These squares have complementary numbers that are all adjacent corner paired.
There are no odd order adjacent corner paired magic squares.
Consider an aspect of the square in which the top 2 rows do not contain the middle value, (n²+1)/2. Going from left to right, the complement of cell (1,i) must be (2,i+1) for i = 1, 3, .., n. But then there is no place for the complement of cell (1,n). Similarly, going from right to left, there is no place for the complement of cell (1,1).
If wrap-around is allowed, squares are possible. With wrap-around, there are just 2 order 5 adjacent corner pair magic squares!
There are no adjacent corner pair magic squares of singly-even order.
Consider the top 2 rows. Going from left to right, the complement of cell (1,i) must be (2,i+1) for i = 1, 3, .., n-1. Then, going from right to left, the complement of cell (1,i) must be (2,i-1) for i = n, n-2, .., 2. Similarly for the remainder of the rows.
So, alternate forward diagonals, (and alternate back diagonals), have the magic sum; and Planck's proof that there are no pandiagonal squares of singly-even order, also applies to these squares.
There are 48 order 4 adjacent corner pair magic squares. These are TYPE II in the classification by Dudeney. See references.
AdjacentCornerSquares makes these squares. The squares are made by starting with a bones of order n = 4 as center and building a bones of size n + 4k, k≥1, around it. Double borders of adjacent corner paired cells are added repeatedly up to the required order.
These squares have complementary numbers that are all adjacent side paired.
There are squares for all orders greater than 3.
There are 96 order 4 adjacent side pair magic squares. These are TYPE IV in the classification by Dudeney. See references.
There are 6216 order 5 adjacent side pair magic squares.
AdjacentSideSquares makes these squares. The squares are made by starting with a bones of order n = 4, 5, 6, or 7 as center and building a bones of size n + 4k, k≥1, around it. Double borders of adjacent side paired cells are added repeatedly up to the required order.
Dudeney, Henry E. "Magic Square Problems"
http://www.web-books.com/Classics/Books/B0/B873/AmuseMathC14P1.htm
Dudeney, Henry E. "Magic Square Problems"
http://www.scribd.com/doc/49756911/Amusments-in-Mathematics, page 287.
Heinz, Harvey "Order 4 Magic Squares"
http://www.magic-squares.net/order4list.htm#The 12 Groups
Planck, C. "PANDIAGONAL MAGICS OF ORDERS 6 AND 10 WITH MINIMAL NUMBERS."
http://www.archive.org/stream/monistquart29hegeuoft#page/306/mode/2up