Even Order Construction

(consecutively concentric squares)

Constraints

The bones 4(n-1) border numbers form 2(n-1) ± pairs. The + numbers are in the range:

((n-2)²+1)/2 to (n²-1)/2

The sum of each bones row, column, and main diagonal is 0.
Therefore, the border constraints are:

• the sum of row 1, (row n), is 0
• the sum of column 1, (column n), is 0
• the other rows and columns, and the main diagonals have matching ±numbers at each end

Note that the center 4x4 square is not a bordered square; its center 2x2 square is not magic.

Algorithn

The square is divided into the following regions:

• center 4x4, (special code or stored)
• NW,SE main diagonal
• NE,SW main diagonal
• NNW sector, SSW sector
• (N-E, S-E), the first column east of center
• NNE sector, SSE sector, (not including N-E, S-E)
• (W-N, E-N), the first row north of center
• WNW sector, ENE sector (not including W-N, E-N)
• (W-S, E-S), the first row south of center
• WSW sector, ESE sector, (not including W-S, E-S)

Define rrel = |row - (n+1)/2|, crel = |column - (n+1)/2|.
Then, the basic formula used here to compute a cell number is 2x² + y + c, where, for each region:

• x is rrel, or crel
• y is crel + rrel, crel - 2rel, ±rrel, ±crel, 2crel, 2rrel, or 0
• c is a constant

The algorithm is:

• assign x: rrel
• assign y: crel
• determine the region of the square
• swap x and y in WNW, W-N, ENE, E-N, WSW, W-S, ESE, E-S
• reset y to 0 in NW-SE, NE-SW
• reset y to y-2x in NNW, SSW
• reset y to y+x in NNE, SSE
• reset y to -x in (N-E, S-E) when rrel MOD 2 = ½, (W-S, E-S) when crel MOD 2 ≠ ½
• reset y to 2x in (N-E, S-E) when rrel MOD 2 ≠ ½, (W-S, E-S) when crel MOD 2 = ½
• reset y to -y in ENE, WNW
• reset y to x in E-N, W-N
• assign c:
• 0 in NW-SE, NNE, SSE
• 0 in (N-E, S-E) when rrel MOD 2 ≠ ½, (W-S, E-S) when crel MOD 2 = ½
• ½ in NNW, SSW, WNW, ENE, WSW, ESE, W-N, E-N
• ½ in (N-E, S-E) when rrel MOD 2 = ½, (W-S, E-S) when crel MOD 2 ≠ ½
• 1 in NE-SW
• compute 2x² + y + c
• SE, SW, N-E, W-S when crel ≠ 3½, E-S when crel = 3½
• W-N when crel MOD 2 ≠ ½, E-N when crel MOD 2 = ½
• NNW when crel MOD 2 = ½, SSW when crel MOD 2 ≠ ½
• NNE when crel MOD 2 ≠ ½, SSE when crel MOD 2 = ½
• WNW if rrel = 2½: when crel MOD 2 ≠ ½, if rrel = 3½: when crel MOD 2 = ½, otherwise: when rrel MOD 2 ≠ ½
• ENE if rrel = 2½: when crel MOD 2 = ½, if rrel = 3½: when crel MOD 2 ≠ ½, otherwise: when rrel MOD 2 = ½
• WSW when rrel ≠ 1½ and (rrel + crel) MOD 2 ≠ ½
• ESE when rrel = 1½ or (rrel + crel) MOD 2 = ½

Summary:

```    region          x              y             c          sign
--------    -----------    -----------     -------    ----------
NW            rrel            0             0           +
SE            rrel            0             0           -

NNW           rrel        crel-2rrel        ½           -+
SSW           rrel        crel-2rrel        ½           +-

N-E           rrel       -rrel,2rrel       ½,0          -
S-E           rrel       -rrel,2rrel       ½,0          +

NNE           rrel        crel+rrel         0           +-
SSE           rrel        crel+rrel         0           -+

NE            rrel            0             1           +
SW            rrel            0             1           -

ENE           crel          -rrel           ½      (-+)(+-)-+
WNW           crel          -rrel           ½      (+-)(-+)+-

E-N           crel           crel           ½           -+
W-N           crel           crel           ½           +-

E-S           crel       2crel,-crel       0,½         (-)+
W-S           crel       2crel,-crel       0,½         (+)-

ESE           crel           rrel           ½         (-)-+
WSW           crel           rrel           ½         (+)+-
```

Correctness

Is the algorithm correct for all even order, (> 4), squares?

