sum = 3[(9+1)/2]{\displaystyle 3[(9+1)/2]} sum = 3(10/2){\displaystyle 3(10/2)} sum = 3(5){\displaystyle 3(5)} sum = 15 Hence, the magic constant for a 3×3 square is 15. All rows, columns, and diagonals must add up to this number.

If the movement takes you to a “box” above the magic square’s top row, remain in that box’s column, but place the number in the bottom row of that column. If the movement takes you to a “box” to the right of the magic square’s right column, remain in that box’s row, but place the number in the furthest left column of that row. If the movement takes you to a box that is already occupied, go back to the last box that has been filled in, and place the next number directly below it.

A singly even square has a number of boxes per side that is divisible by 2, but not 4. [2] X Research source The smallest possible singly even magic square is 6×6, since 2×2 magic squares can’t be made.

sum = [6(62+1)]/2{\displaystyle [6(62+1)]/2} sum = [6(36+1)]/2{\displaystyle [6(36+1)]/2} sum = (6(37))/2{\displaystyle (6(37))/2} sum = 222/2{\displaystyle 222/2} sum = 111 Hence, the magic constant for a 6×6 square is 111. All rows, columns, and diagonals must add up to this number.

So, for a 6x6 square, each quadrant would be 3×3 boxes.

In the example of a 6×6 square, Quadrant A would be solved with the numbers from 1-9; Quadrant B with 10-18; Quadrant C with 19-27; and Quadrant D with 28-36.

Treat the first number of each quadrant as though it is the number one. Place it in the center box on the top row of each quadrant. Treat each quadrant like its own magic square. Even if a box is available in an adjacent quadrant, ignore it and jump to the “exception” rule that fits your situation.

Using a pencil, mark all the squares in the top row until you read the median box position of Quadrant A. So, in a 6×6 square, you would only mark Box 1 (which would have the number 8 in it), but in a 10×10 square, you would mark Boxes 1 and 2 (which, in that case, would have the numbers 17 and 24 in them, respectively). Mark out a square using the boxes you just marked as the top row. If you only marked one box, your square is just that one box. We’ll call this area Highlight A-1. So, in a 10×10 magic square, Highlight A-1 would consist of Boxes 1 and 2 in Rows 1 and 2, creating a 2×2 square in the top left of the quadrant. In the row directly below Highlight A-1, skip the number in the first column, then mark as many boxes across as you marked in Highlight A-1. We’ll call this middle row Highlight A-2. Highlight A-3 is a box identical to A-1, but placed in the bottom left corner of the quadrant. Highlight A-1, A-2, and A-3 together comprise Highlight A. Repeat this process in Quadrant D, creating an identical highlighted area called Highlight D.

Here are two images of a 14×14 Magic Square before and after doing both swaps. Quadrant A swap area is highlighted blue, Quadrant D swap area is highlighted green, Quadrant C swap area is highlighted yellow, and Quadrant B swap area is highlighted orange. 14×14 Magic Square before making swaps (steps 6, 7, & 8) 14×14 Magic Square after making swaps (steps 6, 7, & 8)

The smallest doubly-even box that can be made is a 4×4 square.

sum = [4(42+1)]/2{\displaystyle [4(4^{2}+1)]/2} sum = [4(16+1)]/2{\displaystyle [4(16+1)]/2} sum = (4(17))/2{\displaystyle (4(17))/2} sum = 68/2{\displaystyle 68/2} sum = 34 Hence, the magic constant for a 4×4 square is 68/2, or 34. All rows, columns, and diagonals must add up to this number.

In a 4x4 square, you would simply mark the four corner boxes. In an 8x8 square, each Highlight would be a 2x2 area in the corners. In a 12x12 square, each Highlight would be a 3x3 area in the corners, and so on.

In a 4x4 square, the Central Highlight would be a 2x2 area in the center. In an 8x8 square, the Central Highlight would be a 4x4 area in the center, and so on.

1 in the top-left box and 4 in the top-right box 6 and 7 in the center boxes in Row 2 10 and 11 in the center boxes in Row 3 13 in the bottom-left box and 16 in the bottom-right box.

15 and 14 in the center boxes in Row 1 12 in the left-most box and 9 in the right-most box in Row 2 8 in the left-most box and 5 in the right-most box in Row 3 3 and 2 in the center boxes in Row 4 At this point, all your columns, rows, and diagonals should up to your magic constant you calculated.