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

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:

x_{0}, x_{2}, x_{4}, x_{6}, x_{8}, x_{10}, x_{12}, x_{14}x_{1}, x_{3}, x_{5}, x_{7}, x_{9}, x_{11}, x_{13}, x_{15}y_{0}, y_{2}, y_{4}, y_{6}, y_{8}, y_{10}, y_{12}, y_{14}y_{1}, y_{3}, y_{5}, y_{7}, y_{9}, y_{11}, y_{13}, y_{15}

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

1. x_{0}= y_{0}= 0 2. x_{0}< x_{2}< x_{4}< x_{6}, x_{1}< x_{3}< x_{5}< x_{7}, x_{8}< x_{10}< x_{12}< x_{14}, x_{9}< x_{11}< x_{13}< x_{15}3. y_{0}< y_{2}< y_{4}< y_{6}, y_{1}< y_{3}< y_{5}< y_{7}, y_{8}< y_{10}< y_{12}< y_{14}, y_{9}< y_{11}< y_{13}< y_{15}4. x_{0}+ x_{2}+ x_{4}+ x_{6}= x_{1}+ x_{3}+ x_{5}+ x_{7}= x_{8}+ x_{10}+ x_{12}+ x_{14}= x_{9}+ x_{11}+ x_{13}+ x_{15}5. y_{0}+ y_{2}+ y_{4}+ y_{6}= y_{1}+ y_{3}+ y_{5}+ y_{7}= y_{8}+ y_{10}+ y_{12}+ y_{14}= y_{9}+ y_{11}+ y_{13}+ y_{15}6. x_{1}< x_{9}7. y_{1}< y_{9}

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.:
**
(x _{2} + x_{4} + x_{6}) + (y_{2} + y_{4} + y_{6}) = MagicSum/4 = 510**.

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**, (**1.76×10 ^{12}**), 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.

**Ultramagic**
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**Adjacent Side Paired**
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