It is sufficient to check the border, since the test applies recursively to the nested squares down to order 6.

Let B be the bones, m = (n+1)/2, and X = 2(1-m)² = 2(n-m)².

Rows - Singly-Even

For row 1: rrel = m-1, crel = m-1, .. ,½,½, .. ,n-m.

```NW:  B(1,1)                            =  X
NNW: B(1,m-1½) =   X + 1½ - 2(m-1) + ½ =  X - 2m + 4
NNW: B(1,m-½)  = -(X + ½ - 2(m-1) + ½) = -X + 2m - 3
N-E: B(1,m+½)  = -(X -(m-1) + ½)       = -X + m - 1½
NNE: B(1,m+1½) = -(X + 1½ + (m-1))     = -X - m - ½
NE:  B(1,n)                            =  X + 1
The sum of these 6 is 0.

The remaining NNW numbers are ±(X + (crel - 2rrel) + ½).
NNW: B(1,2) =   X + (m-2) - 2(m-1) + ½  =  X - m + ½
NNW: B(1,3) = -(X + (m-3) - 2(m-1) + ½) = -X + m + ½
…
The sum of each of these (n-6)/4 pairs is 1.

The remaining NNE numbers are ±(X + (crel + rrel)).
…
NNE: B(1,n-2) =   X + (n-2-m) + (m-1)     X + n - 3
NNE: B(1,n-1) = -(X + (n-1-m) + (m-1)) = -X - n + 2
The sum of each of these (n-6)/4 pairs is -1.
```

Row 1 total is 0 + (n-6)/4 - (n-6)/4 = 0.
Rows 2 to n-1 have ± the same numbers at each end.
Row n total is 0, (it contains ± the same numbers as row 1).

Columns - Singly-Even

For column 1: rrel = m-1, .. ,½,½, .. ,n-m, crel = m-1.

```NW:  B(1,1)                      =  X
WNW: B(m-1½,1) = -(X - 1½ + ½)   = -X + 1
W-N: B(m-½,1)  =   X + (m-1) + ½ =  X + m - ½
W-S: B(m+½,1)  = -(X + 2(m-1))   = -X -2m + 2
WSW: B(m+1½,1) =   X + 1½ + ½    =  X + 2
SW:  B(n,1)    = -(X + 1)        = -X - 1
The sum of these 6 is -(n - 6)/2.

The remaining WNW numbers are ±(X - rrel + ½)
WNW: B(2,1) = -(X - (m-2) + ½) = -X + m - 2½
WNW: B(3,1) =   X - (m-3) + ½  =  X - m + 3½
…
The sum of each of these (n-6)/4 pairs is 1.

The remaining WSW numbers are ±(X + rrel + ½)
…
WSW: B(n-2,1) = -(X + (n-2-m) + ½) = -X - n + m + 1½
WSW: B(n-1,1) =   X + (n-1-m) + ½  =  X + n - m - ½
The sum of each of these (n-6)/4 pairs is 1.
```

Column 1 total is -(n-6)/2 + (n-6)/4 + (n-6)/4 = 0.
Columns 2 to n-1 have ± the same numbers at each end.
Column n total is 0, (it contains ± the same numbers as column 1).

Rows - Doubly-Even

For row 1: rrel = m-1, crel = m-1, .. ,½,½, .. ,n-m.

```NW:  B(1,1)                           =  X
NNW: B(1,m-½) = -(X + ½ - 2(m-1) + ½) = -X + 2m - 3
N-E: B(1,m+½) = -(X + 2(m-1))         = -X - 2m + 2
NE:  B(1,n)                           =  X + 1
The sum of these 4 is 0.

The remaining NNW numbers are ±X + (crel - 2rrel) + ½
NNW: B(1,2) = -(X + (m-2) - 2(m-1) + ½) = -X + m - ½
NNW: B(1,3) =   X + (m-3) - 2(m-1) + ½  =  X - m - ½
…
The sum of each of these (n-4)/4 pairs is -1.

The remaining NNE numbers are ±(X + (crel + rrel)).
…
NNE: B(1,n-2) = -(X + (n-2-m) + (m-1)) = -X - n + 3
NNE: B(1,n-1) =   X + (n-1-m) + (m-1)  =  X + n - 2)
The sum of each of these (n-4)/4 pairs is 1.
```

Row 1 total is 0 - (n-4)/4 + (n-4)/4 = 0.
Rows 2 to n-1 have ± the same numbers at each end.
Row n total is 0, (it contains ± the same numbers as row 1).

Columns - Doubly-Even

For column 1: rrel = m-1, .. ,½,½, .. ,n-m, crel = m-1.

```NW:  B(1,1)                       =  X

These 2, if n > 8.
WNW: B(m-4½,1) =   X - 4½ + ½     =  X - 4
WNW: B(m-3½,1) =   X - 3½ + ½     =  X - 3

WNW: B(m-2½,1) = -(X - 2½ + ½)    = -X + 2
WNW: B(m-1½,1) = -(X - 1½ + ½)    = -X + 1
W-N: B(m-½,1)  = -(X + (m-1) + ½) = -X - m + ½

If n = 8:
W-S: B(m+½,1)  =   X - (m-1) + ½  =  X - m + 1½
If n > 8:
W-S: B(m+½,1)  = -(X - (m-1) + ½) = -X + m - 1½

WSW: B(m+1½,1) =   X + 1½ + ½     =  X + 2
WSW: B(m+2½,1) =   X + 2½ + ½     =  X + 3

These 2, if n > 8.
WSW: B(m+3½,1) = -(X + 3½ + ½)    = -X - 4
WSW: B(m+4½,1) =   X + 4½ + ½     =  X + 5

SW: B(n,1)    = -(X + 1)          = -X - 1
The sum of these 8 or 12 is 0.

For n > 12, the remaining numbers are:
The remaining WNW numbers are ±(X - rrel + ½).
WNW: B(2,1) =   X - (m-2) + ½  =  X - m + 2½
WNW: B(3,1) = -(X - (m-3) + ½) = -X + m - 3½
…
The sum of each of these (n-12)/4 pairs is -1.

The remaining WSW numbers are ±(X + rrel + ½).
…
WSW: B(n-2,1) = -(X + (n-2-m) + ½) = -X - n + m + 1½
WSW: B(n-1,1)     X + (n-1-m) + ½  =  X + n - m - ½
The sum of each of these (n-12)/4 pairs is 1.
```

Column 1 total is 0 - (n-12)/4 + (n-12)/4 = 0.
Columns 2 to n-1 have ± the same numbers at each end.
Column n total is 0, (it contains ± the same numbers as column 1).

Border Numbers

Are the border numbers correct, that is, in the right range with no duplicates?

There are 4(n-1) border numbers or 2(n-1) ±pairs.
There is 1 pair in each of the main diagonals.
The remaining 2(n-1) - 2 pairs are in the (NNW,SSW), (NNE,SSE), (WNW,ENE),
and (WSW,ESE) sectors.

The placing of the 2(n-1) positive numbers is:

