Fill out the missing part (the grey area) and explain the pattern behind the colors of this $9 \times 9$ checkerboard tiles!
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.
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.
The answer is:
Split the 9x9 checkerboard into 3x3 smaller checkerboards(I'll call these boards). Then there's 9 in total. Each and one of them have a different number of black tiles ranging from 1 - 9. If we look at board1, it consists of 8 black tiles and 1 white. If we look at board2, it consists of 1 black tile and 8 white tiles. If you rotate board2 180 degrees and place it ontop of board 1 they create a complete black board. So they are complements of each other. The same logic goes for every one of the other boards(they all have "matching" boards) except for the only one that consists of 9 black tiles(this one is already complete). "Matches" are: [Board1, Board2][Board3, Board4][Board5, Board7][Board6, Board9]. Also, if we take a look at board2(1 black tile) and then take a look at board9(2 black tiles) and then take a look at board4(3 black tiles) and so on up to board8(9 black tiles) we notice that they are continuations of the boards prior to the current one.