Office Applications and Entertainment, Magic Squares

14.0    Special Magic Squares, Prime Numbers

14.15   V type ZigZag

When a Magic Square contains repeated wave type patterns (see below) summing to the Magic Sum s1, it is called a V type ZigZag Magic Square.

As illustrated above the repetition can be in the directions:

- Top / Bottom with wrap around and reverse
- Left / Right with wrap around and reverse

If one of these options occurs, the Magic Square is referred to as a Two Way V type ZigZag Magic Square.

If both options are applicable, the Magic Square is referred to as a Four Way V type ZigZag Magic Square.
Such squares occur only for orders larger than 5.

14.15.1 Magic Squares (4 x 4)
        Two Way, V type ZigZag

The defining equations for a Two Way (Top/Bottom) V type ZigZag Magic Square of order 4 can be written as:

a(11) =  s1 - a(12) -   a(15) - a(16)
a( 7) = -s1 + a( 8) + 2*a(12) + a(15) + a(16)
a( 4) =  s1 - a( 8) -   a(12) - a(16)
a( 3) =  s1 - a( 8) -   a(12) - a(15)

a(14) = 0.5 * s1 - a(16)        a(9) = 0.5 * s1 - a(11)
a(13) = 0.5 * s1 - a(15)        a(5) = 0.5 * s1 - a( 7)
a(10) = 0.5 * s1 - a(12)        a(2) = 0.5 * s1 - a( 4)
a( 6) = 0.5 * s1 - a( 8)        a(1) = 0.5 * s1 - a( 3)

which illustrate the consequential symmetry (column symmetrical).

Subject equations can be incorporated in a routine to generate Two Way V type ZigZag Magic Squares (ref. ZigZag4).

Attachment 17.01 shows for miscellaneous Magic Sums the first occurring Two Way V type ZigZag Magic Square.

Each square shown corresponds with 384 (96 unique) squares for the applicable Magic Sum and variable values {a_i}.

14.15.2 Magic Squares (5 x 5)
        Two Way, V type ZigZag

The defining equations for a Two Way (Top/Bottom) V type ZigZag Magic Square of order 5 can be written as:

a( 1) =  0.6 * s1 - a( 3) - a( 5)
a( 2) =  0.4 * s1 - a( 4)
a( 3) =           2*a(13) - a(23)
a( 4) =    2 * s1 - a(10) + a(13) - a(14) - a(15) - 2 * a(19) - a(20) - a(23) - a(24) - 2 * a(25)
a( 5) =        s1 - a(10) - a(15) - a(20) - a(25)
a( 6) =  0.6 * s1 - a( 8) - a(10)
a( 8) =        s1 - 3 * a(13) - a(18)
a( 7) =  0.4 * s1 - a( 9)
a( 9) =      - s1 + a(10) - a(13) + a(15) + a(19) + a(20) + a(23) + 2 * a(25)
a(12) =  0.4 * s1 - a(14)
a(11) =  0.6 * s1 - a(13) - a(15)
a(17) =  0.4 * s1 - a(19)
a(16) =  0.6 * s1 - a(18) - a(20)
a(22) =  0.4 * s1 - a(24)
a(21) =  0.6 * s1 - a(23) - a(25)

The consequential symmetry (Columns 2 - 4) is worth to be noticed and the reason that Two Way V type ZigZag Magic Squares of the fifth order can't be Associated or Pan Magic.

Subject equations can be incorporated in a routine to generate Two Way V type ZigZag Magic Squares (ref. ZigZag5).

Attachment 17.02 shows for miscellaneous Magic Sums the first occurring Two Way V type ZigZag Magic Square.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

14.15.3 Magic Squares (6 x 6)

        Two Way V type ZigZag

Rectangular Compact, Row Symmetric, Pan Magic Squares as deducted and discussed in Section 14.4.34 are Two Way V type ZigZag.

Attachment 6.10.5 shows for miscellaneous Magic Sums the first occurring Rectangular Compact, Row Symmetric, Pan Magic Squares (ref. MgcSqr6105).

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

        Four Way V type ZigZag

The defining equations for a Four Way V type ZigZag Pan Magic Square of order 6 can be written as:

a(19) =   - s1 / 3 + a(28) + a(33) + a(35)
a(20) =   - s1 / 3 + a(29) + a(34) + a(36)
a(21) =     s1 / 6 + a(30) - a(33)
a(22) = 2 * s1 / 3 - a(27) - a(29) - a(34)
a(23) = 2 * s1 / 3 - a(28) - a(30) - a(35)
a(24) =     s1 / 6 + a(27) - a(36)
a(25) =     s1 / 2 - a(27) - a(29)
a(26) =     s1 / 2 - a(28) - a(30)
a(31) =     s1 / 2 - a(33) - a(35)
a(32) =     s1 / 2 - a(34) - a(36)

which illustrate the consequential symmetry (complete).

Subject equations can be incorporated in a routine to generate Four Way V type ZigZag Pan Magic Squares (ref. ZigZag6).

