A New Procedure for Magic Squares (Part I) - New Magic Constant Equation

Loubère Block Modified Squares

A stairs

A Discussion of the New Method

An important general principle for generating odd magic squares by the De La Loubère 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 Loubère 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 Loubère square is shown below as an example:

17 24 1 8 15
2357 14 16
4613 20 22
101219 21 3
11 18 25 2 9

Normally the Loubère method involves a stepwise approach of consecutive numbers, i.e., 1,2,3...n2. For example, A new generalized Loubère 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 Loubère, having numbers greater than n2 by modifying a block or grid of numbers. The initial number is added either at the normal Loubère 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.

This is done by taking the non-magic squares and converting them into magic ones using a variety of means. These squares are depicted below in methods I and II.

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 Block Modified Loubère Squares

Method IA: Start at lower left hand corner (1, 2 ⇒ break)
  1. Place 1 into the lower left hand corner.
  2. Add consecutive numbers up to the middle cell.
  3. Move, i.e. break, one cell down.
  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. Since the numbers 1 and 2 sit on a diagonal which sums to 75, these numbers remain unchanged.
  8. To square 3 adding n2 = 25 to both 20 and 3 (color change to light green) gives the magic square 4 where S = ½(n3 + 5n).
  9. To square 3 adding -n2 = -25 outside the block to 23, 18 and 6 (color change to light green) gives the magic square 5 where S = ½(n3 - 5n).
1
 
 
 
2
13
2
7 9 16 18
81522 6
1421 5 12
2024 11 13
1310 17 19
3
75
7 9 16 182575
81522 24 675
142123 5 12 75
2024 11 1350
1310 17 1950
505075 75 7575
4
75
7 9 16 182575
81522 24 675
142123 5 12 75
4524 11 1375
12810 17 1975
757575 75 7575
+
5
50
7 9 16 -72550
81522 24 -1950
1421-2 5 12 50
2024 11 1350
1310 17 1950
505050 50 5050
Method II: Start on the main diagonal one cell up from the center cell (1, 2 ⇒ break)
  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, one cell down.
  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. Since the numbers 1 and 2 sit on a diagonal which sums to 75, these numbers remain unchanged.
  8. Adding n2 = 25 to both 20 and 3 gives the magic square 4 where S = ½(n3 + 5n).
  9. Similarly the square breaking to the left results a square identical to 4 except for rotation (not shown).
<
1
2
1 3
 
 
 
2
4 11 13 202
101719 1 3
1618 7 9
226 8 15
512 14 21
3
75
4 11 13 20250
101719 1 350
161825 7 9 75
22246 8 1575
23512 14 2175
757575 50 5075
4
75
4 11 13 45275
101719 1 2875
161825 7 9 75
22246 8 1575
23512 14 2175
757575 75 7575
Method IB: Start at lower left hand corner (1, 2, 3, 4 ⇒ break)
  1. Place 1 into the lower left hand corner.
  2. Add consecutive numbers up to the n - 1 cell.
  3. Move, i.e. break, one cell down.
  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 80.
  6. Where the sums 60 cross corresponds to the small square 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. Also the left diagonal.
  8. Adding n2 = 25 to both (2,3,13,17) and (3,4,8,20) (color change to light green) gives the magic squares 3 and 4 and where S = ½(n3 + 9n).
1
 
4
3 5
2
1
2
35
7 14 16 232585
131522 4 660
19213 5 12 60
2029 11 1360
1810 17 2460
606060 60 8560
3
85
7 14 16 232585
381522 4 685
192128 5 12 85
20279 11 1385
1810 42 2485
858585 85 8585
+
4
85
7 14 16 232585
131522 29 685
192128 5 12 85
4529 11 1385
13310 17 2485
858585 85 8585
Method IIIA:Start at the center of the last column (1, 2 ⇒ break)
  1. Place 1 in the middle of the last column.
  2. For this 5x5 square add consecutive number 2.
  3. Move, i.e. break, one cell down.
  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 four numbers do not sit on a diagonal, two possible squares are possible.
  8. Adding n2 = 25 to both (2 and 1) or (3 and 20) gives the magic squares 3 and 4.
1
 
2
3 1
 
 
2
75
21 23 5 121475
2411 13 2050
31017 19 1 50
91618 25 775
152224 6 875
507575 75 5075
3
75
21 23 5 121475
27411 13 2075
31017 19 26 75
91618 25 775
152224 6 875
757575 75 7575
+
4
75
21 23 5 121475
2411 13 4575
281017 19 1 75
91618 25 775
152224 6 875
757575 75 7575
Method IIIB:Start at the center of the last column (1, 2, 3 ⇒ break)
  1. Place 1 in the middle of the last column.
  2. For this 5x5 square add consecutive numbers 2 and 3.
  3. Move, i.e. break, one cell down.
  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 55 instead of 80.
  6. Where the sums 55 cross corresponds to the small square in blue.
  7. Since the numbers to be modified cannot sit on a diagonal, two possible squares are possible.
  8. Adding n2 = 25 to both (3, 8 and 20) or (1, 2 and 3) gives the magic squares 3 and 4 and where S = ½(n3 + 7n).
1
3
24
1
 
 
2
80
21 3 5 121455
2411 18 2055
81017 19 1 55
91623 25 780
152224 6 1380
555580 80 5580
3
80
21 28 5 121480
2411 18 4580
331017 19 1 80
91623 25 780
152224 6 1380
808080 80 5580
+
4
80
21 28 5 121480
27411 18 2080
81017 19 26 80
91623 25 780
152224 6 1380
808080 80 8080
Method IVA:Start at the center of the last row (1, 2 ⇒ break)
  1. Place 1 in the middle of the last row.
  2. For this 5x5 square add consecutive number 2.
  3. Move, i.e. break, one cell down.
  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 one of the six number sits on a diagonal which adds to 50 the magic square must modify this number.
  8. Adding n2 = 25 to 1 and 2 gives the magic square 3.
1
 
 
 
2 3
1
2
75
18 25 7 91675
2468 15 2275
51214 21 23 75
111320 2 450
17191 3 1050
757550 50 7550
3
75
18 25 7 91675
2468 15 2275
51214 21 23 75
111320 27 475
171926 3 1075
757575 75 7575
Method IVB:Start at the center of the last row (1, 2, 3 ⇒ break)
  1. Place 1 in the middle of the last row.
  2. For this 5x5 square add consecutive numbers 2 and 3.
  3. Move, i.e. break, one cell down.
  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 55 instead of 80.
  6. Where the sums 50 cross corresponds to the small square in blue.
  7. Since one of the six number sits on a diagonal which adds to 55, two magic square are possible.
  8. Adding n2 = 25 to (1, 2 and 3) or (10, 20 and 21) gives the magic squares 3 and 4.
1
 
 
3
2 4
1
2
80
23 25 7 91680
24613 15 2280
51214 21 3 55
111820 2 455
17191 8 1055
808055 55 5555
3
80
23 25 7 91680
24613 15 2280
51214 21 28 80
111820 27 480
171926 8 1080
808080 80 8080
+
4
80
23 25 7 91680
24613 15 2280
51214 46 3 80
111845 2 480
17191 8 3580
808080 80 8080

This completes this section on the new block Loubère Method (Part I). The next section deals with Continuation of block Loubère Method (Part II). To return to homepage.


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