question_answer 7)

In a magic square each row, column and diagonal have the same sum. Check which of the following is a magic square. (i)                                                                           

5	? 1	? 4
? 5	? 2	7
0	3	? 3

(ii)                

1	? 10	0
? 4	? 3	? 2
? 6	4	? 7

               

Answer:

                               

5	? 1	? 4
? 5	? 2	7
0	3	? 3

I. Row,  Sum \[=\text{5}+\left( -\text{l} \right)+\left( -\text{4} \right)=\text{5}+\left( -\text{5} \right)=0\] II. Row, Sum \[=\left( -\text{5} \right)+\left( -\text{2} \right)+\text{7}=\left( -\text{7} \right)+\text{7}=0\] III. Row, Sum \[=0+\text{3}+\left( -\text{ 3} \right)=0+0=\text{ }0\] I. Column, Sum \[=\text{5}+\left( -\text{ 5} \right)+0=0+0=\text{ }0\] II. Column, Sum\[=\left( -\text{ 1} \right)+\left( -\text{ 2} \right)+\text{3}\]\[=\left( -\text{ 3} \right)+\text{3 }\]\[~=\text{ }0\] III. Column, Sum\[=\left( -\text{ 4} \right)+\text{ 7}+\left( -\text{ 3} \right)=\]\[~\text{7}+\left( -\text{ 4} \right)\text{ }+\text{ }(-3)\] \[=\text{7}+\left( -\text{ 7} \right)=0\] One Diagonal,   Sum \[=\text{5}+\left( -\text{ 2} \right)+\left( -\text{ 3} \right)\]\[=\text{5}+\]\[\left( -\text{ 5} \right)=0\] Other Diagonal, Sum \[=0+(-2)+(-4)\text{ }\]\[=0+(-6)\]\[=-6\ne 0\] Therefore, the given square is not a magic square.  (ii)

1	? 10	0
? 4	? 3	? 2
? 6	4	? 7

I.   Row,   Sum \[=\text{1}+\left( -\text{1}0 \right)+0=-\text{ 9}\] II. Row,    Sum\[=\left( -4 \right)+\left( -3 \right)+\left( -2 \right)=-9\] III. Row, Sum \[=\left( -\text{ 6} \right)+\text{4}+\left( -\text{ 7} \right)=\text{4}+\]\[\left( -\text{ 6} \right)+\]\[(-~\text{7})=\text{4}+\left( -\text{ 13} \right)=-\text{ }\left( \text{13}-\text{4} \right)=-\text{ 9}\] I. Column, Sum \[=\text{1}+\left( -\text{ 4} \right)+\left( -\text{ 6} \right)=\text{1}\]\[+\]\[\left( -\text{ 1}0 \right)\]\[=-\left( \text{1}0-\text{1} \right)=-\text{ 9}\] II. Column, Sum \[=\left( -\text{ 1}0 \right)+\left( -\text{ 3} \right)+\text{4}\]\[=\left( -\text{ 13} \right)\]\[+\text{ 4}\]\[~=-\left( \text{13}-\text{4} \right)=-\text{ 9}\] III. Column, Sum \[=0+\left( -\text{ 2} \right)+\left( -\text{ 7} \right)=0\]\[+\left( -\text{ 9} \right)=-\text{ 9}\] One Diagonal Sum \[=\text{1}+\left( -\text{ 3} \right)+\left( -\text{ 7} \right)\]\[=\text{1}+\left( -\text{1}0 \right)=-\]\[\text{ }\left( \text{1}0-\text{1} \right)=-\text{ 9}\] Other Diagonal Sum\[=\left( -\text{ 6} \right)+\left( -\text{ 3} \right)+0=\]\[\left( -\text{ 9} \right)+0\]\[~=-\text{ 9}\] Since each row, column and diagonal have the same sum, therefore, the given square is a magic square.