Attachment 17.03 shows for miscellaneous Magic Sums the first occurring Four Way V type ZigZag Pan Magic Square.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

        Construction Method

Four Way V type ZigZag Magic Square of order 6 can be constructed by transformation of Composed Magic Squares (ref. Section 14.4.10).

Following Composed Magic Squares can be used:

A1 Attachment 14.8.4, Simple Magic Squares, Composed of Semi Magic Sub Squares
A2 Attachment 14.8.5, Associated Magic Squares, Composed of Anti Symmetric Semi Magic Sub Squares
A3 Attachment 14.8.6, Pan Magic Complete Squares, Composed of Anti Symmetric Semi Magic Sub Squares

and will return respectively:

Four Way V type ZigZag Simple Magic Squares B1 as illustrated below for MC = 4878:

A1 (Simple) 1289 1013 137 367 1429 643 953 167 1319 613 337 1489 197 1259 983 1459 673 307 617 1049 773 1123 967 349 659 503 1277 739 463 1237 1163 887 389 577 1009 853

B1 (Simple) 1289 367 1013 1429 137 643 617 1123 1049 967 773 349 953 613 167 337 1319 1489 659 739 503 463 1277 1237 197 1459 1259 673 983 307 1163 577 887 1009 389 853

Each suitable set of Semi Magic Sub Squares corresponds with 24 * 12⁴ = 497664 Prime Number Four Way V type ZigZag Simple Magic Squares of order 6.

Four Way V type ZigZag Associated Magic Squares B2 as illustrated below for MC = 2850:

A2 (Associated) 643 709 73 127 727 571 769 43 613 541 97 787 13 673 739 757 601 67 883 349 193 211 277 937 163 853 409 337 907 181 379 223 823 877 241 307

B2 (Associated) 643 127 709 727 73 571 883 211 349 277 193 937 769 541 43 97 613 787 163 337 853 907 409 181 13 757 673 601 739 67 379 877 223 241 823 307

Each suitable set of Anti Symmetric Semi Magic Sub Squares corresponds with 8 * 12² = 1152 Prime Number Four Way V type ZigZag Associated Magic Squares of order 6.

Four Way V type ZigZag Crosswise Symmetric Magic Squares B3 as illustrated below for MC = 1638:

A3 (PM Complete) 443 239 137 353 317 149 263 383 173 419 233 167 113 197 509 47 269 503 193 229 397 103 307 409 127 313 379 283 163 373 499 277 43 433 349 37

B3 (Cross Symm) 443 353 239 317 137 149 193 103 229 307 397 409 263 419 383 233 173 167 127 283 313 163 379 373 113 47 197 269 509 503 499 433 277 349 43 37

Each suitable set of Anti Symmetric Semi Magic Sub Squares corresponds with 8 * 12² = 1152 Prime Number Four Way V type ZigZag Crosswise Symmetric Magic Squares of order 6.

Notes:
For the Associated Magic Squares B2 also the Semi Diagonals sum to the Magic Sum.
For the Crosswise Symmetric Magic Squares B3 also half of the Broken Diagonals sum to the Magic Sum.

14.15.4 Magic Squares (7 x 7)

Although Two and Four Way V type ZigZag Simple Magic Squares of order 7 do exist, this section is limited to seventh order Ultra and Associated Magic Squares.

        Two Way V type ZigZag

The defining equations for a Two Way (Top/Bottom) V type ZigZag Ultra Magic Square of order 7 can be written as:

a(44) =  3*s1/7 - a(46) - a(48)
a(43) =  4*s1/7 - a(45) - a(47) - a(49)
a(38) =    s1/7 - a(40) + a(45) - a(46) + a(47)
a(37) =  3*s1/7 - a(39) - a(41)
a(36) =  3*s1/7 - a(42) - a(45) + a(46) - a(47)
a(35) =  6*s1/7 - a(41) - a(42) - a(47) - a(48) - a(49)
a(34) = 12*s1/7 - a(39) - a(40) - 2 * a(41) - a(42) + a(45) - 2 * a(46) - a(47) - 3 * a(48) - a(49)
a(33) = 10*s1/7 - a(39) - 2 * a(40) - a(41) + a(45) - 2 * a(46) - a(47) - 2 * a(48) - a(49)
a(32) =  2*s1/7 - a(45) + a(46) - a(47)
a(31) =- 5*s1/7 + 2 * a(40) + a(41) - 2 * a(45) + 2 * a(46) + 2 * a(48) + a(49)
a(30) =-11*s1/7 + a(39) + a(40) + 2 * a(41) + a(42) + a(46) + 2 * a(47) + 3 * a(48) + a(49)
a(29) =-   s1   + a(39) + a(41) + a(42) + a(45) + 2 * a(47) + a(48) + a(49)
a(28) =- 5*s1/7 + a(39) + 2 * a(41) - a(45) + a(46) + a(47) + 2 * a(48)
a(27) =-16*s1/7 + a(39) + 2 * a(40) + 2 * a(41) + 2 * a(42) - a(45) + 2*a(46) + 3*a(47) + 4*a(48) + 2*a(49)
a(26) =-13*s1/7 + a(39) + 2 * a(40) + 2 * a(41) - a(45) + 3 * a(46) + a(47) + 4 * a(48) + 2*a(49)
a(25) =    s1/7

Subject equations can be incorporated in a routine to generate Two Way V type ZigZag Ultra Magic Squares (ref. ZigZag7a).

