order square will have a total of 305 + 24 = 329 and two rows and column will have a total of 305- 24 = 281. We now have to arrange numbers 1 to 24 and 98 to 121 in such a way that all rows and columns total 671. One arrangement that gives these totals is;
——————- TOTAL ——————- TOTAL
117 116 005 006 = 244; 113 112 009 010 = 244
017 018 105 104 = 244; 021 022 101 100 = 244
115 114 007 008 = 244; 111 110 011 012 = 244
019 020 103 102 = 244; 023 024 099 098 = 244
121 118 002 003 = 244; 106 109 014 015 = 244
001 004 120 119 = 244; 016 013 108 107 = 244
390 390 342 342 TOTAL- 390 390 342 342
After inserting these our final square is;
060 085 042 117 116 065 005 006 037 080 058
079 053 047 017 018 074 105 104 068 067 039
040 046 073 115 114 072 007 008 066 052 078
009 101 011 033 027 094 088 087 099 014 108
010 100 012 026 093 092 086 032 098 015 107
059 077 071 097 091 061 031 025 051 045 063
113 021 111 090 036 030 029 096 023 106 016
112 022 110 035 034 028 095 089 024 109 013
082 070 056 019 020 050 103 102 049 076 044
043 055 054 121 118 048 002 003 075 069 083
064 041 084 001 004 057 120 119 081 038 062
Now let us just examine the possible combinations of these 32 numbers just transposed in the main 11×11 order square. The six rows of numbers totaling 244 each inserted in rows 1 to 3 and 9 to 11 are interchangeable giving 6! combinations. Further two columns totaling 390 are interchangeable as also the two columns totaling 322. The same holds good for the 6 columns each totaling 244 inserted in columns 1 to 3 and 9 to 11. So in all we have 6!x6!x4×4 combinations. We are also free to use the 2nd set of 24 in place of the 1st set and vice-versa. So the final number of combinations is 4,147,200, for this set and the central 5×5 square. When we consider the possible combinations of the 7×7 order square and the 5×5 order squares the number is stupendous. Besides the combinations of 32 numbers themselves are capable of giving a number of solutions.
The main advantage of this method is that since we have taken care of diagonal totals we are not concerned with diagonal totals while arranging other numbers and this advantage will be more apparent when we examine even number order squares.
We now examine construction of 9×9 order square starting with 5×5 order square. The procedure is exactly identical. We first construct the 5 order square with central number as the central number of the 9×9 order square i.e. 41. We as such use numbers from 29 to 53 and total of 28×5 +65 = 205. We again construct an 5×5 order square, with numbers from 29 to 53. We next decrease all numbers less than 41 by 12 and add 12 to all numbers greater than 41, giving a total of 205 for the central row, central column and the two diagonals and total of 193 for two columns and two rows and 217 for the other two rows and columns. We also have 32 numbers from 1 to 16 and 66 to 81 to be arranged to give total of 164 one way for all 8 rows and 4sets to give 152 and 4 sets to give 176. One such set is;
———— TOTAL —————- TOTAL
66 16 67 15 = 164; 68 14 69 13 =164
08 74 07 75 = 164; 09 73 10 72 =164
77 05 76 06 = 164; 71 11 70 12 =164
01 81 02 80 = 164; 04 78 03 79 =164
152 176 152 176 TOTAL 152 176 152 176
We are now ready to prepare the final 9×9 square.
37 31 16 75 50 66 07 44 43
30 49 74 15 48 08 67 42 36
68 09 25 19 62 56 55 71 04
10 69 18 61 60 54 24 03 70
53 47 65 59 41 23 17 35 29
14 73 58 28 22 21 64 11 78
72 13 27 26 20 63 57 79 12
46 40 05 80 34 77 02 33 52
39 38 81 06 32 01 76 51 45
Without going into the number of squares that can be generated we will now go ifor construction of 8×8 order square from a 4×4 order square.
First of all we will generate 4 sets of squares each having a total of 130, this is simply done by adding 0 to all numbers from 1 to 8 and 48 to all numbers from 9 to 16, for the second by adding 8 to all numbers from 1 to 8 and 40 to all numbers from 9 to 16, for the third by adding 16 to all numbers from 1 to 8 and 32 to all numbers from 9 to 16 and for the fourth by adding 24 to all numbers. If we call these squares A,B,C,D, one way would be to place them as
A B
C D
This arrangement generates a very large number of squares as we can use all combinations of A,B,C,D and there are 712 squares each under A,B,C, and 880 under D ( there are 168 squares which have three numbers less than 9 in one diagonal and 3 numbers greater than 8 in the other diagonal and these squares cannot be used for obvious reasons.). However let us see how we can follow a method similar to that for odd order squares and the advantages of our method.
After we have constructed the four squares let us split any one of the squares into four parts and place it in the corners. Next place one of the squares in the center, this ensures that diagonal totals are correct and 260. We are now free to place the other squares in the four vacant rows and four vacant columns. The advantage comes in here, we can rotate the rows and columns as we are not concerned with diagonal totals. I will illustrate; let us take the square:
01 06 11 16
12 15 02 05
14 09 08 03
07 04 13 10
We are free to place this as
15 02 05 12
06 11 16 01
09 08 03 14
04 13 10 07
We have used the same numbers in each row arranged differently, the rows and column totals are still 34 but not the diagonal totals which however do not affect the 8×8 order square totals. We therefore have the choice of 528×6 + 284×72 = 3168 +20448 =23616 squares instead of the normal 712 squares. ( if we had the option of full 880 squares, we would have had the choice of 528×6 + 352×72 =3168 +25344 = 28512 squares, this being the total number of 4×4 order squares which have the same total for all rows and columns. The advantages of this method of construction thus become obvious.) One setting is given below;
01 06 09 14 51 56 59 64
60 63 52 55 10 13 02 05
39 26 17 22 43 48 29 36
30 35 44 47 18 21 40 25
33 32 46 41 24 19 27 38
28 37 23 20 45 42 34 31
62 57 54 49 16 11 08 03
07 04 15 12 53 50 61 58
And lastly we come to the construction of a 10×10 order square having numbers from 1 to 100 and magic sum of (100+1) x 5 = 505, from a 6×6 order square having numbers from 1 to 36 and magic sum of 65. As a first step we select anyone of 6×6 order square and increase all numbers by (100 – 36)/2 = 32. This square will now have numbers from 33 to 68 and magic sum of 32×6+65 = 303. Next we split this square in 4 parts and place them in corners. Next we form a 4×4 order square with magic sum of 202. This can be done by using numbers from 1 to 8 and 93 to 100; or 9to16 and 85 to 92; or 17 to 24 and 77 to 84 ; or 25 to 32 and 69 to 76. After selecting one of this square we place it in the center to get this position of the square. Let us select the square with numbers from 1 to 8 and 93 to 100 for the purpose of illustration.
033 062 056 ——————- 045 044 063
067 040 055 ——————- 049 058 034
066 060 047 ——————- 053 041 036
————– 098 001 099 004
————– 095 008 094 005
————– 002 097 003 100
————– 007 096 006 093
035 059 048 ——————- 054 042 065
064 043 046 ——————- 052 061 037
038 039 051 ——————- 050 057 068
We now have 24 numbers to be filled in 6 rows in columns C4 to C7 each row to have a total of 202 and column total of 303 and 24 numbers to be filled in 6 columns in rows R4 to R8 each column to have a total of 202 and row total of 303.
We prepare 12 pairs of 2 rows such that rows total 202 and columns total 101.
82 17 83 20;
19 84 18 81;
90 10 87 15;
11 91 14 86;
78 21 79 24;
23 80 22 77;
92 13 12 85;
09 88 89 16;
74 25 75 28;
27 76 26 73;
70 29 71 32;
31 72 30 69;
We are free to use any six pairs out of 12 pairs to the 6 rows in columns C4 to C7 as we have already taken care of the diagonal totals for the 10×10 order square. The remaining 6 to be used in 6 columns in rows R4 to R8. We now give the final square.
