Question

Fill the squares with all the numbers from \[21\] to\[29\]. Given the total of each side is \[75\].

Answer 1

Hint:
As we are given that the sum of the numbers is\[75\], we are going to take the two numbers between \[21\] to \[29\] and then subtract their sum from \[75\]and so on. Taking the numbers column wise, we need to check the total row wise too.

Complete step by step solution:
It is given that the sum of each row and column is\[75\]and we can use the numbers from \[21\]to\[29\].
Considering the first row, take two numbers for the first two squares as: \[26\]and\[27\]. For filling the third square, we subtract the sum of these two squares from\[75\]. i.e.
\[\begin{align}
  & 75-\left( 26+27 \right) \\
 & \Rightarrow 75-53 \\
 & \Rightarrow 22 \\
\end{align}\]

Now only, the numbers \[23,24,28,29\] can be filled.
Now taking second row, in which second square is already filled with\[25\], if we take \[21\] in the first square of row two, then third square is filled with:
\[\begin{align}
  & 75-\left( 25+21 \right) \\
 & \Rightarrow 75-46 \\
 & \Rightarrow 29 \\
\end{align}\]

Considering the third and the final row, for this the only numbers that are left are \[23,24,28\]
Now we have to check the sum column wise too.
So, in first box:
\[\begin{align}
  & 75-\left( 26+21 \right) \\
 & \Rightarrow 75-47 \\
 & \Rightarrow 28 \\
\end{align}\]
In second:
\[\begin{align}
  & 75-\left( 27+25 \right) \\
 & \Rightarrow 75-52 \\
 & \Rightarrow 23 \\
\end{align}\]
In third:
\[\begin{align}
  & 75-\left( 29+22 \right) \\
 & \Rightarrow 75-51 \\
 & \Rightarrow 24 \\
\end{align}\]
So the complete grid becomes:

Formula used:
Taking the value of two boxes arbitrarily between \[21\]and \[29\] and subtracting them from \[75\]. i.e.
\[75-\left( A1+A2 \right)\]
Where, \[A1\]: value of first square
\[A2\]: value of second square

Note:
It is to be noted that numbers only between \[21\] and \[29\] are to be taken and not anything apart from this range. Also, the sum of the three squares of each row and column as well should be \[75\]. Ignoring any one of them will cause errors.