Fill up the grid,

enter image description here,

so that the letters '$P$', '$U$', and '$Z$' occur the same number of times both column-wise and row-wise, as well as diagonally for all diagonals with a number of cells that is a multiple of three (two examples are indicated above in color).

  • $\begingroup$ I assume that also includes the diagonals going the other direction? $\endgroup$ – benzene Aug 25 '17 at 22:48
  • $\begingroup$ No assumption: "all diagonals" $\endgroup$ – user39732 Aug 25 '17 at 23:03
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    $\begingroup$ "occur the same number of times both column-wise and row-wise" - what does this mean? Every column and row has three P's, three U's, and three Z's? $\endgroup$ – Rand al'Thor Aug 25 '17 at 23:14
  • $\begingroup$ @Randal'Thor, consider the column individually. This column must have the same number of each P, U, and Z. Example: PUZZUPUPZ This goes for rows also. All columns. All rows. All indicated diagonals. Does that help? $\endgroup$ – user39732 Aug 25 '17 at 23:27

This is akin to:

IMO 2016 question 2

So the grid can only be filled if:

The side length is a multiple of 9

And can be filled by tessellating the following tile in the normal fashion:


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    $\begingroup$ Aww. It's not as interesing a puzzle as I'd hoped. $\endgroup$ – Rand al'Thor Aug 25 '17 at 23:28
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    $\begingroup$ I upvoted your comment. $\endgroup$ – user39732 Aug 25 '17 at 23:28

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