33 62 56 70 29 71 32 45 44 63
67 40 55 31 72 30 69 49 58 34
66 60 47 82 17 83 20 53 41 36
74 27 90 98 01 99 04 11 92 09
25 76 10 95 08 94 05 91 13 88
75 26 87 02 97 03 100 14 12 89
28 73 15 07 96 06 93 86 85 16
35 59 48 19 84 18 81 54 42 65
64 43 46 78 21 79 24 52 61 37
38 39 51 23 80 22 77 50 57 68
So in this square first we have the choice of choosing the 6×6 order square, then one of the 712 squares for the central place, than the pairs that can be formed have choices, thereafter we have the choice of selecting 6pairs out of the available 12. And once the selection have been made we have the choice of placing the 6 lines in rows according to our choice and for these further choice of changing the column numbers as well. Similarly we have the choice of placing the six lines in columns according to our choice and then changing the rows. Just work out the independent solutions and don’t be surprised if they run into trillions.
To sum up the procedure requires:
Selection of a square of order less by 4 and making adjustments to it, so that it consists of central numbers.
To split the square in 4 parts and transfer to corners.
To select 5×5 order square for odd number square or a 4×4 order square for even order square.
Place this square in the center space after making adjustments for starting number such that in all cases the diagonal totals give the magic sum, noting that for 5×5 order square we use only 24 numbers and not 25 ( central number is common for both the odd number squares ). Fill up the blank rows with appropriate numbers without bothering about the diagonal totals to get the full square.
MAGIC CUBES:
While Magic Squares are 2 dimensional, Magic Cubes are three dimensional. Instead of only one face they have to begin with have n faces for a ‘n’ order cube. In addition some faces get formed when we look into the various conditions that follow.
Condition for a perfect Magic Cube:
The n faces having nxn numbers must cover all numbers from 1 to nxnxn, (for convenience I will assume that we are going to start from 1) once and once only. The total of each row, each column and the 2 main diagonals of each face must have the same total of n(n3+1)/2.
Also the faces formed as under must have the same property as the main face mentioned above:
a) nth row of nth face.
b) nth column of the nth face.
c) nth row of each face.
d) nth column of each face.
e) nth row of the nth face and nth column of the nth face in reverse order, i.e. first row of last face, 2nd row of last but one face etc..
Multiple Magic Cubes can be generated by very simple instructions summarised in one line and two numbers, particularly for odd order cubes.
i) Write a combination of three alphabets, A,’a’ and ‘n’ , starting with Aan and proceed alphabetically, when the last alphabet is reached, depending on the size of the cube, go back to the first alphabet.
ii) Instructions for start of next row and start of next face.
I will illustrate this by constructing 7×7x7 Magic Cube.
i) Write Aan Bbo Ccp Ddq Eer Ffs Ggt.
ii) For next row start with 536 of earlier row. (536 refers to place of A’s,a’s and n’s.)
iii) Next face start with 635 of 1st row of earlier face.
Construction of 7×7x7 order Magic Cube.
————-1st Face ——————————2nd Face
Aan Bbo Ccp Ddq Eer Ffs Ggt ——— Fcr Gds Aet Bfn Cgo Dap Ebq
Ecs Fdt Gen Afo Bgp Caq Dbr ——— Cep Dfq Egr Fas Gbt Acn Bdo
Beq Cfr Dgs Eat Fbn Gco Adp ——— Ggn Aao Bbp Ccq Ddr Ees Fft
Fgo Gap Abq Bcr Cds Det Efn ——— Dbs Ect Fdn Geo Afp Bgq Car
Cbt Dcn Edo Fep Gfq Agr Bas ——— Adq Ber Cfs Dgt Ean Fbo Gcp
Gdr Aes Bft Cgn Dao Ebp Fcq ——— Efo Fgp Gaq Abr Bcs Cdt Den
Dfp Egq Far Gbs Act Bdn Ceo ——— Bat Cbn Dco Edp Feq Gfr Ags
————3rd Face ———————————4th Face
Deo Efp Fgq Gar Abs Bct Cdn ——— Bgs Cat Dbn Eco Fdp Geq Afr
Agt Ban Cbo Dcp Edq Fer Gfs ——— Fbq Gcr Ads Bet Cfn Dgo Eap
Ebr Fcs Gdt Aen Bfo Cgp Daq ——— Cdo Dep Efq Fgr Gas Abt Bcn
Bdp Ceq Dfr Egs Fat Gbn Aco ——— Gft Agn Bao Cbp Dcq Edr Fes
Ffn Ggo Aap Bbq Ccr Dds Eet ——— Dar Ebs Fct Gdn Aeo Bfp Cgq
Cas Dbt Ecn Fdo Gep Afq Bgr ——— Acp Bdq Cer Dfs Egt Fan Gbo
Gcq Adr Bes Cft Dgn Eao Fbp ——— Een Ffo Ggp Aaq Bbr Ccs Ddt
————-5th Face ——————————-6th Face
Gbp Acq Bdr Ces Dft Egn Fao ——— Edt Fen Gfo Agp Baq Cbr Dcs
Ddn Eeo Ffp Ggq Aar Bbs Cct ——— Bfr Cgs Dat Ebn Fco Gdp Aeq
Afs Bgt Can Dbo Ecp Fdq Ger ——— Fap Gbq Acr Bds Cet Dfn Ego
Eaq Fbr Gcs Adt Ben Cfo Dgp ——— Ccn Ddo Eep Ffq Ggr Aas Bbt
Bco Cdp Deq Efr Fgs Gat Abn ——— Ges Aft Bgn Cao Dbp Ecq Fdr
Fet Gfn Ago Bap Cbq Dcr Eds ——— Dgq Ear Fbs Gct Adn Beo Cfp
Cgr Das Ebt Fcn Gdo Aep Bfq ——— Abo Bcp Cdq Der Efs Fgt Gan
————7th Face ———-
Cfq Dgr Eas Fbt Gcn Ado Bep
Gao Abp Bcq Cdr Des Eft Fgn
Dct Edn Feo Gfp Agq Bar Cbs
Aer Bfs Cgt Dan Ebo Fcp Gdq
Egp Faq Gbr Acs Bdt Cen Dfo
Bbn Cco Ddp Eeq Ffr Ggs Aat
Fds Get Afn Bgo Cap Dbq Ecr
Now let us see first row of each Face:
Aan Bbo Ccp Ddq Eer Ffs Ggt
Fcr Gds Aet Bfn Cgo Dap Ebq
Deo Efp Fgq Gar Abs Bct Cdn
Bgs Cat Dbn Eco Fdp Geq Afr
Gbp Acq Bdr Ces Dft Egn Fao
Edt Fen Gfo Agp Baq Cbr Dcs
Cfq Dgr Eas Fbt Gcn Ado Bep
And now first column of each Face:
Aan Fcr Deo Bgs Gbp Edt Cfq
Ecs Cep Agt Fbq Ddn Bfr Gao
Beq Ggn Ebr Cdo Afs Fap Dct
Fgo Dbs Bdp Gft Eaq Ccn Aer
Cbt Adq Ffn Dar Bco Ges Egp
Gdr Efo Cas Acp Fet Dgq Bbn
Dfp Bat Gcq Een Cgr Abo Fds
Now nth row of nth Face:
Aan Bbo Ccp Ddq Eer Ffs Ggt
Cep Dfq Egr Fas Gbt Acn Bdo
Ebr Fcs Gdt Aen Bfo Cgp Daq
Gft Agn Bao Cbp Dcq Edr Fes
Bco Cdp Deq Efr Fgs Gat Abn
Dgq Ear Fbs Gct Adn Beo Cfp
Fds Get Afn Bgo Cap Dbq Ecr
And now the nth column of nth Face:
Aan Gds Fgq Eco Dft Cbr Bep
Ecs Dfq Cbo Bet Aar Gdp Fgn
Baq Aao Gdt Fgr Ecp Dfn Cbs
Fgo Ect Dfr Cbp Ben Aas Gdq
Cbt Ber Aap Gdn Fgs Ecq Dfo
Gdr Fgp Ecn Dfs Cbq Beo Aat
Dfp Cbn Bes Aaq Gdo Fgt Ecr
And now 1st row of 7th Face, 2nd row of 6th Face…….