Attachment 17.04 shows for a few Magic Sums the first occurring Two Way V type ZigZag Ultra Magic Square.

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

        Four Way V type ZigZag

Associated Magic Squares of order 7 with Square Inlays of Order 3 and 4 - as deducted and discussed in Section 14.5.7 - are Four Way V type ZigZag.

Attachment 14.6.13 shows for miscellaneous Magic Sums the first occurring Associated Magic Square with order 3 and 4 Square Inlays (Priem7e2).

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

14.15.5 Magic Squares (8 x 8)
        Four Way V type ZigZag

        Complete Pan Magic Squares

Four Way V type ZigZag Magic Squares of order 8 can be constructed by transformation of Composed Magic Squares (ref. Section 14.6.1).

A self explanatory numerical example (Pan Magic Sub Squares) is shown below for MC = 3360:

A 71 89 733 787 83 97 739 761 691 829 29 131 673 827 17 163 107 53 769 751 101 79 757 743 811 709 149 11 823 677 167 13 227 193 599 661 293 337 443 607 439 821 67 353 349 701 199 431 241 179 613 647 397 233 547 503 773 487 401 19 641 409 491 139

B 71 83 89 97 733 739 787 761 227 293 193 337 599 443 661 607 691 673 829 827 29 17 131 163 439 349 821 701 67 199 353 431 107 101 53 79 769 757 751 743 241 397 179 233 613 547 647 503 811 823 709 677 149 167 11 13 773 641 487 409 401 491 19 139

If the 4 Sub Squares of Square A are Pan Magic, the resulting Square B will be Pan Magic, Complete and 4 x 4 Compact.

Each suitable set of Pan Magic Sub Squares corresponds with 24 * 384⁴ = 0,5 10¹² Prime Number Four Way V type ZigZag Pan Magic Squares of order 8.

17.15.6 Magic Squares (9 x 9)
        Four Way V type ZigZag

Associated Magic Squares of order 9 with Square Inlays of Order 4 and 5 - as deducted and discussed in Section 14.7.13 - are Four Way V type ZigZag.

Attachment 14.7.12 shows miscellaneous Associated Magic Squares with order 4 and 5 Square Inlays, based on transformation of Composed Magic Squares (Priem9f3).

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

14.15.7 Magic Squares (10 x 10)
        Four Way V type ZigZag

Four Way V type ZigZag Magic Square of order 10 can be constructed by transformation of Composed Magic Squares (ref. Section 14.8.13).

Following Composed Magic Squares can be used:

A1 Attachment 14.8.66, Associated Magic Squares, Composed of Anti Symmetric Magic Sub Squares
A2 Attachment 14.8.67, Pan Magic Complete Squares, Composed of Anti Symmetric Magic Sub Squares

and will return respectively:

Four Way V type ZigZag Associated Magic Squares B1 as illustrated below for MC = 26950:

Each suitable set of Anti Symmetric Magic Sub Squares corresponds with numerous Prime Number Four Way V type ZigZag Assiciated Magic Squares of order 10.

Four Way V type ZigZag Croswise Symmetric Magic Squares B2 as illustrated below for MC = 26950:

Each suitable set of Anti Symmetric Magic Sub Squares corresponds with numerous Prime Number Four Way V type ZigZag Crosswise Symmetric Magic Squares of order 10.

Notes:
For the Associated Magic Squares B1 also the Semi Diagonals sum to the Magic Sum.
For the Crosswise Symmetric Magic Squares B2 also half of the Broken Diagonals sum to the Magic Sum.

14.15.8 Magic Squares (11 x 11)
        Four Way V type ZigZag

Associated Magic Squares of order 11 with Square Inlays of Order 5 and 6 - as deducted and discussed in Section 14.9.9 - are Four Way V type ZigZag.

Attachment 14.8.16 shows miscellaneous Associated Magic Squares with order 5 and 6 Square Inlays, based on transformation of Composed Magic Squares (Prime11c1).

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

14.15.9 Magic Squares (13 x 13)
        Four Way V type ZigZag

Associated Magic Squares of order 13 with Square Inlays of Order 6 and 7 - as deducted and discussed in Section 14.10.4 - are Four Way V type ZigZag.

Attachment 14.10.6 shows miscellaneous Associated Magic Squares with order 5 and 6 Square Inlays, based on transformation of Composed Magic Squares (Priem13c).

Each square shown corresponds with numerous squares for the applicable Magic Sum and variable values {a_i}.

14.16   Summary

The obtained results regarding the miscellaneous types of ZigZag Magic Squares as deducted and discussed in previous sections are summarized in following table:

Comparable routines as listed above, can be used to generate Bent Diagonal Magic Squares, which will be described in following sections.