Question

Solve the problem/puzzle​

Answer 1

The problem has no answer.

$\Large\textrm{Explanation}$

We have four equations.

Does the answer exist? It is a system equation of four variables and four equations. It seems, but not actually.

To show that the answer does not exist, let's add equations two by two.

$\;$

Add the upper and below columns for the sum of differences.

Add the left and right rows for the sums of all four numbers.

We add the results together to get twice the left row, which becomes $\rm37$. But actually, it should be $\rm24$.

There are no such pair of numbers!

$\;$

The correct answer to the puzzle is no answer.

Answer 2

$\large\underline{\sf{Solution-}}$

Given puzzle is

$\boxed{ \rm{ \: \: \: }} - \boxed{ \rm{ \: \: \: }} \: = \: 9 \\ \: + \: \: \: \: \: \: \: \: \: + \: \: \: \: \: \: \: \: \: \\ \: \: \boxed{ \rm{ \: \: \: }} - \boxed{ \rm{ \: \: \: }} \: = \: 14 \\ \: \: \: \: \parallel \: \: \: \: \: \: \: \: \parallel \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \rm \: 12 \: \: \: \: \: \: \: \: 2 \: \: \: \: \: \: \: \: \: \: \:$

Let fill the missing boxes with unknown variables as

$\boxed{ \rm{ \: \: x \: }} - \boxed{ \rm{ \: \: y \: }} \: = \: 9 \\ \: + \: \: \: \: \: \: \: \: \: + \: \: \: \: \: \: \: \: \: \\ \: \: \boxed{ \rm{ \: \: z \: }} - \boxed{ \rm{ \: \: w \: }} \: = \: 14 \\ \: \: \: \: \parallel \: \: \: \: \: \: \: \: \parallel \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \rm \: 12 \: \: \: \: \: \: \: \: 2 \: \: \: \: \: \: \: \: \: \: \:$

So, we have

$\rm \: x - y = 9- - - - (1)$

$\rm \: x + z = 12 - - - - (2)$

$\rm \: y + w = 2- - - - (3)$

$\rm \: z - w = 14- - - - (4)$

On adding equation (3) and (4), we get

$\rm \: y + z = 16 - - - (5)$

On Subtracting equation (5) from equation (2), we get

$\rm \: x - y = - 4 - - - (6)$

So, from equation (1) and (6), we concluded that equations have no solution.

So, the above puzzle have no real solution.

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$