A New Procedure for Magic Squares (Part III)

Méziriac Block Modified Squares

A stairs

A Discussion of the New Method

An important general principle for generating odd magic squares by the Méziriac method is that the center cell must always contain the middle number of the series of numbers used, i.e. a number which is equal to one half the sum of the first and last numbers of the series, or ½(n2 + 1). The properties of these regular or associated Méziriac squares are:

  1. That the sum of the horizontal rows, vertical columns and corner diagonals are equal to the magic sum S.
  2. The sum of any two numbers that are diagonally equidistant from the center (DENS) is equal to n2 + 1, i.e., or twice the number in the center cell and are complementary to each other.

The 5x5 regular Méziriac square is shown below as an example:

3 16 9 22 15
20821 14 2
72513 1 19
24125 18 6
11 4 17 10 23

Normally the Méziriac method involves a stepwise approach of consecutive numbers, i.e., 1,2,3...n2. For example, A new generalized Méziriac procedure where the initial number is placed anywhere on any n - 1 cells of the first row. However, in this new method although the numbers are added consecutively starting with 1 and ending with n2, the final square is transformed into a modified Méziriac, having numbers greater than n2 by modifying a block or grid of numbers. The initial number is added either at the normal Méziriac sites, i.e., (the center of the last row or last column) or on the main diagonal. Every number on the diagonal can be modified, up to n - 1. Replacement of all n numbers on the diagonal leads to semi-magic squares.

The generation of these magic squares involves an almost normal approach:
Numbers are added consecutively starting out with 1 and ending with numbers greater than n2 after modification. A break involves translational moves (up, down or sideways). Moreover, it must be stated here that the magic sum has been modified from the known equation S = ½(n3 + n) to the general equation:

S = ½(n3 ± an)

which takes into account these new squares. The variable a, an odd number, is equal to 1,3,5,7 or ... and may take on + or - values. For example when a = 1 the normal magic sum S is implied. When a takes on different odd values S gives the magic sum of a modified magic square. It will be shown that the addition or subtraction of n2 to some of the cells in the square gives rise to a new magic square.

Construction of 5x5 and 7x7 Block Modified Méziriac Squares

Method IA: Start at right of center (1, 2 ⇒ break (2))
  1. Place 1 into the first cell right of center.
  2. Add the next consecutive number.
  3. Move, i.e. break, two cells left.
  4. Repeat the process until the square is filled as shown below in squares 1-3.
  5. As shown below this square is not magic because the two columns and two rows (in grey) sum to 50 instead of 75.
  6. Where the sums 50 cross corresponds to the small square in blue.
  7. Only the numbers 1 and 2 may be changed to 75.
  8. Adding n2 = 25 to both 1 and 2 (color change to light green) gives the magic square 3 where S = ½(n3 + 5n).
1
 
3 2
  1
 
 
2
75
23 10 22 41675
9213 15 250
20714 1 8 50
61325 12 1975
172411 18 575
757575 50 5075
3
75
23 10 22 41675
9213 15 2775
20714 26 8 75
61325 12 1975
172411 18 575
757575 75 7575
Method II: Start on the main diagonal one cell up from the center cell (1, 2 ⇒ break (2))
  1. Place 1 one cell up from the center of the main diagonal.
  2. Add consecutive numbers up to the right hand corner.
  3. Move, i.e. break, two cells left.
  4. Repeat the process until the square is filled as shown below in squares 1-2.
  5. As shown below this square is not magic because the two columns and two rows (in grey) sum to 50 instead of 75.
  6. Where the sums 50 cross corresponds to the small square in blue.
  7. Since the numbers 1 and 2 sit on a diagonal which sums to 75, these numbers remain unchanged.
  8. Adding n2 = 25 to both 8 and 15 gives the magic square 3 where S = ½(n3 + 5n).
  9. Starting over and breaking two cells right instead of left gives another magic square 4.
1
3 2
1
 
 
 
2
75
9 21 3 15250
20714 1 850
61325 12 19 75
172411 18 575
231022 4 1675
757575 50 5075
3 Break left
75
9 21 3 40275
20714 1 3375
61325 12 19 75
172411 18 575
231022 4 1675
757575 75 7575
 
