What will be the missing tile?
The colorblind version is available here.
I think the missing tile should be:
By observation, consider the following patterns:
1) Every tile is composed with 4 different pillar included 1x1, 1x2, 1x3 and 1x4 unit grid. And there also has overlapping between these pillars.
2) Every tile is also composed with different colors: Blue, Red, Green and Yellow. And those pillars assign different color for each.
3) For each tile, there has 2 horizontal pillars and 2 vertical pillars (The 1x1 pillar could be counted either way).
4) The pillar's shorter edge(the 1-unit side) must connect to the tile's boundary(That is, you won't see any 1x1 or 1x2 pillar in the center of pile).
5) For each color, you can find that the pillar will be horizontal in every odd-th tiles(e.g. 1st, 3rd, 5th, etc. The upper-left-most pile is 1st, lower-right-most is 16th) and be vertical in every even-th tiles, or vise versa.
6) Also checking the White grids, you'll find there has 4 different patterns for the White after you rotate properly.
7) Let's check to the color precedence. Define A overlaps B means A has higher precedence than B. For the total 4x4 piles, each color has highest precedence on a arbitrary row and a column.
By summary the above pattern, I could find out the 16th pile will be composed with:
a) 1x4 Blue pillar, 1x3 Green pillar, 1x2 Yellow pillar, 1x1 Red pillar.
b) Blue and Yellow are horizontal, Green and Red are vertical.
c) Check the White grids pattern, the missing 16th pile is same as the 1st pile.
d) Check the precedence, Blue is higher than Green in 16th pile, so overlaps the Green.
Found another pattern. By extending the observation 4), Each side of the pile will connect to a pillar(e.g. Side where the pillar grown from). And for same type(1x1, 1x2, 1x3 or 1x4) pillar, it will connect to 4 different sides in any pile row(e.g. 1~4th, 5~8th, 9~12th or 13~16th piles).
Thus by checking to the 1x3 Green pillar and 1x2 Yellow pillar and 13~15th piles, the former answer should be rotated 180 degrees as the final correct answer.
@Conifers has correctly determined the missing tile with some rules a bit different from my intended ones but still got it uniquely determined so I'm giving him a checkmark. This is the intended answer, :D
Observe that, for each tile:
It consists of $4$ different "pillars" with some properties and rules being followed.
The properties are:
- The color of the pillar, in rainbow order: Red, Yellow, Green, Blue.
- The height of the pillar, in increasing order: $1, 2, 3, 4$.
- The ground location (side) of the pillar, in clockwise order: Up, Right, Down, Left.
- The column position (left-to-right) of the pillar, in increasing order: 1st, 2nd, 3rd, 4th.
For the first tile:
The order and properties of the pillars can be determined from the starting pillar, which are in the order given in properties (e.g. color in rainbow order, etc.) For example, the next tile after "Red, $1$, Up, 1st" will be "Yellow, $2$, Right, 2nd", and so on.
Thus, we need to find the starting pillar for each tile.
And we can have the following as the answer for it (above spoiler):
(The starting pillars will be:)
Notice the patterns that:
- The color will be the same in diagonal "/".
- The height will be the same in the horizontal.
- The ground location (side) will be the same in diagonal "\".
- The column position (left-to-right) will be the same in the vertical.
The starting pillar for the missing tile will be "Red, $1$, Up, 4th".
So the full order of $4$ pillars will be:
- Red, $1$, Up, 4th
- Yellow, $2$, Right, 1st
- Green, $3$, Down, 2nd
- Blue, $4$, Left, 3rd
Thus the missing tile is: