# Order 16 Franklin Squares

## Description

A great many order 16 principal Franklin magic and semimagic squares can be made from principal reversible squares by the method described below.

## From Reversible

Below are numbers of squares made as outlined by Hurkens in the search for order 12 Franklin squares, except that here the first row and first column tuples:

```  x0, x2, x4, x6, x8, x10, x12, x14    x1, x3, x5, x7, x9, x11, x13, x15

y0, y2, y4, y6, y8, y10, y12, y14    y1, y3, y5, y7, y9, y11, y13, y15
```

are those that can be made from numbers in the first row and first column of the principal reversible squares. The constraints are:

```  1. x0 = y0 = 0

2. x0 < x2  < x4  < x6,   x1 < x3  < x5  < x7,
x8 < x10 < x12 < x14,  x9 < x11 < x13 < x15

3. y0 < y2  < y4  < y6,   y1 < y3  < y5  < y7,
y8 < y10 < y12 < y14,  y9 < y11 < y13 < y15

4. x0 + x2  + x4  + x6  = x1 + x3  + x5  + x7 =
x8 + x10 + x12 + x14 = x9 + x11 + x13 + x15

5. y0 + y2  + y4  + y6  = y1 + y3  + y5  + y7 =
y8 + y10 + y12 + y14 = y9 + y11 + y13 + y15

6. x1 < x9

7. y1 < y9
```

For the square to be magic, a sum from constraint 4 plus a sum from constraint 5 above equals 1/4 of the magic sum, e.g.: (x2 + x4 + x6) + (y2 + y4 + y6) = MagicSum/4 = 510.

### Squares

The principal reversible squares are numbered in ascending order of their Frénicle standard form. Each principal Franklin square can be transformed into 1,761,205,026,816, (1.76×1012), distinct Franklin squares. Note: Numbers of semimagic squares are exclusive of magic squares.

For each principal reversible square there are 16 principal Franklin ultramagic squares, (total 560). There are a total of 673,740 principal Franklin adjacent side paired magic squares and 7,186,560 principal Franklin adjacent side paired semimagic squares.

### Examples

Ultramagic .txt   