 ## 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: ## 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: 