Cfq Dgr Eas Fbt Gcn Ado Bep
Bfr Cgs Dat Ebn Fco Gdp Aeq
Afs Bgt Can Dbo Ecp Fdq Ger
Gft Agn Bao Cbp Dcq Edr Fes
Ffn Ggo Aap Bbq Ccr Dds Eet
Efo Fgp Gaq Abr Bcs Cdt Den
Dfp Egq Far Gbs Act Bdn Ceo
And finally 1st column of 7th Face, 2nd column of 6th Face……..
Cfq Fen Bdr Eco Abs Dap Ggt
Gao Cgs Ffp Bet Edq Acn Dbr
Dct Gbq Can Fgr Bfo Ees Adp
Aer Ddo Gcs Cbp Fat Bgq Efn
Egp Aft Deq Gdn Ccr Fbo Bas
Bbn Ear Ago Dfs Gep Cdt Fcq
Fds Bcp Ebt Aaq Dgn Gfr Ceo
Here also Aan means A+a+n. The A’s can be given values from 1,2,3,5,6,7
a’s values from 0,7,14,28,35,42 and n,o,p etc., 0,49,98,196,245,294, but there is one condition – C,f and p have to be given central value only. C has to be given value 4, f – 21 and p – 147. If we decide to give A’s 0,7 etc., suitable changes will have to be made in values of C,f and p.
From the arrangement it will be noticed that all numbers from 1 to 343 will be covered as required.
It may be noted that there are other arrangements possible. For example, if without altering the original arrangement we decide to have next Row from
533 and new Face from 354, we will have to give central values to A,a & r.
8×8x8 Magic Cube:
Even order cubes are a bit difficult to put in simple words. I will first give the order in which the face is to be constructed by relating numerals to the order which A’s, a’s & n’s will follow. Next i will give the complete first Face and then the first lines(rows) of all Faces, I will not be constructing all the faces.
111 222 333 444 555 666 777 888
534 643 712 821 178 287 356 465
657 568 875 786 213 124 431 342
276 185 458 367 632 541 814 723
862 751 684 573 426 315 248 137
483 374 261 152 847 738 625 516
328 417 146 235 764 853 582 671
745 836 527 618 381 472 163 254
As a check please note the first and last number add upto 999, same for the 2nd and 7th etc.
First Face:
Aan Bbo Ccp Ddq Eer Ffs Ggt Hhu
Ecq Fdp Gao Hbn Agu Bht Ces Dfr
Fet Efu Hgr Ghs Bap Abq Dcn Cdo
Bgs Ahr Deu Cft Fco Edn Haq Gbp
Hfo Gen Fhq Egp Dbs Car Bdu Act
Dhp Cgq Bfn Aeo Hdt Gcu Fbr Eas
Cbu Dat Ads Bcr Gfq Hep Eho Fgn
Gdr Hcs Ebt Fau Chn Dgo Afp Beq
First Rows
Aan Bbo Ccp Ddq Eer Ffs Ggt Hhu
Gfu Het Ehs Fgr Cbq Dap Ado Bcn
Bhq Agp Dfo Cen Fdu Ect Hbs Gar
Hcr Gds Fat Ebu Dgn Cho Bep Afq
Cdt Dcu Abr Bas Ghp Hgq Efn Feo
Ego Fhn Geq Hfp Acs Bdr Cau Dbt
Des Cfr Bgu Aht Hao Gbn Fcq Edp
Fbp Eaq Hdn Gco Bft Aeu Dhr Cgs.
A’s will have values from 1 to 8, a’s from 0,8,16,24,32,40, 48 & 56 and n’s from 0,64,128,192,256,320,384 & 448. This combination alone will give
8!X8!x8!/4 solutions.
9×9x9 Magic Cube:
One combination is 533 for the next Row and 356 for the next Face. For this combination we are required to have A+D+G=B+E+H=C+F+I=15, IF A’s are
given values fro 1 to 9; a+d+g=b+e+h=c+f+i=108; and n+q+t=o+r+u=
p+s+v=972. This combination will itself yield 9!x9!x9!/4 solutions, and there are other combinations too.
10×10x10 Magic Cube:
As far as my knowledge goes nobody has been, so far, successful in getting this Cube.
11×11x11 Magic Cube:
For best results for next Row take 5-5-3 and next Face take 6-10-8.
Values to be taken are:
1 to 11; 0,11,22,33,44,55,66,77,88,99,110;
0,121,242,363,484,605,726,847,968,1089, 1210.
The only restriction would be to give central value to G – 6,55, or 605 as the case maybe.
FRANKLIN SQUARE
Franklin square differ from regular magic squares. Firstly the diagonals do not add upto the magic sum, instead it is the bent diagonals that give the required total. Next all 2×2 squares give the same total of 2(n^2+1). Given below are 2 examples of Franklin Squares:
60 13 24 33 28 45 56 01 …….. 52 61 04 13 20 29 36 45
07 50 43 30 39 18 11 62 …….. 14 03 62 51 46 35 30 19
57 16 21 36 25 48 53 04 …….. 53 60 05 12 21 28 37 44
06 51 42 31 38 19 10 63 …….. 11 06 59 54 43 38 27 22
59 14 23 34 27 46 55 02 …….. 55 58 07 10 23 26 39 42
08 49 44 29 40 17 12 61 …….. 09 08 57 56 41 40 25 24
58 15 22 35 26 47 54 03 …….. 50 63 02 15 18 31 34 47
05 52 41 32 37 20 09 64 …….. 16 01 64 49 48 33 32 17
Here again it is possible to generate more squares from these squares with some precautions as diagonal totals should not be equal. Here are some simple procedures:
Add 16 to numbers from 1 to 16 and numbers from 33 to 48 and subtract 16 from numbers 17 to 32 and 49 to 64. Similarly add 32 to numbers from 1 to 32 and subtract 32 from numbers 33 to 64. Similar combinations will work if we add 8 to numbers from 1 to 8, 17 to 24, 33 to 40, 49 to 56, and subtract 8 from numbers from 9 to 16, 25 to 32, 41 to 48 and 57 to 64. Also turn all odd numbers to even by increasing them by 1 and all even numbers to odd by subtracting 1. Try some more by adding qand subtracting 4 and 2 suitably selecting the number combinations.
Another possibility is to transfer entire row or column. For example in the right hand side square 2nd and 3rd column can be replaced by 6th and 7th column and vice-versa. All the 4 rows, of course, normally cqan be moved and so can columns, but care has to be exercised to see that diagonals maintain their property.
16×16 Franklin Square:
252 013 056 193 188 077 120 129 124 141 184 065 060 205 248 001
007 242 203 062 071 178 139 126 135 114 075 190 199 050 011 254
249 016 053 196 185 080 117 132 121 144 181 068 057 208 245 004
006 243 202 063 070 179 138 127 134 115 074 191 198 051 010 255
220 045 024 225 156 109 088 161 092 173 152 097 028 237 216 023
039 210 235 030 103 146 171 094 167 082 107 158 231 018 043 222
217 048 021 228 153 112 085 164 089 176 149 100 025 240 213 036
038 211 234 031 102 147 170 095 166 083 106 159 230 019 042 223
219 046 023 226 155 110 087 162 091 174 151 098 027 238 215 034
040 209 236 029 104 145 172 093 168 081 108 157 232 017 044 221
218 047 022 227 154 111 086 163 090 175 150 099 026 239 214 035
037 212 233 032 101 148 169 096 165 084 105 160 229 020 041 224
251 014 055 194 187 078 119 130 123 142 183 066 059 206 247 002
008 241 204 061 072 177 140 125 136 113 076 189 200 049 012 253
250 015 054 195 186 079 118 131 122 143 182 067 058 207 246 003
005 244 201 064 069 180 137 128 133 116 073 192 197 052 009 256
And now the 24×24 square, due to paucity of space I am giving first 12 columns and then next 12 columns.
First 12 columns:
572 013 088 481 380 205 184 385 476 109 280 289
007 562 491 094 199 370 395 190 103 466 299 286
569 016 085 484 377 208 181 388 473 112 277 292
006 563 490 095 198 371 394 191 102 467 298 287
540 045 056 513 348 237 152 417 444 141 248 321
039 530 523 062 231 338 427 158 135 434 331 254
537 048 053 516 345 240 149 420 441 144 245 324
038 531 522 063 230 339 426 159 134 435 330 255
508 077 024 545 316 269 120 449 412 173 216 353
071 498 555 030 263 306 459 126 167 402 363 222
505 080 021 548 313 272 117 452 409 176 213 356
070 499 554 031 262 307 458 127 166 403 362 223
507 078 023 546 315 270 119 450 411 174 215 354
072 497 556 029 264 305 460 125 168 401 364 221
506 079 022 547 314 271 118 451 410 175 214 355
069 500 553 032 261 308 457 128 165 404 361 224
539 046 055 514 347 238 151 418 443 142 247 322
040 529 524 061 232 337 428 157 136 433 332 253
538 047 054 515 346 239 150 419 442 143 246 323
037 532 521 064 229 340 425 160 133 436 329 256
571 014 087 482 379 206 183 386 475 110 279 290
008 561 492 093 200 369 396 189 104 465 300 285
570 015 086 483 378 207 182 387 474 111 278 291
005 564 489 096 197 372 393 192 101 468 297 288
Now the 13th to 24th columns:
284 301 472 097 188 397 376 193 092 493 568 001
295 274 107 478 391 178 203 382 487 082 011 574
281 304 469 100 185 400 373 196 089 496 565 004
294 275 106 479 390 179 202 383 486 083 010 575
252 333 440 129 156 429 344 225 060 525 536 033
327 242 139 446 423 146 235 350 519 050 043 542
249 336 437 132 153 432 341 228 057 528 533 036
326 243 138 447 422 147 234 351 518 051 042 543
220 365 408 161 124 461 312 257 028 557 504 065
359 210 171 414 455 114 267 318 551 018 075 510
217 368 405 164 121 464 309 260 025 560 501 068
358 211 170 415 454 115 266 319 550 019 074 511
219 366 407 162 123 462 311 258 027 558 503 066
360 209 172 413 456 113 268 317 552 017 076 509
218 367 406 163 122 463 310 259 026 559 502 067
357 212 169 416 453 116 265 320 549 020 073 512
251 334 439 130 155 430 343 226 059 526 535 034
328 241 140 445 424 145 236 349 520 049 044 541
250 335 438 131 154 431 342 227 058 527 534 035
325 244 137 448 421 148 233 352 517 052 041 544
283 302 471 098 187 398 375 194 091 494 567 002
296 273 108 477 392 177 204 381 488 081 012 573
282 303 470 099 186 399 374 195 090 495 566 003
293 276 105 480 389 180 201 384 485 084 009 576
MAGIC SQUARES OF THE SEVEN PLANETS:
I had already mentioned some, here is the complete chart:
SATURN……………..JUPITER…………….MARS
………………………………………….11 24 07 20 03
…………………..04 14 15 01 ………04 12 25 08 16
04 O9 02 ……….09 07 06 12 ………17 05 13 21 09
03 05 07 ……….05 11 10 08 ………10 18 01 14 22
08 01 06 ……….16 02 03 13 ………23 06 19 02 15
……….SUN………………………….. VENUS
………………………………….22 47 16 41 10 35 04
06 32 03 34 35 01 ……………05 23 48 17 42 11 29
07 11 27 28 08 30 ……………30 06 24 49 18 36 12
19 14 16 15 23 24 ……………13 31 07 25 43 19 37
18 20 22 21 17 13 ……………38 14 32 01 26 44 20
25 29 10 09 26 12 ……………21 39 08 33 02 27 45
36 05 33 04 02 31 ……………46 15 40 09 34 03 28
……………..MERCURY………………………..MOON
………………………….. …………37 78 29 70 21 62 13 54 05
08 58 59 05 04 62 63 01 ………..06 38 79 30 71 22 63 14 46
49 15 14 52 53 11 10 56 ………..47 07 39 80 31 72 23 55 15
41 23 22 44 45 19 18 48 ………..16 48 08 40 81 32 64 24 56
32 34 35 29 28 38 39 25 ………..57 17 49 09 41 73 33 65 25
40 26 27 37 36 30 31 33 ………..26 58 18 50 01 42 74 34 66
17 47 46 20 21 43 42 24 ………..67 27 59 10 51 02 43 75 35
09 55 54 12 13 51 50 16 ………..36 68 19 60 11 52 03 44 76
64 02 03 61 60 06 07 57 ………..77 28 69 20 61 12 53 04 45
Talismans bearing Magic Squares:
Talismans can be made using planetary magic squares (zeroes before numbers need not be taken) – each of the seven planets have their own corresponding magic square which consists of a table of numbers, as must already have been noticed. The simplest table is that of Saturn – which has a grid of nine numbers and the most complex is the moon with its grid of eighty-one numbers.
The numbers of each square, in any vertical or horizontal division – always adds to give the same sum. This sum is the magical number of the planetary daemon or spirit.
Saturn’s table when engraved on a piece of lead brings good fortune to its wearer and is effective “to help child-birth and to make a man powerful.”
Jupiter’s table engraved on a piece of silver was believed to bring favour and love to he who wears it. “It will dissolve witchcraft, engraven on coral.”
Mars table “engraven on iron or swords makes him that bears it valiant in wars and terrible to his adversaries. Cut in carnelian, it stops bleeding.”
“The Sun engraven on gold, makes the bearer fortunate and beloved, and to be a companion of Kings.”
“Venus engraven on silver, brings good fortune and love of women. It makes the wearer powerful and dissolves witchcraft, also generates peace between man and wife.”
“Mercury engraven on silver, tin or brass, or written on virgin parchment, will make him that wears it obtain what he desires. It brings gain, gives memory and understanding, and knowledge of occult things by dreams.”
“The Moon engraven on silver, brings cheerfulness takes away ill will, makes him secure when travelling and expels enemies and all evil things. Made in lead and buried, it shall bring misfortune to the inhabitants of a city, also ships and mills.”
_The use of Magic Squares were popularised by the magician Agrippa _in his book D.E. Occulta Philosophia, 1534.
Back to Magic Cubes:
12×12x12 magic cubes:
………………………………1st Face
1652 0761 1601 0800 1688 0845 0068 0905 0017 0944 0104 0989
0142 0982 1678 0723 0034 0915 1726 0831 0094 0867 1611 0771
0084 0817 0121 0936 1632 0877 1668 0961 1705 0792 0048 0733
0005 0896 1640 0857 0113 0956 1589 0752 0056 1001 1697 0812
1647 0766 0015 0946 1683 0850 0063 0910 1599 0802 0099 0994
0133 0984 1680 0721 0025 0924 1717 0840 0096 0865 1609 0780
1664 0821 0125 0932 1628 0737 0080 0965 1709 0788 0044 0881
0010 0891 1642 0855 0118 0951 1594 0747 0058 0999 1702 0807
1656 0757 0013 0948 1692 0841 0072 0901 1597 0804 0108 0985
1721 0980 1676 0725 0029 0776 0137 0836 0092 0869 1613 0920
1659 0819 0123 0934 1623 0742 0075 0970 1707 0790 0046 0 886
0001 0900 0060 0997 0109 0960 1585 0756 1644 0853 1693 0816
…………………………….. 2nd Face
1370 0539 0410 1223 1334 0455 0362 1259 1418 0503 0326 1175
0292 1185 1348 0573 0400 1245 1300 0465 0340 1293 1408 0525
1362 0475 0306 1231 1398 0559 0354 1195 1314 0511 0390 1279
0431 1262 1391 0434 0323 1202 1439 0542 0383 1154 1331 0482
1377 0532 0417 1216 1341 0448 0369 1252 1425 0496 0333 1168
0295 1182 1351 0571 0403 1242 1303 0462 0342 1290 1411 0522
1358 0479 0311 1235 1394 0563 0350 1199 1310 0506 0386 1283
0424 1269 1384 0441 0316 1209 1432 0549 0376 1161 1324 0489
1374 0535 0414 1218 1338 0451 0366 1255 1423 0499 0330 1171
0299 1178 1346 0566 0407 1238 1307 0458 0347 1295 1415 0518
1365 0472 0309 1228 1401 0556 0357 1192 1317 0508 0393 1276
0427 1266 1387 0438 0319 1206 1435 0546 0379 1158 1327 0486
……………………………… 3rd Face
1088 0188 1133 0212 1109 0161 0656 1484 0701 1508 0677 1457
0711 1551 1059 0154 0603 1534 1011 0262 0663 1582 1042 0202
0660 0252 0697 1512 1045 1453 1092 1548 1129 0216 0613 0157
0581 1469 1064 0281 0692 1532 1013 0173 0632 1577 1124 0236
1078 0183 0706 1515 1114 0159 0658 1479 1030 0207 0682 1563
0577 1549 1068 0277 0612 1536 1009 0253 0636 1573 1044 0240
1076 0248 0593 1520 1049 0269 0644 1544 1025 0224 0617 1565
0579 1474 1095 0286 0687 1498 1143 0178 0627 1450 1119 0238
1080 0192 0589 1524 1105 0265 0648 1488 1021 0228 0673 1561
1145 1553 1100 0149 0608 0200 0713 0257 0668 1445 1040 1496
1090 0250 0598 1503 1054 0267 0646 1539 1138 0219 0615 1455
0709 1465 0672 1441 0696 1500 1141 0169 1104 0145 1128 0204
…………………………….. 4th Face
0074 0962 0131 0926 0047 0887 1658 0818 1715 0782 1631 0743
1593 0753 0057 1000 1701 0808 0009 0892 1641 0856 0112 0952
1650 0906 1603 0798 0103 0847 0066 0762 0019 0942 1687 0991
1727 0839 0086 0875 1610 0770 0143 0983 1670 0731 0026 0914
0076 0969 1708 0789 0040 0885 1660 0825 0124 0933 1624 0741
1591 0751 0054 1003 1698 0810 0007 0895 1638 0859 0114 0954
0062 0902 1607 0794 0107 0995 1646 0758 0023 0938 1691 0851
1725 0832 0093 0868 1617 0772 0141 0976 1677 0724 0033 0916
0078 0966 1711 0786 0043 0883 1662 0822 0127 0930 1627 0739
0011 0755 0050 1007 1694 0950 1595 0899 1634 0863 0110 0806
0064 0904 1600 0801 0100 0993 1648 0765 0016 0945 1689 0849
1723 0835 1674 0727 1614 0774 0139 0979 0090 0871 0030 0918
……………………………… 5th Face
0356 1193 1316 0509 0392 1277 1364 0473 0308 1229 1400 0557
1438 0543 0382 1155 1330 0483 0430 1263 1390 0435 0322 1203
0372 1249 1428 0493 0336 1165 1380 0529 0420 1213 1344 0445
1301 0464 0341 1292 1409 0524 0293 1184 1349 0572 0401 1244
0351 1198 1311 0514 0387 1282 1359 0478 0303 1234 1395 0562
1429 0552 0373 1153 1321 0492 0421 1272 1392 0444 0313 1212
0368 1253 1421 0497 0332 1169 1376 0533 0416 1220 1340 0449
1306 0459 0346 1287 1414 0519 0298 1179 1354 0567 0406 1239
0360 1189 1320 0516 0396 1273 1368 0469 0301 1225 1404 0553
1433 0548 0380 1160 1325 0488 0425 1268 1385 0437 0317 1208
0363 1258 1419 0502 0327 1174 1371 0538 0411 1222 1335 0454
1297 0468 0337 1296 1405 0528 0289 1188 1345 0576 0397 1248
…………………………….. 6th Face
0650 1547 0707 1502 0614 1463 1082 0251 1139 0206 1046 0167
1012 0172 0628 1581 1120 0237 0580 1473 1060 0285 0693 1533
1086 1483 1135 0210 0678 0163 0654 0187 0703 1506 1110 1459
1019 1550 0626 1583 0611 0230 0587 0254 1058 0287 1043 1526
0657 1540 1137 0208 0621 1456 1089 0244 0705 1504 1053 0160
1015 0174 0630 1579 1123 0234 0583 1470 1062 0283 0691 1530
0638 1487 1031 0218 0674 1571 1070 0191 0599 1514 1106 0275
1144 0261 0664 1449 1036 0201 0712 1557 1096 0153 0604 1497
0642 0247 1072 0222 1050 1567 1074 1543 0595 1518 0618 0271
0719 0170 0662 1451 1127 1490 1151 1466 1094 0155 0695 0194
0645 1485 1029 0220 0681 1564 1077 0184 0597 1516 1108 0268
1147 0258 1098 0151 1039 0198 0715 1554 0666 1447 0607 1494
…………………………….. 7th Face
1649 0764 1604 0797 1685 0848 0065 0908 0020 0941 0101 0992
0135 0975 1671 0730 0027 0922 1719 0838 0087 0874 1618 0778
0061 0972 0024 0937 0037 0996 1645 0828 1608 0793 1621 0852
0140 0893 1673 0728 0116 0917 1724 0749 0089 0872 1700 0773
1654 0759 0022 0939 1690 0843 0070 0903 1606 0795 0106 0987
0144 0973 1669 0732 0036 0913 1728 0829 0085 0876 1620 0769
1661 0824 0128 0929 1625 0740 0077 0968 1712 0785 0041 0884
0003 0898 1635 0862 0111 0958 1587 0754 0051 1006 1695 0814
1657 0768 0132 0925 1681 0744 0073 0912 1716 0781 0097 0888
1592 0833 1637 0860 1616 0809 0008 0977 0053 1004 0032 0953
1666 0826 0130 0927 1630 0735 0082 0963 1714 0783 0039 0879
0012 0889 0049 1008 0120 0949 1596 0745 1633 0864 1704 0805
…………………………….. 8th Face
1379 0530 0419 1214 1343 0446 0371 1250 1427 0494 0335 1166
0297 1180 1353 0568 0405 1240 1305 0460 0345 1288 1413 0520
1363 0474 0307 1230 1399 0558 0355 1194 1315 0510 0391 1278
0422 1271 1382 0443 0314 1211 1430 0551 0374 1163 1322 0491
1372 0537 0412 1221 1336 0453 0364 1257 1420 0501 0328 1173
0294 1183 1350 0570 0402 1243 1302 0463 0343 1291 1410 0523
1367 0470 0302 1226 1403 0554 0359 1190 1319 0515 0395 1274
0429 1264 1389 0436 0321 1204 1437 0544 0381 1156 1329 0484
1375 0534 0415 1219 1339 0450 0367 1254 1422 0498 0331 1170
0290 1187 1355 0575 0398 1247 1298 0467 0338 1286 1406 0527
1360 0477 0304 1233 1396 0561 0352 1197 1312 0513 0388 1281
