# 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 /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.

### Doubly-Even Order

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

#### Construction

AdjacentCornerSquares makes these squares
using the double border method:

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

## 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.

There are 6216 order 5 adjacent side pair magic squares.

#### Construction

AdjacentSideSquares makes these squares
using the 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

##### REFERENCES

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