New dimension to look at Magic Square - T.R. Jothilingam

Fun with numbers – Magic Squares

The sweeter side of the MATHS…

        Magic squares are a set of numbers, arranged in a set of 3 x 3 or 4 x 4 or 5 x 5 squares, in which all the Vertical, Horizontal and both diagonal totals are same, called as the “Magic Sum”.

Normally in a 3 x 3 magic square natural numbers 1 to 9 are used, and for 4 x 4 numbers 1 to 16 are to be used.  In general, for a “n x n” magic square numbers, “1 to n²” numbers are to be used.  And the “Magic Sum” of the magic square must be          {(1 + n ²)/2}* n. (First number plus last number to be added, and the sum divided by two, and then multiplied by the number of rows). These magic squares can be done from 3 x 3, 4 x 4… up to any highest number in the world.

Magic squares can also be made with using fractions, zeros, negative numbers, with squares and cubes. We can make magic squares for a given numbers, or size, or for a given date or date of birth also.

 Example:

For 3 x 3 magic square, numbers 1 to 9 are to be used.

The row total will be {(1 + n ²)/2}* n ={(1 + 9)/ 2}*3 = 5 x 3 = 15.

3 x 3 Magic square (Total 15)                     3 x 3 (Total 2019)

8	1	6
3	5	7
4	9	2

 

676	669	674
671	673	675
672	677	670

 

                 

 4 x 4 Magic square (Total 34)    

   {(1+ 16) /2} x 4 = 34                                     4 x 4      (Total 2018)

16	2	3	13
5	11	10	8
9	7	6	12
4	14	15	1

 

512	498	499	509
501	507	506	504
505	503	502	508
500	510	511	497

 

 

                                                                                              

 

6 x 6 Magic Square (Total 111)

35	1	6	26	19	24
3	32	7	21	23	25
31	9	2	22	27	20
8	28	33	17	10	15
30	5	34	12	14	16
4	36	29	13	18	11

 

 

 

 

 

There are many interesting magic squares to enjoy:

 

Smallest Prime number magic square: All numbers in the magic square are prime numbers         

                  (Total 177)

71	89	17
5	59	113
101	29	47

 

 

  

                                      

                        Reverse Magic squares: 

All numbers in a magic square are written from the reverse and that also form a magic square and with the same total.

         (Total 1515)                               (Total 1515)

806	107	602
301	505	709
408	903	204

 

608	701	206
103	505	907
804	309	402

                 

 

Upside-down magic square: 

A magic square when turned to 180⁰    also giving the same total 24.                      

                                                (Total 24)

9	1	8	0	6
0	6	9	1	8
1	8	0	6	9
6	9	1	8	0
8	0	6	9	1

 

 

 

 

This article is written by Mr. T.R. Jothilingam, Retd. Rly. Station Supt., Madurai, He is a poly mathematician and magic square wizard with several record in his name. He is also a winner of Ramanujan Award of All India Ramanujan Maths Club, Gujarat

Contact him at :-  +91 94428 10486

email: mailtojothi@gmail.com

 http://jollymaths.com/

IMPORTANT PROPERTY OF SQUARE NUMBERS

IMPORTANT PROPERTY OF SQUARE NUMBERS : useful for finding square, square root , last 2 digits and many more.

 

Same property is very useful in many other problems which would be shared in next blog.