Question

In a magic square each row, column and a diagonal have the same sum. Then the values of A and B respectively are _______.

1	-10	0
A	-3	-2
-6	4	B

Answer 1

Hint: This is a simple question with basic algebra and logic required to be solved. We will first check the sum of any row, column or diagonal where all the three elements are known. Then we will consider a row, column or a diagonal containing the variables and equate their sum to the sum calculated previously, which will give us the answer.

Complete step-by-step answer:

We need to find the unknown variables A and B missing in the magic square. It is given that the sum of each of the rows, columns and diagonals is the same. So, we will start by finding a row which contains all the three elements, which will give us the sum.

We can see that the first row has all the three elements known to us. So, we will add these to get the sum which will be used throughout the table. The sum, say s, can be obtained as-
${\text{s}} = 1 + \left( { - 10} \right) + 0 = - 9$
Now that we have the sum, we can select any arbitrary row, column or a diagonal involving the variable and equate its sum to -9, which will give us the values of A and B.

On selecting the first column of the table, we can write that-
$1 + {\text{A}} + \left( { - 6} \right) = {\text{s}}$
$1 + {\text{A}} - 6 = - 9$
${\text{A}} = - 9 + 6 - 1 = - 4$
Similarly, when we consider the third column we can write that-
$0 + \left( { - 2} \right) + {\text{B}} = {\text{s}}$
$0 - 2 + {\text{B}} = - 9$
${\text{B}} = - 9 + 2 = - 7$
Hence, the values of A and B are -4 and -7 respectively.

Note: A common practice to follow here is that we should always check our answers by using the property of a magic square along any other row or column. For example, A lies on the second row, we can find its sum as: - 4 - 3 - 2 = -9 = s, which verifies this value. For B, we can consider the third row and find the sum as: -6 + 4 - 7 = -9 = s, which verifies the answer again.