# Fill up the grid with the letters 'P', 'U', and 'Z' in the following way

Fill up the grid,

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).

• I assume that also includes the diagonals going the other direction? – benzene Aug 25 '17 at 22:48
• No assumption: "all diagonals" – user39732 Aug 25 '17 at 23:03
• "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? – Rand al'Thor Aug 25 '17 at 23:14
• @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? – user39732 Aug 25 '17 at 23:27

## 1 Answer

This is akin to:

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:

PPPUUUZZZ
UUUZZZPPP
ZZZPPPUUU


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