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A 20x20 Magic Square

Scroll down to see a magic square
in which all rows, columns, and both diagonals
add up to the same magic sum (4010)

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 20 by 20 magic square shown below has a magic sum equal to 4010.

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 20x20 Magic Square
with a Magic Sum of 4010

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 20x20 blank grid and color in the two main diagonals.
  2. Also, color in the two main diagonals of each 4x4 box within this 20x20 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 400 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 400, but this time, only write down the numbers in the non-shaded squares.

Step One: Draw a 20x20 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 (400).

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

This time, you can either
  • Start counting backward, starting with 400 (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 400 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 400 is already written in that square), write down 2 and 3, then skip 4, etc.
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A 20-by-20 Magic Square

(Magic Sum = 4010)

139939845395394893913901213387386161738338220
38022233773762627373372303136936834353653643839361
36042433573564647353352505134934854553453445859341
61339338646533533468693313307273327326767732332280
813193188485315314888931131092933073069697303302100
300102103297296106107293292110111289288114115285284118119281
280122123277276126127273272130131269268134135265264138139261
141259258144145255254148149251250152153247246156157243242160
161239238164165235234168169231230172173227226176177223222180
220182183217216186187213212190191209208194195205204198199201
200202203197196206207193192210211189188214215185184218219181
221179178224225175174228229171170232233167166236237163162240
241159158244245155154248249151150252253147146256257143142260
140262263137136266267133132270271129128274275125124278279121
120282283117116286287113112290291109108294295105104298299101
30199983043059594308309919031231387863163178382320
32179783243257574328329717033233367663363376362340
60342343575634634753523503514948354355454435835941
40362363373636636733323703712928374375252437837921
3811918384385151438838911103923937639639732400
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