Find missing number in grid. I'm solving a missing number in square problem. I have tried multiple ways but haven't been able to find any common rule. 
My attempts are-


*

*$((5+7+6)+1) \times 6=114$

*$4 \times ((3+5)+1) = 36$


Please refer to Question number $39$ in below image. 

 A: The following rule gives a unique answer which is among the choices. Whether it is the rule they were looking for....who knows. 
The rule is (not including the bottom row): 
Multiply all numbers in a column. Add to this value the sum of all numbers in the column which are unique. Divide the resulting value by $2$ to get the value at the bottom row. 
This gives the answer $4$, i.e. A. 
Let's check if this rule works for each column.
Column 1, value in last row:
$$Value = \frac{3*5*4+3+5+4}{2} = \frac{72}{2} = 36$$
Column 2, value in last row:
$$Value = \frac{4*4*4}{2} = \frac{64}{2} = 32$$
Column 3, value in last row:
$$Value = \frac{5*7*6+5+7+6}{2} = \frac{228}{2} = 114$$
Works! 
A: I can not ####ing believe they still teach math this way!
The hallmark of an intelligent student should be the realization that assumed rules use assumptions that may or may not be valid and therefore any conclusion can not be certain.  Therefore the development of any good mathematician is to stop making assumptions.  These "find the rules" do the exact opposite.
And any mathematician eventually learns that rules may be constructed for any finite set of points... which means I can put in ANY answer into the square an define a rule that will fit them all. (That was actually the tongue in cheek intent of Jan's answer--- you can algebraicly create a polynomial that will give any answer you want.) Indeed this is fundamental to the concept of "function".  The "mathematicians" who make these exercises should know that but instead they teach absolute falsehoods and sloppy thinking to innocent students.  It ####es me off so much!
Okay.... calm down.
It's just a puzzle.  And puzzles are fun.  And the goal is to find a simple rule (Jan's rules are not "simple".)  ....
But that's not MATH!  That's puzzles.  "Simple" is not well-defined and it isn't at all clear what the consider simple.  And to find a rule counts on intuition and .... when you get down to it, you can not teach intuition.  And if you try, you end up teaching students rote rules that leave them totally helpless and with recourse if the question takes even the slightest altercation.  
There was study done in the 80's about how students do word problems.  The gave students the question: "Malcolm left has a car.  He drove is car 7 miles.  Then he turned right and drove 2 miles.  How many miles did Mr. Left drive?"  Many students answered ... 5.  Why?  Because the man's name was Mr. Left and "left" means subtraction, of course!
This type of math teaching is no different.
Anyway... So I look at your square.... I don't see it either.  I think the line with 36, 32, and 114 and so much larger than the other rows that the rule must lie in the columns.  Somehow 3,5,and 4 make 36; ?,4,4 make 32, and 5, 7 and 6 make 114.  But I'm damned if I see it.  (I do see that 4 divides 36.  4 divides 32 and 6 divides 114 but I don't see any way that relates.)
So don't worry.  If you can't do this it doesn't mean you can't do math.  It means you can't do puzzles.  ... Well, no it doesn't.  It means you can't do this puzzle.  And neither can many other people here.
A: There is nothing mathematical in your question except that numbers are being added. There probably exists infinity of solutions to your problem. One way to solve it is to write the 'rule' your question mentions (note that the rule is completely arbitrary) as
$$\begin{aligned}
3a+5b+4c=36\\
xa+4b+4c=32\\
5a+7b+6c=114\\
\end{aligned}$$
and solve this system of equations for fixed $x$. The solution is $a=\frac{124}{4-x}$, $b=-120+\frac{124}{4-x}$ and $c=159-\frac{248}{4-x}$. When, for example, $x=1$ this collapses to $a=124$, $b=4$ and $c=-89$.
