# A Math Puzzle with a Pattern

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}

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

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.

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