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Inspired from a puzzle marathon, here is a math puzzle. There is a pattern with this set of numbers (not in the creation though). According to the pattern, what is the logical replacement for the question marks?

$$\begin{align} 3 & 5 & 1 & 6 & 8 & 2 \\ 1 & 3 & 1 & 3 & 3 & 1 \\ \\ 5 & 1 & 3 & 8 & 1 & 4\\ 7 & 2 & 1 & 2 & 2 & 9\\ \\ 6 & 7 & 3 & 2 & 4 & 1\\ ? & ? & 5 & 3 & 8 & 2\\ \end{align}$$

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    $\begingroup$ What was the name of the puzzle marathon, by the way? Are they annual? Just curious is all :D $\endgroup$ – Mr Pie Sep 18 '18 at 22:13
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    $\begingroup$ It's not annual... it's just a thread here: artofproblemsolving.com/community/c4h1666263 $\endgroup$ – Jason Kim Sep 19 '18 at 0:05
  • $\begingroup$ Oh, at the art of problem solving! Should have expected that. Thank you :P $\endgroup$ – Mr Pie Sep 19 '18 at 0:31
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It's:

34

because:

read each $2\times2$ entry as a determinant (ad-bc), where: 

 a   b
 c   d
and we get 
 4  -3  2
 3  -2  1
 ?  -1  0
So $?=2$, and $34$ is the first positive integer that fits, although I can't see any logic in the choice of numbers on the top - there are several that would do the same trick.

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  • $\begingroup$ That is correct! $\endgroup$ – Jason Kim Sep 18 '18 at 23:00
  • $\begingroup$ Also, I didn't mean to choose any logical sequence of generating numbers... just a few examples :) $\endgroup$ – Jason Kim Sep 19 '18 at 0:09

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