Is it possible to arrange the integers $1,2,3,\ldots,169$ in a $13\times13$ square, so that in every $2\times2$ square the sum of the four numbers is divisible by $170$?
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The answer is
yes
Basically we just have ordered pairs of 170 alternating with high-low left/right. The pattern to figure out which pairs go where is very simple as well: We start with the square 1 169, 168 2. If we take 2 steps to the right, we do -13 if the original was bigger then 85, and +13 if it was smaller. If we take 2 steps down, we do -2 if the original was bigger then 85, and +2 if it was smaller. This way the whole 13x13 square can be filled as in the image. The numbers below are all possible sums and they are (ofcourse) all 340.
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1$\begingroup$ Great, the pattern formed is intriguing... $\endgroup$ Commented Dec 20, 2015 at 14:36