0426 1267 1386 0439 0318 1207 1434 0547 0378 1159 1326 0487
……………………………… 9th Face
1073 0185 1028 0221 1112 0272 0641 1481 0596 1517 0680 1568
0718 1558 1102 0147 0610 1491 1150 0255 0670 1443 1035 0195
0649 0241 0708 1501 1056 1464 1081 1537 1140 0205 0624 0168
0584 0176 1061 0284 1121 1529 1016 1472 0629 1580 0689 0233
1071 0190 0591 1522 1107 0274 0639 1486 1023 0226 0675 1570
0720 1560 1093 0156 0601 1489 1152 0264 0661 1452 1033 0193
1085 0245 0704 1505 1052 0164 0653 1541 1136 0209 0620 1460
0586 1467 1066 0279 0694 1527 1018 0171 0634 1575 1126 0231
1069 1477 0600 1513 0684 0276 0637 0181 1032 0217 1116 1572
1148 1556 1097 0152 0605 0197 0716 0260 0665 1448 1037 1493
1083 0243 0699 1510 1047 0166 0651 1546 1131 0214 0622 1462
0588 1476 0625 1584 0685 1525 1020 0180 1057 0288 1117 0229
……………………………… 10th Face
0071 0971 0014 0947 0038 0986 1655 0827 1598 0803 1622 0842
1588 0748 0088 1005 1696 0777 0136 0897 1636 0729 0117 0957
1651 0763 1602 0799 1686 0846 0067 0907 0018 0943 0102 0990
1718 0830 0095 0866 1619 0779 0134 0974 1679 0722 0035 0923
0081 0964 1605 0784 0045 0988 1653 0820 0129 0940 1629 0736
1722 0750 0091 0870 1699 0775 0138 0894 1675 0726 0115 0919
0083 0911 1706 0791 0098 0878 1667 0767 0122 0935 1682 0734
1720 0837 0052 0873 1612 0813 0004 0981 1672 0861 0028 0921
0079 0967 1710 0787 0042 0882 1663 0823 0126 0931 1626 0738
0002 0890 0059 0998 0119 0959 1586 0746 1643 0854 1703 0815
0069 0909 1713 0796 0105 0880 1665 0760 0021 0928 1684 0844
1590 0834 1639 0858 1615 0811 0006 0978 0055 1002 0031 0955
……………………………… 11th Face
0353 1196 1313 0512 0389 1280 1361 0476 0305 1232 1397 0560
1431 0550 0375 1162 1323 0490 0423 1270 1383 0442 0315 1210
0361 1260 1417 0504 0325 1176 1369 0540 0409 1224 1333 0456
1304 0461 0344 1289 1412 0521 0296 1181 1352 0569 0404 1241
0358 1191 1318 0507 0394 1275 1366 0471 0310 1227 1402 0555
1440 0541 0384 1164 1332 0481 0432 1261 1381 0433 0324 1201
0365 1256 1424 0500 0329 1172 1373 0536 0413 1217 1337 0452
1299 0466 0339 1294 1407 0526 0291 1186 1347 0574 0399 1246
0349 1200 1309 0505 0385 1284 1357 0480 0312 1236 1393 0564
1436 0545 0377 1157 1328 0485 0428 1265 1388 0440 0320 1205
0370 1251 1426 0495 0334 1167 1378 0531 0418 1215 1342 0447
1308 0457 0348 1285 1416 0517 0300 1177 1356 0565 0408 1237
……………………………… 12th Face
0659 1538 0698 1511 0623 1454 1091 0242 1130 0215 1055 0158
1017 0177 0633 1576 1125 0232 0585 1468 1065 0280 0688 1528
1075 1482 1026 0223 0679 0270 0643 0186 0594 1519 1111 1566
1142 0263 0671 1442 1034 0203 0710 1559 1103 0146 0602 1499
0652 1545 1132 0213 0616 1461 1084 0249 0700 1509 1048 0165
1014 0175 0631 1578 1122 0235 0582 1471 1063 0282 0690 1531
0647 1478 1022 0227 0683 1562 1079 0182 0590 1523 1115 0266
1149 0256 0669 1444 1041 0196 0717 1552 1101 0148 0609 1492
0655 1542 1134 0211 0619 1458 1087 0246 0702 1507 1051 0162
0578 0179 0635 1574 1118 1535 1010 1475 1067 0278 0686 0239
0640 1480 1024 0225 0676 1569 1072 0189 0592 1521 1113 0273
1146 0259 1099 0150 1038 0199 0714 1555 0667 1446 0606 1495
Now try 13×13x13, by taking next row as 553 and next face as 786.
………………………………………1st Face
0001 2392 1912 2137 1082 0476 1504 0002 2391 1925 2334 1081 0475 1503
2327 0896 0664 1500 0207 2597 1718 2132 0895 0467 1303 0208 2584 1717
1509 0203 2390 1907 2141 1083 0463 1510 0022 2389 1908 2142 1084 0464
1722 2343 0883 0670 1296 0009 2593 1721 2330 0870 0669 1309 0206 2594
0666 1508 0003 2394 1910 2130 1089 0469 1507 0004 2407 1909 2129 1090
2596 1709 2153 0875 0668 1306 0209 2595 1710 2336 0890 0667 1291 0210
1088 0465 1512 0215 2381 1915 2136 1087 0466 1511 0006 2382 1916 2135
0197 2588 1716 2333 0885 0672 1308 0198 2587 1729 2138 0886 0671 1307
2131 1078 0468 1304 0025 2387 1914 2328 1091 0663 1499 0012 2402 1913
1313 0007 2586 1712 2351 0887 0659 1314 0204 2585 1711 2338 0888 0660
1917 2147 1065 0474 1506 0205 2383 1918 2134 1080 0473 1505 0024 2384
0470 1312 0199 2590 1714 2326 0894 0665 1311 0200 2589 1727 2325 0893
2386 1905 2349 1085 0472 1502 0013 2385 1906 2140 1086 0471 1501 0014
0892 0662 1315 0019 2577 1720 2332 0891 0661 1316 0202 2578 1719 2331
…………………………………….. 2nd Face
0560 1559 0085 2420 1884 1983 1011 0559 1560 0086 2419 1883 2180 1026
2612 1689 1977 0840 0748 1360 0291 2611 1690 2160 0839 0747 1359 0292
1028 0548 1565 0091 2418 1880 1987 1027 0743 1566 0092 2417 1879 1974
0289 2608 1693 2175 0827 0739 1366 0290 2411 1708 2176 0828 0740 1365
1961 1034 0554 1563 0088 2618 1882 1976 1033 0553 1564 0087 2421 1881
1361 0294 2610 1682 2167 0833 0752 1362 0293 2413 1681 2182 0834 0751
1887 2177 1032 0536 1567 0089 2410 1888 2178 0835 0549 1568 0090 2409
0756 1349 0282 2616 1702 2179 0829 0755 1350 0281 2615 1687 1984 0830
2416 1885 2173 1035 0551 1556 0095 2415 1886 1964 1036 0552 1555 0096
0832 0744 1369 0287 2614 1684 2183 0817 0547 1370 0288 2613 1683 2184
0094 2412 1889 1979 1023 0558 1562 0093 2607 1890 1966 1024 0557 1561
2171 0837 0736 1367 0284 2422 1685 2172 0838 0749 1368 0283 2617 1686
1557 0098 2414 1878 1971 1029 0556 1558 0097 2609 1877 1986 1030 0555
1691 1981 0836 0746 1371 0285 2606 1692 1982 1031 0745 1358 0286 2605
……………………………………..3rd Face
2068 1054 0531 1377 0029 2714 1940 2067 1039 0532 1378 0030 2517 1939
1191 0236 2696 1745 2257 0867 0720 1178 0235 2695 1746 2258 0868 0523
1935 2072 1056 0520 1383 0035 2712 1936 2071 1055 0519 1384 0036 2501
0711 1379 0038 2705 1749 2246 0856 0030 1198 0233 2706 1750 2259 0855
2506 1951 2059 1062 0526 1381 0031 2505 1938 2060 1061 0525 1382 0228
0862 0723 1193 0238 2694 1738 2251 0861 0724 1194 0237 2693 1737 2070
0034 2494 1944 2065 1060 0522 1385 0033 2493 1943 2066 1059 0717 1400
2264 0844 0727 1182 0240 2518 1744 2263 0843 0728 1181 0239 2699 1743
1387 0040 2500 1941 2061 1063 0524 1374 0039 2499 1942 2062 1064 0719
1739 2267 0860 0716 1187 0231 2516 1740 2268 0859 0715 1188 0232 2697
0529 1183 0234 2495 1945 2063 1051 0726 1394 0037 2496 1946 2064 1052
