A trick with Magic Square

The other day I saw a Steve Harvey Show that went viral in the social media. A boy accepted a number (between 65 and 300 and divisible by 5, he wanted and he got 265) from the audience and started filling a magic square of the order 5 (5 * 5 square) from numbers 41 to 65. When finished, he showed that the cells of every column, row and diagonals added up to 265.

 

This trick is easy to perform and children should be encouraged to know the fact behind it and also to improvise it.

 

What is a magic square?

 

Well, a square arrangement of n * n cells where each row, column and diagonal adds up to the same sum is known as a magic square.

 

 

The following is a 3 * 3 square filled with 1 to 9.

 

8	1	6
3	5	7
4	9	2

 

It is easy to find that for 3 * 3 square the sum of the row (column or diagonal) will be 15, if the numbers start from 1.

Surely the sum of all numbers will be 1+2+… +9 = 9*10/2 = 45. The three rows (or columns) should add to 45. So, one row (or column) will be 45 / 3 = 15.

In general, all the cells together of  n * n matrix should add to n^2 * (n^2 + 1) / 2. Since this is distributed among n rows, the sum of one row will be (n^2 * (n^2 + 1) / 2) / n = n * (n^2 + 1) / 2.

For a 5 * 5 matrix it will be 5 * 26 / 2 = 65.

It is easy to fill the cells of a n * n matrix where n is odd. (When n is even the procedure is different). The following steps can be used.

1.       Enter 1 in the top row middle cell.

2.       After filling any cell with a number, put the next number in the cell above and right (upper row, right column). If there is no upper row, take the bottom-most row. If there is now right column take the left-most column.

3.       If there is no right or top, put the next number in the cell below the current cell.

4.       If the top right is already full, put the next number in the cell below the current cell.

 

 

 

Example 5 * 5.

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

 

1.       Put 1 in row 1, column 3.

2.       Since there is no row above put 2 in the same column lowest row.

3.       After 3, since there is no right column, put 4 in the same row left-most column.

4.       After 5, since the top right column is already filled, put 6 just below.

5.       After 10, same case as above.

6.       After 15, same case as above.

Variations.

What happens if we start with 2 instead of 1 and end with 26 instead of 25?

Now the total will be 25 * (2 + 26) / 2 = 350 and 1 row ill add up to 350 / 5 = 70.

So if we add 1 to all numbers by starting with 2 instead of 1, a row sum will be 5 more than 65.

If we add 40 to all numbers (start with 41 instead of 1) there will be an increase of 40 * 5 = 200; the sum will be 265.

This is what the boy did. He starts the numbering with 41.

57	64	41	48	55
63	45	47	54	56
44	46	53	60	62
50	52	59	61	43
51	58	65	42	49

 

Suppose you want a total of 500 in each row. What number you will start with? You should leave (500-65)/5 = 435 / 5 = 87 and start with 88.

104	111	88	95	102
110	92	94	101	103
91	93	100	107	109
97	99	106	108	90
98	105	112	89	96