# A missing part of a checkerboard

Fill out the missing part (the grey area) and explain the pattern behind the colors of this $$9 \times 9$$ checkerboard tiles!

• There is an rot13(boivbhf flzzrgel gb gur vagreany 7k7 tevq). Do you mean that there is some rule that can generate the desired pattern through successive application of the rule? – Thomas Markov Dec 19 '19 at 16:18
• @ThomasMarkov the symmetry might be intentional (or coincidence), but the pattern/rule must be applied to all 9x9 i.e. after filling the missing part, explain why should the overall 9x9 be colored in such way. – athin Dec 19 '19 at 22:04

I think Prim3numbah got the right answer, but there's more to it than the explanation given:

It's based on the classic 3x3 magic square.

816
357
492

When mapped to the 3x3 sections, this gives the number of black squares in each. Example: the lower left 3x3 section has 4 black squares.

Then, when mapped to the individual squares within each 3x3 section, this gives the specific black squares. Example: the lower left 3x3 section has black squares at upper middle (1), lower right (2), middle left (3), and lower left (4).

This obviously produces the accumulation effect from one 3x3 section to the next. Also, the symmetries of the magic square are translated through this process to become the symmetries in the final pattern, i.e. each 3x3 section with N black squares (N = 1 to 4) can be inverted and rotated 180 degrees to match the 3x3 section with 9-N black squares. Example:

For the 3x3 section with 2 black squares, the black squares are 1 (upper middle) and 2 (lower right).
For the 3x3 section with 9-2=7 black squares, the black squares are everything except 9 (lower middle) and 8 (upper left).
Note that 1 and 9 are mirror images across the center square, and similar for 2 and 8, so a 180-degree rotation will swap each pair.

• Yep, this is the correct explanation, very well done! :) -- Also please upvote @Prim3numbah 's as well! – athin Dec 21 '19 at 1:01