2701 1755 2255 0866 0722 1185 0227 2702 1742 2256 0865 0721 1186 0032
1058 0527 1375 0042 2498 1934 2069 1057 0528 1376 0055 2497 1933 2266
0230 2690 1747 2261 0864 0718 1189 0243 2703 1748 2248 0863 0521 1190
……………………………………..4th Face
2475 1785 2012 1124 0587 1461 0058 2490 1786 1997 1123 0784 1476 0057
0937 0565 1457 0068 2682 1591 2201 1148 0580 1262 0263 2667 1592 2202
0064 2473 1782 2016 1126 0772 1467 0063 2474 1781 2015 1139 0575 1468
2204 0940 0781 1267 0262 2664 1595 2189 0939 0782 1268 0065 2663 1596
1466 0060 2673 1587 2003 1145 0582 1465 0059 2674 1784 2004 1132 0581
1584 2014 0932 0779 1264 0266 2665 1583 2013 1141 0780 1263 0265 2666
0577 1483 0061 2466 1775 2010 1144 0578 1470 0062 2661 1776 2009 1143
2671 1575 2208 0941 0783 1265 0254 2672 1576 2207 0942 0588 1280 0253
1147 0761 1261 0264 2486 1773 2005 0952 0776 1458 0067 2471 1788 2006
0260 2669 1586 2212 0944 0576 1271 0259 2670 1585 2211 0943 0757 1272
2008 1136 0585 1463 0066 2468 1791 1993 1135 0586 1464 0261 2467 1792
1270 0256 2477 1783 2199 0949 0778 1269 0255 2478 1588 2200 0950 0763
1779 2210 1128 0583 1459 0070 2470 1780 2209 0945 0584 1460 0069 2469
0773 1288 0272 2662 1579 2205 0934 0774 1273 0257 2465 1594 2192 0947
…………………………………….. 5th Face
1406 0142 2545 1855 2236 1110 0419 1405 0141 2546 1870 2039 1095 0406
1676 2229 0923 0593 1205 0348 2528 1675 2230 0924 0594 1206 0347 2737
0407 1411 0148 2529 1865 2043 1112 0394 1412 0147 2726 1866 2044 1111
2719 1876 2036 0912 0599 1225 0346 2734 1665 2035 0911 0600 1212 0345
1117 0399 1410 0354 2533 1867 2032 1118 0400 1409 0143 2534 1868 2031
0349 2721 1667 2238 0918 0611 1207 0154 2722 1668 2237 0917 0598 1208
2234 1116 0395 1218 0146 2522 1873 2233 1115 0396 1413 0145 2535 1874
1210 0352 2727 1673 2040 0899 0615 1209 0351 2728 1674 2235 0900 0602
1858 2033 1120 0397 1416 0152 2724 1871 2034 1119 0398 1415 0151 2527
0603 1216 0344 2725 1669 2240 0916 0590 1215 0343 2530 1670 2239 0915
2524 1680 2232 1108 0403 1407 0150 2523 1875 2231 1107 0404 1422 0149
0921 0596 1214 0158 2729 1672 2227 0922 0595 1213 0339 2730 1671 2228
0153 2525 1863 2041 1114 0401 1403 0350 2540 1864 2042 1113 0402 1404
2038 0919 0591 1414 0342 2718 1663 2037 0920 0592 1231 0341 2731 1678
……………………………………. 6th Face
1165 0686 1434 0113 2364 1813 2124 1166 0489 1447 0114 2363 1814 2123
0320 2569 1619 2314 0979 0677 1233 0123 2570 1620 2299 0980 0678 1234
2127 1154 0673 1454 0134 2375 1613 2128 1153 0674 1439 0119 2376 1796
1240 0317 2565 1610 2316 0968 0487 1239 0318 2566 1623 2315 0967 0684
1811 2115 1173 0483 1438 0116 2365 1812 2312 1174 0484 1437 0115 2380
0682 1236 0321 2357 1611 2322 0974 0681 1235 0322 2568 1612 2321 0973
2367 1818 2318 1171 0479 1442 0118 2368 1621 2303 1172 0676 1245 0117
0969 0490 1238 0324 2559 1603 2320 0970 0685 1237 0323 2574 1604 2319
0124 2373 1816 2118 1175 0481 1430 0319 2374 1815 2103 1176 0482 1429
2309 0971 0477 1244 0316 2571 1809 2310 0972 0478 1243 0329 2572 1614
1436 0121 2370 1820 2120 1164 0683 1435 0122 2369 1819 2105 1163 0488
1616 2311 0963 0679 1242 0311 2575 1615 2116 0978 0680 1241 0312 2576
0486 1432 0126 2553 1807 2126 1170 0485 1431 0125 2372 1808 2125 1169
2563 1622 2122 0975 0675 1246 0314 2564 1817 2107 0976 0480 1441 0313
……………………………………. 7th Face
1842 2095 0983 0630 1518 0170 2461 1841 2096 0998 0433 1517 0169 2462
0622 1332 0376 2653 1648 2286 0797 0621 1317 0375 2458 1647 2285 0798
2460 1837 2099 0985 0617 1538 0176 2459 1838 2100 0790 0618 1523 0175
0785 0432 1324 0373 2650 1651 2288 0786 0627 1323 0388 2649 1652 2287
0171 2660 1840 2088 0991 0427 1536 0172 2449 1825 2087 1006 0428 1535
2293 0791 0626 1516 0391 2455 1640 2294 0792 0625 1319 0378 2456 1639
1525 0173 2451 1845 2094 0989 0423 1526 0384 2452 1846 2093 0990 0424
1646 2291 0787 0434 1322 0366 2657 1645 2292 0802 0629 1321 0365 2658
0426 1514 0193 2457 1844 2090 0993 0425 1513 0180 2654 1843 2089 0994
2656 1642 2295 0789 0421 1328 0372 2655 1641 2296 0986 0422 1341 0371
0981 0628 1520 0191 2439 1847 2092 0982 0445 1519 0178 2454 1848 2091
0382 2464 1644 2284 0795 0624 1326 0381 2645 1643 2283 0796 0623 1325
2097 0987 0430 1320 0195 2651 1835 2098 0988 0429 1515 0182 2652 1836
1329 0369 2647 1650 2290 0793 0619 1330 0188 2648 1649 2289 0794 0620
…………………………………… 8th Face
0015 2405 1926 2151 1067 0462 1490 0016 2406 1911 2348 1068 0461 1489
2145 0882 0650 1289 0221 2583 1732 2342 0881 0649 1290 0222 2598 1731
1495 0021 2404 1922 2155 1069 0449 1496 0008 2403 1921 2156 1070 0646
1736 2329 0869 0460 1310 0219 2579 1735 2344 0884 0459 1491 0220 2580
0456 1494 0018 2408 1924 2144 1075 0651 1493 0017 2393 1923 2143 1076
2582 1724 2335 0889 0654 1292 0027 2581 1723 2350 0876 0653 1305 0224
1074 0452 1497 0201 2395 1929 2150 1073 0451 1498 0020 2396 1930 2149
0211 2601 1730 2347 0871 0658 1294 0212 2602 1715 2152 0872 0657 1293
2341 1092 0453 1486 0011 2401 1928 2146 1077 0454 1485 0026 2388 1927
1299 0217 2600 1726 2337 0873 0645 1300 0218 2599 1725 2352 0874 0450
1931 2133 1079 0656 1492 0023 2397 1932 2148 1066 0655 1295 0010 2398
0652 1298 0214 2604 1728 2340 0880 0455 1297 0213 2603 1713 2339 0879
2400 1920 2139 1071 0458 1488 0223 2399 1919 2154 1072 0457 1487 0028
0878 0647 1301 0005 2592 1734 2346 0877 0648 1302 0216 2591 1733 2345
…………………………………… 9th Face
0545 1545 0100 2434 1897 1969 1025 0546 1546 0295 2433 1898 1970 1012
2626 1703 1963 0826 0734 1346 0306 2625 1704 2174 0825 0733 1345 0305
1014 0534 1551 0301 2432 1894 1973 1013 0533 1552 0106 2431 1893 1988
0108 2426 1903 2161 0813 0753 1352 0303 2621 1694 2162 0814 0754 1351
1975 1020 0540 1549 0297 2436 1896 1962 1019 0539 1550 0298 2435 1699
1347 0308 2624 1696 2181 0819 0738 1348 0307 2427 1695 2168 0820 0737
1901 1967 1017 0550 1554 0299 2423 1902 1968 1018 0535 1553 0104 2424
0741 1363 0296 2630 1688 2165 0815 0742 1364 0099 2629 1701 2166 0816
2430 1899 2159 1022 0538 1542 0109 2429 1900 1978 1021 0537 1541 0110