4 Break right
75
15 3 21 34275
7208 1 3975
191225 13 6 75
112417 5 1875
23164 22 1075
757575 75 7575
Method IB: Start on the main diagonal one cell up from the center cell (1, 2, 3, 4 ⇒ break)
  1. Place Place 1 one cell up from the center of the main diagonal.
  2. Add consecutive numbers up to the n - 1 cell.
  3. Move, i.e. break, two cells left.
  4. Repeat the process until the square is filled as shown below in squares 1-2.
  5. As shown below this square is not magic because the two columns and two rows (in grey) sum to 35 and 60 instead of 85.
  6. Where the sums 60 cross corresponds to the small squares in blue.
  7. Since the numbers 1, 2, 3 and 4 sit on a diagonal which sums to 35, two of these numbers will be modified.
  8. Adding n2 = 25 to either (3,4,13,15) or (10,17,13,15) (color change to light green) gives the magic squares 3 and 4 where where S = ½(n3 + 9n).
1
2
1
 
4 5
3
2
35
14 21 8 15260
20719 1 1360
61825 12 24 85
17411 23 560
31022 9 1660
606085 60 6085
3
85
14 21 8 40285
20719 1 3885
61825 12 24 85
172911 23 585
281022 9 1685
858585 85 8585
+
4
85
14 21 8 152785
20719 26 1385
61825 12 24 85
42411 23 585
33522 9 1685
858585 85 8585

Construction of 7x7 Block Modified Méziriac Squares

Method IA: Start at right of center (1, 2, 3 ⇒ break (2))
  1. Generate the square and its sums.
  2. As shown below this square is not magic because the three columns and three rows (in grey) sum to 147 instead of 196.
  3. Where the sums 147 cross corresponds to the small square in blue.
  4. For example two squares 2 and 3 where the set (1,2,3) or (2,4,42) are modified by the addition of n2 = 49 (color change to light green).
  5. The magic sum S = ½(n3 + 7n).
1
196
46 21 4513 37529 196
20 44 1236 4283 147
43 11 3510 27219 147
17 34 926 11842 147
33 8 2549 244116147
7 31 4823 401532147
30472239 14 386147
196196196 196 147147 147 147
2
196
46 21 4513 37529 196
20 44 1236 42852 196
43 11 3510 275119 196
17 34 926 501842 196
33 8 2549 244116196
7 31 4823 401532196
30472239 14 386196
196196196 196 196196 196 196
+
3
196
46 21 4513 37529 196
20 44 1236 53283 196
43 11 3510 275119 196
17 34 926 11891 196
33 8 2549 244116196
7 31 4823 401532196
30472239 14 386196
196196196 196 196196 196 196
Method IB: Start at right of center (1, 2, 3, 4, 5 ⇒ break (2))
  1. Generate the square and its sums.
  2. As shown below this square is not magic because the three columns and three rows (in grey) sum to 112 and 161 instead of 210.
  3. Where the sums 112 and 161 cross corresponds to the small squares in blue.
  4. For example two squares 2 and 3 where the set (2,3,14,17,21) or (4,5,11,25,26) are modified by the addition of n2 = 49 (color change to light green).
  5. The magic sum S = ½(n3 + 11n).
1
112
20 44 1936 11283 161
43 18 3510 27226 161
17 34 933 12542 161
40 8 3249 244116 210
7 31 4823 471539210
30 5 2246 14 386 161
4214513 37 1229 161
161161210 210 161161 161 210
2
210
20 44 1936 112851 210
43 18 3510 275026 210
66 34 933 12542 210
40 8 3249 244116 210
7 31 4823 471539210
30 5 2246 63 389 210
4704513 37 1229 210
210210210 210 210210 210 210
+
3
210
20 44 1936 60283 210
43 18 3510 27275 210
17 34 933 17442 210
40 8 3249 244116 210
7 31 4823 471539210
30 54 2246 14 389 210
53214513 37 1229 210
210210210 210 210210 210 210

This completes this section on the new block Méziriac Method (Part III). To return to Part II. To return to homepage.


Copyright © 2009 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com