Singly-even:

```----- (n-2)/2 numbers alternating SSW,NNW
SSW: B(n,m-½)  = X + ½ - 2(n-m) + ½     = X - n + 2
NNW: B(1,m-1½) = X + 1½ - 2(m-1) + ½    = X - n + 3
…
SSW: B(n,3)    = X + (m-3) - 2(n-m) + ½ = X - (n+2)/2
NNW: B(1,2)    = X + (m-2) - 2(m-1) + ½ = X - n/2

S-E: B(n,m+½)  = X - (n-m) + ½          = X - (n-2)/2

----- (n-4)/2 numbers alternating ENE, WNW
ENE: B(2,n)    = X - (m-2) + ½          = X - (n-4)/2
WNW: B(3,1)    = X - (m-3) + ½          = X - (n-6)/2
…
WNW: B(m-2½,1) = X - 2½ + ½             = X - 2
ENE: B(m-1½,n) = X - 1½ + ½             = X - 1

NW: B(1,1)                              = X
NE: B(1,n)                              = X + 1

----- (n-4)/2 numbers alternating WSW, ESE
WSW: B(m+1½,1) = X + 1½ + ½             = X + 2
ESE: B(m+2½,n) = X + 2½ + ½             = X + 3
…
ESE: B(n-2,n)  = X + (n-2-m) + ½        = X + (n-4)/2
WSW: B(n-1,1)  = X + (n-1-m) + ½        = X + (n-2)/2

W-N: B(m-½,1)  = X + (m-1) + ½          = X + n/2

----- (n-4)/2 numbers alternating SSE, NNE
SSE: B(n,m+1½) = X + (n-m) + 1½         = X + (n+2)/2
NNE: B(1,m+2½) = X + (m-1) + 2½         = X + (n+4)/2
…
NNE: B(1,n-2)  = X + (m-1) + (n-2-m)    = X + n - 3
SSE: B(n,n-1)  = X + (n-m) + (n-1-m)    = X + n - 2

E-S: B(m+½,n)  = X + 2(n-m)             = X + n - 1
```

Doubly-even:

```----- (n-2)/2 numbers alternating SSW,NNW
SSW: B(n,m-½)  = X + ½ - 2(n-m) + ½     = X - n + 2
NNW: B(1,m-1½) = X + 1½ - 2(m-1) + ½    = X - n + 3
…
NNW: B(1,3)    = X + (m-3) - 2(m-1) + ½ = X - (n+2)/2
SSW: B(n,2)    = X + (m-2) - 2(n-m) + ½ = X - n/2

If n = 8:
W-S: B(m+½,1)  = X - (m-1) + ½          = X - 3
If n > 8:
E-S: B(m+½,n)  = X - (n-m) + ½          = X - (n-2)/2

If n > 8:
----- (n-12)/2 + 1 numbers alternating WNW, ENE
WNW: B(2,1)    = X - (m-2) + ½          = X - (n-4)/2
ENE: B(3,n)    = X - (m-3) + ½          = X - (n-6)/2
…
WNW: B(m-4½,1) = X - 4½ + ½             = X - 4

If n > 8:
WNW: B(m-3½,1) = X - 3½ + ½             = X - 3

ENE: B(m-2½,n) = X - 2½ + ½             = X - 2
ENE: B(m-1½,n) = X - 1½ + ½             = X - 1

NW: B(1,1)                              = X
NE: B(1,n)                              = X + 1

WSW: B(m+1½,1) = X + 1½ + ½             = X + 2

----- (n-6)/2 numbers alternating WSW, ESE
WSW: B(m+2½,1) = X + 2½ + ½             = X + 3
ESE: B(m+3½,n) = X + 3½ + ½             = X + 4
…
WSW: B(n-1,1)  = X + (n-1-m) + ½        = X + (n-2)/2

E-N: B(m-½,n)  = X + (n-m) + ½          = X + n/2

----- (n-4)/2 numbers alternating SSE, NNE
SSE: B(n,m+1½) = X + 1½ + (n-m)         = X + (n+2)/2
NNE: B(1,m+2½) = X + 2½ + (m-1)         = X + (n+4)/2
…
SSE: B(n,n-2)  = X + (n-2-m) + (n-m)    = X + n - 3
NNE: B(1,n-1)  = X + (n-1-m) + (m-1)    = X + n - 2

S-E: B(n,m+½)  = X + 2(n-m)             = X + n - 1
```

The next smaller even order square is order k = n - 2.
Its biggest border number is 2(1-(k+1)/2)² + k - 1 = X - n + 1.
The smallest border number of the order n is X - n + 2.
That is, the smallest border number of the order n is 1 greater
than the biggest border number of the order n-2.