A $5\times 5$ grid of single-digit numbers in $\mathbb N$, with one cell empty. What number should be in the cell? 
16.$$\begin{array}{|c|c|c|c|c|}
\hline
2 & 7 & 4 & 3 & 5
\\\hline
7 & 3 & 4 & 5 & 4
\\\hline
1 & 3 & 2 & 2 & 6
\\\hline
2 & 4 & 5 & 4 & \mathbf{?}
\\\hline
8 & 3 & 6 & 3 & 5
\\\hline
\end{array} $$
Is the answer 1, 2, 3, 4, or 5?
Photograph of the problem source

Really I don't understand. How does such a question relate to logic? For me, it's a game about numbers.Not Logic. It really annoys me to solve such a question. Anyway, I took 12 minutes for this question in exam. I came home. I could not even "solve" it at home ether. I think, such a question is nonsense. No science has anything to do with it. Please help me with the question and please explain me, what does it really mean to solve such a question?


$$\begin{array}{c}
x \\
y \\
p \\
q \\
r
\end{array} \qquad\to\qquad x^y = pqr\quad\text{(concatenated)} $$


"[W]hat does it really mean to solve such a question?" 
Well, such a question challenges us to find order amid chaos. That's what mathematics ---as the study of pattern--- is all about, so it's not a completely irrelevant mental exercise. As others mention, it could be instructive to find ways to justify any answer. (It's always possible to do that in a puzzle like this one, although not all rules are particularly "nice" ... but "nice" is subjective and not actually required.)
That said, this kind of thing is a horrible exercise for an exam. Not only is it unreasonable to expect someone to notice any pattern in a fixed time-frame, it is unreasonably-unreasonable to expect someone to notice the (so-called) "correct" pattern at all, since that's often akin to mind-reading. ("What was the author thinking?")
In any case, here's a walk-through of my thought process: After a minute or so of skimming the grid, and just before abandoning the whole thing, I happened to recognize the powers "27", "128", and "256" amid the columns of digits; then ---oh, yeah!--- "243" and "343" (which aren't always on the tip of my brain); and then ---hey!--- that could be "625" in the last column! I must be onto something! Yet ... "73" and "44" and friends aren't powers, so maybe "27" was a red herring. Then, the "aha!" moment: $2^7 = 128$. Done.
