If a 4x4 grid (where each cell contains an arrow in an orthogonal direction) conforms to a special rule, I call it an Awry Grid. If it does not conform to this rule, I call it a Aright Grid.
Here are some examples (click on an image to view larger version):
For a full answer, you must figure out two things:
- A rule to determine if a given grid is awry or aright
- A way to make each awry* grid into an aright grid, only by changing the value of a single cell within the grid, which is not the first cell marked by red (0,0). Once you figure out 1), there might seem to be multiple ways to do this, at first. But for each grid, there is only one correct operation in order to achieve this. Finding this may require digging a little bit deeper into the structure of the grids.
(*) There are special cases of awry grids, I call them Lost Grids. You cannot (properly) turn a lost grid into an aright grid. There are no cases of lost grids in this puzzle, but this note might serve as a little hint.
If you want to create an awry/aright grid, you can fill most of the cells with random values. That doesn't mean that they're not important, since their interplay with the cells which are not random is the crux of this puzzle.
Hint #2 (fairly big):
Here's another aright right. The presentation reveals more about the structure:
Every 4x4 awry and aright grid can be presented in this way (i.e. the orange mark & yellow marks are always in the same place). And I'm going to spoil it right away, continuing from hint #1: When creating such a grid, all of the 11 unmarked cells can be random. Marked cells have to be filled with fitting values, depending on which type of grid you want to make.
The position of the yellow marked cells were not chosen arbitrarily, which is why I think it's fair to not include them in the original puzzle right away.
Furthermore, another hint/clarification: I already said that an awry/aright grid can be created with something different than four different arrows. I could've also used a cat, the number 42, a toilet and a nice picture of Richard Hammond as the four different values. The symbols are simply there to distract you from the nature of the puzzle. To make it more abstract, there are just 4 different states, and that's all you need to know about the symbols.
For posterity: xhienne and Gareth correctly pointed out the grids resemble Hamming Codes
This is more of a public answer to Gareth's comment:
It's clear that this is something like the correct answer. I've tried various different ways of interpreting the arrows (values mod 4, that have to add up to 0? values in GF(4), which for our purpose is just C2 x C2? ...) and so far I haven't found anything that seems to make the Aright Grids right and the Awry Grids wrong
Extremely close, I think Gareth intuitively chose the arguably more elegant (for a few reasons) way to 'transform' ___ into base 4. I chose 'another' way, which is less beautiful, but I thought it would be the easier one to come up with. Apparently not :) In that case, it may be helpful to think about how to determine the value of parity cell when creating such a grid, rather than 'checking' a filled grid.