0818 0730 1355 0105 2628 1698 2169 0831 0729 1356 0302 2627 1697 2170
0304 2622 1707 1965 1010 0544 1548 0107 2425 1904 1980 1009 0543 1547
2157 0823 0750 1353 0101 2632 1700 2158 0824 0735 1354 0102 2631 1895
1544 0112 2428 1892 1985 1015 0542 1543 0111 2623 1891 1972 1016 0541
1705 2163 0822 0731 1357 0103 2619 1706 2164 0821 0732 1372 0300 2620
……………………………………. 10th Face
2250 1040 0517 1195 0043 2504 1954 2249 1053 0518 1392 0044 2503 1953
1177 0250 2710 1759 2047 0854 0706 1192 0249 2709 1760 2244 0853 0705
1949 2057 1042 0506 1397 0049 2698 1950 2058 1041 0505 1398 0050 2515
0725 1197 0248 2692 1763 2260 0841 0712 1184 0247 2691 1764 2049 0842
2519 1937 2045 1047 0512 1396 0045 2520 1952 2046 1048 0511 1395 0242
0848 0709 1179 0252 2708 1751 2265 0847 0710 1180 0251 2511 1752 2252
0048 2508 1958 2052 1046 0507 1399 0047 2507 1957 2051 1045 0704 1386
2054 0858 0713 1391 0226 2700 1758 2053 0857 0714 1196 0225 2713 1757
1373 0054 2514 1955 2243 1050 0510 1388 0053 2513 1956 2048 1049 0509
1754 2254 0846 0702 1201 0245 2502 1753 2253 0845 0701 1202 0246 2711
0515 1393 0052 2510 1959 2050 1038 0516 1380 0051 2509 1960 2245 1037
2715 1741 2241 0851 0708 1199 0241 2716 1756 2242 0852 0707 1200 0046
1044 0513 1389 0056 2512 1947 2055 1043 0514 1390 0041 2707 1948 2056
0244 2704 1761 2247 0850 0703 1203 0229 2689 1762 2262 0849 0508 1204
……………………………………. 11th Face
2489 1772 2194 1138 0573 1475 0071 2476 1771 2011 1137 0574 1462 0072
0951 0775 1275 0278 2668 1577 2187 0938 0762 1276 0277 2485 1578 2188
0078 2487 1767 2002 1140 0562 1482 0273 2488 1768 2001 1125 0561 1481
2190 0925 0768 1281 0276 2678 1581 2203 0926 0767 1282 0079 2677 1582
1480 0073 2491 1769 1989 1131 0764 1479 0074 2492 1770 1990 1146 0567
1569 2196 0946 0765 1277 0280 2680 1570 2195 0931 0570 1278 0279 2679
0564 1469 0076 2480 1789 1996 1130 0563 1484 0271 2479 1790 1995 1129
2685 1589 1998 0927 0769 1279 0267 2686 1590 2193 0928 0770 1266 0268
1133 0579 1471 0082 2472 1787 1991 1134 0566 1472 0081 2681 1774 1992
0274 2684 1571 2198 0930 0758 1285 0077 2683 1572 2197 0929 0771 1286
1994 1122 0571 1477 0080 2481 1777 2007 1121 0572 1478 0275 2482 1778
1283 0270 2687 1573 2185 0936 0568 1284 0269 2688 1574 2186 0935 0777
1765 2000 1142 0569 1474 0084 2484 1766 1999 1127 0766 1473 0083 2483
0760 1274 0258 2676 1593 2191 0948 0759 1287 0075 2675 1580 2206 0933
……………………………………. 12th Face
1420 0155 2531 1869 2026 1096 0601 1419 0156 2532 1660 2025 1109 0616
1662 2216 0909 0607 1219 0166 2738 1857 2215 0910 0608 1220 0165 2723
0393 1426 0162 2543 1851 2030 1098 0408 1425 0161 2740 1852 2029 1097
2733 1666 2218 0898 0613 1211 0360 2720 1679 2217 0897 0614 1226 0163
1104 0413 1424 0340 2547 1853 2017 1103 0414 1423 0157 2548 1854 2018
0167 2735 1653 2223 0904 0597 1221 0364 2736 1654 2224 0903 0612 1222
2024 1102 0409 1428 0356 2536 1859 2023 1101 0410 1427 0159 2521 1860
1224 0338 2741 1659 2222 0914 0405 1223 0337 2742 1856 2221 0913 0420
1872 2020 1106 0411 1402 0362 2542 1661 2019 1105 0412 1401 0361 2541
0589 1230 0358 2739 1655 2225 0902 0604 1229 0357 2544 1656 2226 0901
2537 1861 2022 1094 0417 1421 0164 2538 1862 2021 1093 0418 1408 0359
0907 0609 1228 0144 2743 1657 2213 0908 0610 1227 0353 2744 1658 2214
0363 2539 1849 2028 1100 0415 1418 0168 2526 1850 2027 1099 0416 1417
2220 0905 0605 1232 0160 2732 1677 2219 0906 0606 1217 0355 2717 1664
…………………………………….. 13th Face
1151 0504 1448 0128 2377 1799 2110 1152 0699 1433 0127 2378 1800 2109
0333 2555 1606 2300 0966 0691 1247 0138 2556 1605 2313 0965 0692 1248
2114 1168 0491 1440 0120 2361 1795 2113 1167 0492 1453 0133 2558 1810
1254 0331 2551 1624 2106 0954 0697 1253 0332 2552 1609 2301 0953 0698
1798 2101 1159 0497 1452 0130 2379 1797 2102 1160 0694 1451 0129 2366
0696 1250 0336 2371 1597 2308 0960 0695 1249 0335 2554 1598 2307 0959
2353 1804 2108 1157 0493 1456 0328 2354 1607 2121 1158 0494 1455 0327
0955 0700 1251 0310 2573 1617 2306 0956 0503 1252 0309 2560 1618 2305
0137 2359 1802 2104 1161 0495 1443 0334 2360 1801 2117 1162 0496 1444
2323 0957 0687 1257 0330 2557 1600 2324 0958 0688 1258 0315 2362 1599
1450 0135 2356 1806 2302 1150 0501 1449 0136 2355 1805 2119 1149 0502
1602 2297 0977 0693 1256 0325 2561 1601 2298 0964 0498 1255 0326 2562
0500 1446 0139 2567 1793 2112 1156 0499 1445 0140 2358 1794 2111 1155
2549 1608 2304 0961 0689 1260 0132 2550 1803 2317 0962 0690 1259 0131
……………………………………. 14th Face
1828 2081 0997 0644 1532 0183 2447 1827 2082 0984 0447 1531 0184 2448
0636 1318 0390 2443 1634 2272 0811 0635 1331 0389 2640 1633 2271 0812
2642 1824 2085 0999 0436 1524 0190 2445 1823 2086 1000 0435 1537 0189
0799 0446 1338 0387 2635 1638 2274 0800 0641 1337 0374 2636 1637 2273
0185 2450 1826 2074 1005 0441 1522 0186 2463 1839 2073 0992 0638 1521
2279 0805 0640 1334 0181 2637 1626 2280 0806 0639 1333 0392 2638 1625
1539 0187 2437 1831 2080 1003 0437 1540 0370 2438 1832 2079 1004 0438
1632 2277 0801 0448 1335 0380 2643 1631 2278 0788 0643 1336 0379 2644
0440 1528 0179 2639 1830 2076 1007 0439 1527 0194 2444 1829 2075 1008
2446 1627 2281 0803 0632 1342 0386 2641 1628 2282 0804 0631 1327 0385
0995 0642 1533 0177 2453 1834 2078 0996 0431 1534 0192 2440 1833 2077
0368 2646 1630 2270 0809 0637 1340 0367 2659 1629 2269 0810 0442 1339
2083 1001 0444 1530 0377 2441 1821 2084 1002 0443 1529 0196 2442 1822
1343 0383 2634 1636 2276 0807 0633 1344 0174 2633 1635 2275 0808 0634
15×15x15 Magic Cube:
One solution is – 334 for the next row and 899 for the next face. Since we require 15 alphabets, we can choose A to O, a to o, and p to z, 1,2,3,4. We will have to ensure that A+D+G+J+M = B+E+H+K+N =C+F+I+L+O = 40, if we are giving values from 1 to 15 to A’s, a+d+g+j+m = b+e+h+k+n = c+f+i+l+o = 525, if we are giving a’s values from 0,15 to 210, and p+s+v+y+1 = q+t+w+z+2 = r+u+x+1+4 = 7875, if we are giving values from 0,225 to 3150.
Of course there are other combinations available, try and see if you get a batter one.
