A New Procedure for Magic Squares (Part IV)

Loubère and Méziriac Knight Block Modified Squares

A Loubere knight square

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 on the left and the regular Méziriac on the right :

17 24 1 8 15
2357 14 16
4613 20 22
101219 21 3
11 18 25 2 9
3 16 9 22 15
20821 14 2
72513 1 19
24125 18 6
11 4 17 10 23

Normally the Loubère and Méziriac methods involve 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 or Méziriac square, having numbers greater than n2. This is done by modifying a block or grid of numbers on the square. The initial number is added either at the normal Loubère sites, 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 same for the Méziriac method.

In this new method the numbers are added consecutively starting out with 1 and ending with numbers greater than n2. Whenever a break occurs a knight move is performed instead of a normal translational move (up, down or sideways). The knight move is either a (2,1) or a variable knight move , e.g (3,2), for Méziriac squares or Loubère squares. Both these square types were covered in Loubère Knight and Méziriac Knight square methods.

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 Knight Modified Loubère and Méziriac Squares

Method I: Start on the first row center (1 ⇒ (2,1) up knight break)
  1. Place 1 in the center cell of the first row.
  2. Add consecutive numbers.
  3. Knight break, two cells up one cell 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 all the columns, rows and diagonals don't sum to a magic sum.
  6. Where the sum 45 crosses corresponds to the small square in blue.
  7. Add 25 to the number 1. At this point all the numbers except for one sum to 70.
  8. Adding n2 = 25 to (3,4,5,15 and 18) gives the magic square 4. Note that all these modified numbers fall outside the blue square. In the previous non-knight Part I method all the modified numbers fall within the blue square.
  9. The magic sum is S = ½(n3 + 13n).
1
1 5
4
3
2
6
2
17 9 1 135
812 4 21
163 20 7
15219 11
61810 14
3
20
17 9 1 13545
82512 4 2170
24163 20 7 70
15219 11 2370
61810 22 1470
707045 70 7070
4
95
17 9 26 133095
82512 29 2195
241628 20 7 95
40219 11 2395
64310 22 1495
959595 95 9595
Method II: Start at lower left hand corner (1, 2, 3, 4 ⇒ (2,1) down knight break)
  1. Place 1 into the lower left hand corner.
  2. Add three consecutive numbers up to the rightmost cell.
  3. Knight break, two cells down one cell 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 all the columns, rows and diagonals don't sum to 90 the magic sum.
  6. Where the sums 60 cross corresponds to the small square in blue.
  7. One way of changing four of the numbers in the block is shown next.
  8. Adding n2 = 25 to both 1, 2, 6 and 21 (color change to light green) gives the square 4.
  9. To convert the square to the fully magic square 5 add 5 to each of the cells (for a total of 25) in the left diagonal, where each of the cells lastly modified are tan. Note that some of the modified numbers fall outside the blue square and that no duplicate numbers are introduced.
  10. The magic sum is S = ½(n3 + 11n).
1
 
4
3
25
1
2
8
10 4 7
3 6
25
19
3
35
8 11 19 222585
101821 4 760
17203 6 14 60
2425 13 1660
1912 15 2360
606060 60 8565
4
85
8 11 19 222585
101846 4 785
17203 31 14 85
24275 13 1685
26912 15 2385
858585 85 8565
5
90
13 11 19 222590
102346 4 790
17208 31 14 90
24275 18 1690
26912 15 2890
909090 90 9090
Method III: Start at lower left hand corner (1, 2, 3, 4 ⇒ (2,1) down knight break)
  1. Place 1 into the center cell of the last row .
  2. Add 2 consecutive numbers.
  3. Knight break, two cells up one cell 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 all the columns, rows and diagonals don't all sum to one number and possibly 105 is the magic sum.
  6. One way to modify is using square 3 which converts all sums except for the main diagonal into 80.
  7. On one of the parallel diagonals (which includes a cell of the left diagonal) add 25 to each of the entries (square 4). Note that some of the modified numbers fall outside the blue square.
  8. The magic square sum for square 4 is S = ½(n3 + 17n).
1
4
 
3
2
1 5
2
105
13 25 17 42180
24168 20 1280
15719 11 3 55
62310 2 1455
2291 18 555
808055 55 5555
3
105
13 25 17 42180
24168 20 1280
15719 11 28 80
62310 27 1480
22926 18 580
808080 80 8080
4
105
13 25 42 421105
24418 20 12105
40719 11 28 105
62310 27 39105
22926 43 5105
105105105 105 105105
Method IV:Start one cell right of center (1 ⇒ (2,1) up knight break)
Interconversion of a Méziriac Knight Square to a Loubère Knight Square
  1. Start out with a Méziriac type square by placing 1 one cell to the right of center.
  2. Knight break, two cells up one cell left.
  3. Repeat the process until the square is filled as shown below in squares 1-2.
  4. As shown below this square is not yet magic.
  5. Where the sums 45 cross corresponds to the small square in blue.
  6. Adding n2 = 25 to both (6,7,11,25,26) gives the magic square 4 where S = ½(n3 + 13n).
  7. To convert this Méziriac Knight Square to a Loubère Knight Square -1 is subtracted from every cell to give square 5 where S is now equal to ½(n3 + 11n).
  8. Subtraction of 25 from those cells colored in tan from square 5 gives the Loubère semi-magic Knight square 2b. This square can be generated using the New De la Loubère Method (Part II), except using a (2,1) up knight break.
1
2
6
5 1
4
3
2
45
23 15 2 191170
14618 10 2270
5179 1 13 45
21825 12 470
72416 3 2070
707070 45 7070
3
45
23 15 2 191170
14618 10 2270
5179 26 13 70
21825 12 470
72416 3 2070
707070 70 7070
4
95
23 15 2 193695
143118 10 2295
5179 51 13 95
21850 12 495
322416 3 2095
959595 95 9595
5
90
22 14 1 183590
133017 9 2190
4168 50 12 90
20749 11 395
312315 2 1990
909090 90 9090
1b
1
5
4
3
6 2
2b
40
22 14 1 181065
13517 9 2165
4168 25 12 45
20724 11 365
62315 2 1965
656565 45 6565
Square 5
Method V:Start at right of center (1, 2, 3 ⇒ (2,1) up knight break)
  1. Place 1 to the right of the center cell.
  2. For this 5x5 square add consecutive numbers 2 and 3.
  3. Knight break, two cells up one cell 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 three columns and three rows (in grey) sum to 55 instead of 80.
  6. Where the sums 55 cross corresponds to the small square in blue.
  7. Adding n2 = 25 to both (1, 2 and3) or (1, 11 and 14) gives the magic squares 3 and 4. Note that some of the modified numbers fall outside the blue square.
  8. The magic sum of both squares is S = ½(n3 + 7n).
1
3 7
6 2
5 1
4
 
2
55
3 15 7 191155
14623 10 255
5229 1 18 55
211325 17 480
122416 8 2080
558080 55 5555
3
80
28 15 7 191180
14623 35 280
5229 1 33 80
211325 17 480
122416 8 2080
808080 80 8080
+
4
80
3 15 7 193680
29623 10 280
5229 26 18 80
211325 17 480
122416 8 2080
808080 80 8080

This completes this section on the new block Loubère Method (Part IV). The next section deals with Loubère and Méziriac Knight Block Modified 7x7 Squares (Part V). To return to homepage.


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