# Introduction – Properties of a Magic Square

Magic squares are made in 3×3, 4×4, 5×5 , etc. A 3×3 magic square is filled with numbers 1 to 9 ( 9 = 3×3), a 4×4 magic square with 1 to 16 ( 16 = 4×4 ) and a 5×5 magic square with 1 to 25 etc. In general a ‘n’ x ‘n’ magic square is filled with 1 to n^{2} numbers.

It will have ‘n’ rows and ‘n’ columns and totally there are n^{2} small squares.

After filling all the numbers, all the VERTICAL, HORIZONTAL AND BOTH THE DIAGONALS’ TOTALS ARE EQUAL. And this total will be {(1+n^{2})/2}*n.

Generally 1 to n^{2} numbers are used to form the magic squares. In special cases we can use fractions, zero, negative numbers and squares, cubes also

We can make 3×3, 4×4, 5×5 etc. magic squares for a particular given number like your birth year and for a given interval of numbers also.

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## How to make a 3 X 3 Magic Square ?

1. Draw an empty 3×3 square. It is the boundary for 3×3 magic square

2. Put number ‘1’ in the upper middle square

3.Always write the next number one square right and one square above, AND WHEN IT GOES

A) Beyond boundary in the upper side, write it vertically downmost square

B) beyond the boundary in the right side, write it in the horizontally leftmost square

C) When the square is already filled with a number , just step down one square vertically and write the number

4. and after the top row right square always write the next number one square vertically below

5. With these simple techniques you can make any ODD NUMBER MAGIC SQUARE like 5×5, 7×7..

# 3×3 Magic square: Total 15

( 1+ 9 = 10/2 = 5×3 = 15 )

8 | 1 | 6 |

3 | 5 | 7 |

4 | 9 | 2 |

# 3×3 Magic square Total 2010

673 | 666 | 671 |

668 | 670 | 672 |

669 | 674 | 667 |

# 5×5 Magic square Total = 65

( 1+25 = 26/2 = 13×5 = 65 )

17 | 24 | 1 | 8 | 15 |

23 | 5 | 7 | 14 | 16 |

4 | 6 | 13 | 20 | 22 |

10 | 12 | 19 | 21 | 3 |

11 | 18 | 25 | 2 | 9 |

## A 5×5 Magic square with magic sum 2010

406 | 413 | 390 | 397 | 404 |

412 | 394 | 396 | 403 | 405 |

393 | 395 | 402 | 409 | 411 |

399 | 401 | 408 | 410 | 392 |

400 | 407 | 414 | 391 | 398 |

## How to make a 4 X 4 Magic Square ?

1. Draw a 4×4 empty square.

2. Draw the two diagonals.

3. Count all squares from the top left towards the right. (Total 16)

4. Now start writing numbers 1 to 16 from the top left square . Count all squares , and write numbers only in the blank squares. Do not write numbers in the squares with diagonal lines. (2,3,5,8,9,12,14,15 are to be filled)

5. Now start writing the numbers from the bottom most right square towards left, count all squares BUT WRITE NUMBERS ONLY IN THE SQUARES HAVING DIAGONAL LINES ONLY. (1,4,6,7,10,11,13,16 are to be written.

6. Magic square 4×4 ready . All vertical, Horizontal and both diagonals are equal.

7. This method will be applicable for 4×4 only.

4×4 Total 2010

510 | 496 | 497 | 507 |

499 | 505 | 504 | 502 |

503 | 501 | 500 | 506 |

498 | 508 | 509 | 495 |

## Any Order – Odd

This procedure is same as that of making a 3 X 3 magic square, except that we start with a n X n square initially and start filling numbers following the same rules that we followed for making a 3 X 3 Magic Square. As a sample filling a 5 X 5 Magic Square has been shown below

17 | 24 | 1 | 8 | 15 |

23 | 5 | 7 | 14 | 16 |

4 | 6 | 13 | 20 | 22 |

10 | 12 | 19 | 21 | 3 |

11 | 18 | 25 | 2 | 9 |

## Any Order – Even

As such there is no known, GENERAL methodology to obtain an even order magic square upto any order.

## Word Magic Square

As such there is no known, GENERAL methodology to obtain an even order magic square upto any order. Anyhow you can make these magic squares with trial and error methods.

An example is given to you to get an idea.

Total 45

25 | 2 | 18 |

8 | 15 | 22 |

12 | 28 | 5 |

In the above 3 x 3 magic square, all totals are 45. Take the upper middle square, 2 (TWO) written in alphabhats will have three letters and in the corresponding square shown down you have to write the number three. Likewise all numbers are to be written and according to the alphabhats available. AMAZINGLY THIS WILL ALSO BE A MAGIC SQUARE of total 21.

Total 21

10 | 3 | 8 |

5 | 7 | 9 |

6 | 11 | 4 |

## OTHER SUCH INTERESTING Magic Squares

As such there is no known, GENERAL methodology to obtain any other Magic Sqaure

1. Diamond magic square,

2. Swastika magic square,

3. Spell magic square,

4. Date of birth magic square,

5. Biography magic square,

6. Reverse magic square,

7. Upside down magic square,

8. Mirror magic square,

9. 180^{0} turned Magic square,

10. Palindrome Magic square,

11. Magic squares inside a magic square

12. Upside down and Mirror Magic square

13. Diabolic magic squares

14. multiplication Magic Squares

15. Spell magic squares

16. Magic squares for a given year

17. Completing a magic square when only any two numbers are given in any 3×3 or 4×4 or any other order

18. 2×2 continuous magic squares

19. 64 moves of a king in a chess board and the final path is a 8×8 magic square

20. 64 moves of a knight in a chess board and the final path is a 8×8 magic square

21. 64 moves of a rook in a chess board and the final paths is a 8×8 magic square