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8.8 Concentric and Eccentric Magic Squares
In general an even concentric magic square consists of a centre of 2 x 2 cells, around which borders can be constructed again
and again.
With the inner 4 x 4 square pan magic, the embedded Magic Squares can be described by following linear equations:
a(50) = 195 - a(51) - a(52) - a(53) - a(54) - a(55)
which can be completed with the equations describing the outer border, which results in following linear equations:
a(57) = 260 - a(58) - a(59) - a(60) - a(61) - a(62) - a(63) - a(64)
Note: The Embedded Magic Square is based on the consecutive integers 15, 16, ... 50.
With the exterior border variables constant, an optimized guessing routine (MgcSqr8i), might produce the 4,54 109 possible Magic Squares, of which 1728, based on one interior Magic Square of the 4th order and the variables a(23) and a(31) constant, are shown in Attachment 8.6.1.
8.8.2 Concentric Magic Squares (2)
Alternatively the border of an 8th order Concentric Magic Square can be described by following equations:
a(58) = 260 - a(57) - a(59) - a(60) - a(61) - a(62) - a(63) - a(64)
The resulting solutions will be unique when following conditions are added to the equations listed above:
a( 8) < a(57) < a(64) prevent rotation and reflection
An optimized guessing routine (MgcSqr8i2) counted,
based on the integers 1 ... 14 and 51 ... 64, 28490 unique borders in 138 seconds.
Also other Magic Squares of the 6th order, as described and constructed in
Section 6.2 thru Section 6.11,
can be used as Center Squares for 8th order Bordered Magic Squares.
Attachment 8.6.4 contains
- based on some of the described Magic Squares of order 6 - examples of Bordered Magic Squares for the first occurring border.
8.8.4 Eccentric Magic Squares (1)
An Eccentric Magic Square can be defined as a Magic Corner Square of order n, supplemented with two or more (i) rows and columns to a Magic Square of order (n + i).
Rather than starting with the equations of the Magic Corner Square, the equations of the supplementary rows and columns can be used as a starting point for the generation of Eccentric Magic Squares.
The supplementary rows and columns can be described by following linear equations:
a( 1) + a( 2) + a( 3) + a( 4) + a( 5) + a( 6) + a( 7) + a( 8) = 260
Which can be reduced, by means of row and column manipulations, to:
a(57) = 65 - a(58)
Note: The Magic Corner Square is based on the consecutive integers 15, 16, ... 50.
It is obvious that the number of Eccentric Magic Squares is determined by the sum (s2) of the values of the key variables
a(43), a(36), a(29) and a(22).
8.8.5 Eccentric Magic Squares (2)
Alternatively the supplementary rows and columns - further referred to as 'border' -
of an 8th order Eccentric Magic Square can be described by following equations:
a( 8) = 260 - a(15) - s2 - a(50) - a(57)
The resulting solutions will be unique when following conditions are added to the equations listed above:
a( 2) < a( 9) prevent mirroring around diagonal a(1) ... a(64)
An optimized guessing routine (MgcSqr8j2) produced,
based on the integers 1 ... 14 and 51 ... 64, while varying s2, 83756336 unique borders.
The breakdown of these borders as a function of s2 is shown in the graph below.
The actual number of borders Nb can be obtained by multiplication with 2 * (4!)2 (= 1152).
On Holger Danielsson's site I found following 8th order Inlaid Magic Square:
This 8th order Magic Square is composed out of an Associated Border and
four each 3th order Magic Center Squares A, B, C and D
with Magic Sums s(1) = 87, s(2) = 111, s(3) = 84 and s(4) = 108.
s(1) = 3 * s8 / 4 - s(4)
With s8 = 260 the Magic Sum of the 8th order Inlaid Magic Square.
Based on the general equations defining a Magic Square of the third order: a'(7) = s1' - a'(8) - a'(9) a'(6) = 4 * s1' / 3 - a'(8) - 2 * a'(9) a'(5) = s1' / 3 a'(4) = 2 * s1' / 3 - a'(6) a'(3) = 2 * s1' / 3 - a'(7) a'(2) = 2 * s1' / 3 - a'(8) a'(1) = 2 * s1' / 3 - a'(9) and the distinct integers 1 ... 64, 35280 Magic Squares of order 3 can be found (= 4410 Unique Squares) with 56 different Magic Sums (s1' = 15 ... 180). The frequency of the Magic Sums is summarised in the graph below (Unique Squares): The Associated Border can be described by following linear equations: a(60) = a(61) - s(3) + s(4) a(59) = a(62) - s(3) + s(4) a(58) = a(63) - s(3) + s(4) a(57) = s8 - 2*a(61) - 2*a(62) - 2*a(63) - a(64) + 3*s(3) - 3*s(4) a(41) = s8 - a(48) - s(3) - s(4) a(40) = 2 * s8 - a(48) - a(56) - a(61) - a(62) - a(63) - a(64) - 3*s(4) a(33) = s8 - a(40) - s(3) - s(4) a(32) = s8/4 - a(33) a(25) = - s8/2 - a(32) + s(3) + s(4) a(24) = -3 * s8/4 + a(48) + s(3) + s(4) a(17) = s8/4 - a(48) a(16) = -3 * s8/4 + a(56) + s(3) + s(4) a( 9) = s8/4 - a(56) a( 8) = s8/4 - a(57) a( 7) = s8/4 - a(63) + s(3) - s(4) a( 6) = s8/4 - a(62) + s(3) - s(4) a( 5) = s8/4 - a(61) + s(3) - s(4) a( 4) = s8/4 - a(61) a( 3) = s8/4 - a(62) a( 2) = s8/4 - a(63) a( 1) = s8/4 - a(64)
Which can be incorporated in an optimised guessing routine MgcSqr8k, together with the defining equations of the four 3th order inlays.
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s(3) s(4) n3 n8 Total s(3) s(4) n3 n8 Total 54
57
81
84
84105
102
111
108
1144096
4096
4096
4096
4096144
144
144
1008
144589824
589824
589824
4128768
58982487
87
90
90
93105
111
108
141
1384096
4096
4096
4096
4096144
1008
144
144
144589824
4128768
589824
589824
589824
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which results for the Inlaid Magic Squares as shown in Attachment 8.6.14 in 4 * 12976128 = 103809024
solutions.
The linear equations deducted above, have been applied in following Excel Spread Sheets:
Only the red figures have to be “guessed” to construct one of the applicable 8th order Magic Squares (wrong solutions are obvious).
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