There are 2 kinds of adjacent pair magic squares: adjacent corner paired and adjacent side paired.

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 /diagonals, (and alternate \diagonals), have the magic sum; and Planck's proof that there are no pandiagonal squares of singly-even order, also applies to these squares.

AdjacentCornerSquares makes these squares.

A double border method:

- start with an adjacent corner center bones of order 4
- repeatedly add double borders of adjacent corner paired bones numbers

A method attributed to Philippe de la Hire from Holger Danielsson

There are 48 order 4 adjacent corner pair magic squares.
These are **TYPE II** in the classification by
Dudeney.

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.

There are 6216 order 5 adjacent side pair magic squares.

AdjacentSideSquares makes these squares using a double border method:

- start with an adjacent side center bones of order 4, 5, 6, or 7
- repeatedly add double borders of adjacent side paired bones numbers

Danielsson, Holger "Magic Squares" https://www.magic-squares.info/en.html

Dudeney, Henry E. "Magic Square Problems"

https://archive.org/details/AmusementsInMathematicspdf

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