enter image description here

What will be the missing tile?
The colorblind version is available here.

  • 5
    $\begingroup$ Upvote for including a colorblind-friendly version. $\endgroup$
    – Brandon_J
    Sep 19 '19 at 15:05
  • $\begingroup$ Just out of curiosity, has this got multiple choice? $\endgroup$ Sep 22 '19 at 21:57
  • $\begingroup$ @Deepthinker101 no there isn't, the answer can be uniquely determined by observing various patterns $\endgroup$
    – athin
    Sep 22 '19 at 22:15
  • $\begingroup$ I personally saw four logically valid answers, that is why I was wondering... I meant in the original version by the way too $\endgroup$ Sep 22 '19 at 22:16
  • 1
    $\begingroup$ @user71981 yes this is my own puzzle btw $\endgroup$
    – athin
    Nov 28 '20 at 7:09

I think the missing tile should be:

enter image description herex
enter image description herev


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.

Reason Update:

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.

  • $\begingroup$ Great observations! :D But I'm afraid there are still some missing points (nevertheless, you cover many of them!) The answer is still incorrect, but I could say that it's "almost" correct, :) $\endgroup$
    – athin
    Sep 19 '19 at 22:44
  • $\begingroup$ If incorrect, the answer might be rot13(punatr gur bireynccvat tevq sebz Oyhr gb Terra, qhr gb gur cerprqrapr ehyr V nz n ovg abg fher.) $\endgroup$
    – Conifers
    Sep 20 '19 at 0:59
  • $\begingroup$ That's actually not quite right either, there is a single definite answer here, including the definite precedences. You may try to find other missing properties ^^ $\endgroup$
    – athin
    Sep 20 '19 at 1:10
  • 1
    $\begingroup$ I just found the answer might be rot13(whfg ebgngr 180 qrterrf. Qhr V'z cerggl fher zl bofreingvba 1~5 vf pbeerpg, naq gur 6 vf nyfb pbeerpg, ohg arrq gb qrpvqr gur ebgngvba. Vs cerivbhf vf vapbeerpg, gura zhfg or nsgre ebgngvbaf gb svg gur bofreingvba 5.) $\endgroup$
    – Conifers
    Sep 20 '19 at 2:14
  • 1
    $\begingroup$ Yep, that's actually the correct answer :) Let's see if you can work out why should it be the case, and not the previous one! $\endgroup$
    – athin
    Sep 20 '19 at 3:28

@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 $4$ pillars are actually put in this order:
- Yellow, $3$, Up, 2nd
- Green, $4$, Right, 3rd
- Blue, $1$, Down, 4th
- Red, $2$, Left, 1st

And the next one will replace the previous one:

enter image description here

Notice that:

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:)

enter image description here

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:

enter image description here

  • $\begingroup$ Oh now I get it! :D $\endgroup$
    – Conifers
    Sep 25 '19 at 1:35

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