Bordering Magic Squares


Bordered Magic Squares are consecutively concentric magic squares with the property that each concentrically inlaid square consists of consecutive integers.

For even order, there are also Bordering Magic Squares. These are concentric magic squares in which the border, instead of the inlaid squares, consists of consecutive integers. The center 4x4 square contains the smallest 8 numbers and the biggest 8. The border of each of the other concentrically inlaid squares then extends these low and high sequences of consecutive numbers. The outermost border completes the sequence with the middle numbers.

An inlaid square of order m can be converted to a normal magic square by changing only the numbers that are bigger than m²/2. Subtract the of the border order n square and add , which is equivalent to subtracting n²-m².


BorderingSquares makes these squares.

Example Squares

Order 4, 6, 8 squares

The center 4x4 square can be any aspect of 712 of the 880 order 4 squares. The main diagonals of these 712 squares have 2 numbers greater than 8 and 2 numbers less than or equal to 8. These squares transform to the centers of the higher order magic squares by adding n² - 4² to numbers greater than 8.

Bones of Above Squares

Order 4, 6, 8 bones

Note that ½ has been ommitted from all the bones numbers displayed here.
Therefore, to convert these displayed bones numbers to the actual square numbers: