A New Procedure for Magic Squares

A Loubère Type Method - The Slant Break

A stairs

A Discussion of the New Method

Magic squares such as the Loubère have a center cell which must always contain the middle number of a series of consecutive numbers, 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.

In this method the numbers on the square are added by the staircase method starting at (one down, one right) to the center cell. The Loubère method is applied (the staircase method). When a break is encountered the move is (2 down, 2 right), i.e., a 2 down slant. The result is the generation of a Loubère type square different from the standard Loubère.

In addition, those numbers divisible by three will not be magic but will give triads of sums, viz. for a 9 x9 this triad is (360,369,379). To convert this square to magic we can use the mask method as shown in Cross and Mask Methods. The sums of these squares is also shown to follow the new sum equation as was shown in the New block Loubère Method:

S = ½(n3 ± an)

Construction of a 5x5 Slant Break Magic Square

  1. Construct Square 1 by adding consecutive numbers at position (one down, one right) to the center cell. On reaching 5 break (2 down,2 right) and continue with 6.
  2. Fill in the rest of the cell in like manner.
1
4  
3 6
   2
1
5
2
7 4 24 1815
32517 14 6
241613 10 2
20129 1 23
1185 22 19

Construction of a slant break 9x9 Magic Square

Use of a mask
  1. Construct Square 1 by adding consecutive numbers at position (one down, one right) to the center cell. On reaching 5 break (2 down,2 right) and continue with 10.
  2. Fill in the rest of the cell in like manner (Squares 2 and 3).
  3. At this point not all rows sum to 369.
  4. 1
    6 18
    517
    16 19 4
    15 3
    14 2
    13 1
    9 12
    8 11
    7 10
    2
    21 6 3318
    53217 20
    3116 19 4
    3015 27 3
    29 14 26 2
    13 25 1 28
    249 36 12
    238 35 11
    37 22 7 34 10
    3
    369
    21 6 7248 3318 756045378-9
    571473217 74 5944 203690
    7046311673 58 4319 43609
    5430158157 4227 369378-9
    29 14 805641 26 268 533690
    13 79 554025 1 6752 283609
    78 63 39249 66 5136 12378-9
    62 38 23865 5035 11773690
    37 22 76449 34 1076 613609
    369369369 369 369369 369 369369 369
  5. Adjust the values in the center row by adding and subtracting the values in the last columns to generate 4. At this point fourduplicates (in light orange) have been generated.
  6. Generate a mask using n2 as a factor to be added to the appropriate cell of square 5. Since two duplicates are present in columns 4 and 6, two n2 factors must be present per row, column and diagonal. In this case to the center cell is added 2n2.
  7. 4
    369
    21 6 7248 2418 756045369
    571473217 74 5944 20369
    7046311682 58 4319 4369
    5430158148 4227 369369
    29 14 805641 26 268 53369
    13 79 554034 1 6752 28369
    78 63 39240 66 5136 12369
    62 38 23865 5035 1177369
    37 22 76458 34 1076 61369
    369369369 369 369369 369 369369 369
    +
    Mask B
    81 81
    8181
    81 81
    81 81
    162
    81 81
    81 81
    81 81
    81 81
  8. Addition of these factors to square 4 gives square 85 with S = 531 = ½(n3 + 37n) and the modification of 17 numbers.
  9. 5
    531
    21 6 72129 2418 7514145531
    86711283217 74 5944 20531
    7046311682 139 43100 4531
    13530158148 42108 369531
    29 14 8056203 26 268 53531
    13 79 1364034 1 6752 109531
    78 144 391050 66 5136 12531
    62 38 23865 50116 11158531
    37 103 76458 115 1076 61531
    531531531 531 531531 531 531531 531

    This completes this section on a new slant break Loubère Type 5x5 and 9x9 Mask-Generated Methods. To return to homepage.


    Copyright © 2010 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com