Question

Complete the following magic square:

16	2	………
………	10	………
……..	………	4

Answer 1

Hint: Here we need to complete the given magic square. We have to fill the numbers in the magic that the sum of all the numbers along rows, columns or diagonals of the magic square is equal. We will first obtain the sum of the numbers along the diagonal as all the numbers are given in that. Then we will find all the missing numbers using this sum.

Complete step by step solution:
Here we need to complete the given magic square.
We know that the sum of all the numbers along rows, columns or diagonals of the magic square is equal.
We can see that all the numbers along the diagonals are given, so, we will find the sum of these numbers first.
Sum of numbers along the diagonal \[ = 16 + 10 + 4 = 30\]
We will consider the second column now.
Only two numbers are given. Let the third number be \[x\].
Sum \[ = 2 + 10 + x\]
We know that sum should be equal to 30.
Therefore, we get
\[ \Rightarrow 30 = 2 + 10 + x\]
On adding the numbers, we get
\[ \Rightarrow 30 = 12 + x\]
Now, we will subtract 12 from both sides.
\[\begin{array}{l} \Rightarrow 30 - 12 = 12 + x - 12\\ \Rightarrow 18 = x\end{array}\]
So we have got

16	2	………
………	10	………
……..	18	4

Now, we will consider the third row.
Only two numbers are given. Let the first number be \[y\].
Sum \[ = y + 18 + 4\]
We know that sum should be equal to 30.
Therefore, we get
\[ \Rightarrow 30 = y + 18 + 4\]
On adding the numbers, we get
\[ \Rightarrow 30 = y + 22\]
Now, we will subtract 22 from both sides.
\[\begin{array}{l} \Rightarrow 30 - 22 = y + 22 - 22\\ \Rightarrow 8 = y\end{array}\]
So we have got

16	2	………
………	10	………
8	18	4

Now, we will consider the first column.
We have only two numbers in the first column. Let the middle number be \[z\].
Sum \[ = 16 + z + 8\]
We know that sum should be equal to 30.
Therefore, we get
\[ \Rightarrow 30 = 16 + z + 8\]
On adding the numbers, we get
\[ \Rightarrow 30 = z + 24\]
Now, we will subtract 22 from both sides.
\[\begin{array}{l} \Rightarrow 30 - 24 = z + 24 - 24\\ \Rightarrow 6 = z\end{array}\]
So we have got

16	2	………
6	10	………
8	18	4

Now, we will consider the first row.
We have only two numbers in the first row. Let the last number be \[k\].
Sum \[ = 16 + 2 + k\]
We know that sum should be equal to 30.
Therefore, we get
\[ \Rightarrow 30 = 16 + 2 + k\]
On adding the numbers, we get
\[ \Rightarrow 30 = 18 + k\]
Now, we will subtract 22 from both sides.
\[\begin{array}{l} \Rightarrow 30 - 18 = 18 + k - 18\\ \Rightarrow 12 = k\end{array}\]
So we have got

16	2	12
6	10	………
8	18	4

Now, we will consider the third column.
We have only two numbers in the first row. Let the middle number be \[l\].
Sum \[ = 12 + l + 4\]
We know that sum should be equal to 30.
Therefore, we get
\[ \Rightarrow 30 = 12 + l + 4\]
On adding the numbers, we get
\[ \Rightarrow 30 = 16 + l\]
Now, we will subtract 22 from both sides.
\[\begin{array}{l} \Rightarrow 30 - 16 = 16 + l - 16\\ \Rightarrow 14 = l\end{array}\]
So we have got

16	2	12
6	10	14
8	18	4

Hence, this is the required complete magic square.

Note:
Here we have completed the magic square by filling the appropriate numbers. We need to remember the key points to solve the magic square. The sum of the numbers along every row, columns and diagonals are equal. So we have used this key point to solve that problem. Here, we can get confused by thinking magic square as square of numbers. A magic square is stable consisting of different numbers.