Adjacent Pair Magic Squares

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

Adjacent Corner Pair Magic Squares

Description

These squares have complementary numbers that are all adjacent corner paired.

Odd Order

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!

Singly-Even Order

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.

Doubly-Even Order

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

Construction

AdjacentCornerSquares makes these squares using the double border method:

Adjacent Side Pair Magic Squares

Description

These squares have complementary numbers that are all adjacent side paired.

Orders

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.

Construction

AdjacentSideSquares makes these squares using the double border method:

REFERENCES

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