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Other 4N+0 Magic Square Sizes:
Detailed Directions for: 4x4 * 8x8 * 12x12 *
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* 16 * 20 * 24 * 28 * 32 * 36 * 40 * 44 * 48 * 52 * 56 *
* 60 * 64 * 68 * 72 * 76 * 80 * 84 * 88 * 92 * 96 *

A 16x16 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (2056)

A Magic Square is a square of numbers in which every row, every column, and both diagonals add up to the same number. This number is often called the magic sum. The 16 by 16 magic square shown below has a magic sum equal to 2056.

Though magic squares can be made with non-consecutive and non-regular sequences, they are usually seen made up of consecutive numbers, possibly because it is harder to be consecutive.

The method described below can be used to create all 4N+0 (multiple-of-four) magic squares. This is just one method for creating magic squares, there are many other methods, and many-many-many magic squares to be found.

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How to Create a 16x16 Magic Square
with a Magic Sum of 2056

The method described on this page can be used to create 4x4, 8x8, 12x12, 16x16 and larger 4N+0 magic squares

Steps:

  1. Draw a 16x16 blank grid and color in the two main diagonals.
  2. Also, color in the two main diagonals of each 4x4 box within this 16x16 square.
  3. Starting in the top left corner, moving left to right through the rows, count the squares, 1, 2, 3, 4, but only write down the numbers you say when you get to one of the colored in squares. If you do this properly, you will write 1 in the top left corner and write 256 in the bottom right corner.
  4. Next, do the same thing, starting in the bottom right corner, moving right-to-left upward through the rows, again count from 1 to 256, but this time, only write down the numbers in the non-shaded squares.

Step One: Draw a 16x16 blank grid and
shade in the two main diagonals

You don't necessarily have to shade in the squares. You could simply draw a light line that passes through these squares. You only need to be able to tell the difference between squares that are on these 4x4 diagonals and those that are not.

Step Two: Shade in all the 4x4 diagonals

If you are making a square larger than 4x4, the entire square can be visualized as a collection of 4x4 squares. Shade in the main diagonals of each 4x4 square within the big square.

If you do this correctly, by the time you are done, you will have shaded in exactly half the squares in your overall grid. If you had simply drawn diagonal lines through those squares, you would see a series of X's.

Step Three: Count the squares
(writing only on the shaded squares)

This is fairly straightforward. Starting in the top left corner, and moving left-to-right through each row, count the squares, but only write down the numbers in the shaded squares, but remember to count all the blank squares. In the first four spaces, you would write down 1, count 2 and 3, and write down 4. Continue until you have written down the last number (256).

Step Four: Count the squares again, counting backward,
writing into the blank squares

This time, you can either
  • Start counting backward, starting with 256 (which you would not write down) and count downward, writing down the number whenever you come to a blank square. The last two numbers you write down would be 3 and 2 just before you get to the 256 which you wrote down in Step Three.
  • Or, if you prefer counting upward, start with 1 in the bottom right corner and move right-to-left through each row until you get to the top left corner. Working this way, you will skip 1 (because 256 is already written in that square), write down 2 and 3, then skip 4, etc.
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A 16-by-16 Magic Square

(Magic Sum = 2056)

12552544525125089247246121324324216
2401819237236222323323226272292283031225
2243435221220383921721642432132124647209
4920720652532032025657199198606119519464
6519119068691871867273183182767717917880
1768283173172868716916890911651649495161
1609899157156102103153152106107149148110111145
113143142116117139138120121135134124125131130128
129127126132133123122136137119118140141115114144
11214614710910815015110510415415510110015815997
9616216393921661678988170171858417417581
1777978180181757418418571701881896766192
1936362196197595820020155542042055150208
4821021145442142154140218219373622222333
3222622729282302312524234235212023823917
241151424424511102482497625225332256
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