What is my favorite brand of pen?

Question

This is an entry to the Monthly Topic Challenge #3: Pencil and Paper Games

Given the following valid "tic-tac-toe" boards:

What is my favorite brand of pen?

Other possible tags to check your work: solvable-with-pen-and-paper the-answer-is-obviously-correct

Hint:

The last line contains a critical clue.

Hint 2:

Curiously, Hints 1 and 2 contain a critical clue.

Hint 3:

Start by making sense of the "Given" section. You can disregard colors, the bottom section, and anything to do with "tic-tac-toe" or "ciphers".

Hint 4:

The wordplay does not give the mechanism, but rather the genre.

Hint 5:

X to play. The boards were selected somewhat arbitrarily; the possibilities are endless. Your task is to determine what makes them special.

Hint 6:

What very well known game has the symmetry found in row 2?

Hint 7:

Added a clue to the image. Why might a teacher use a red pen?

Hint 8:

Stiv has correctly deduced that the first 9 boards are "valid" according to certain rules, but the next 9 are invalid.
Let's focus on row 3. Stiv has correctly guessed that it is not a pawn. In fact, the board is already solved for you (it is an example solution-imagine if it first looked like a row 4 board).
Can you solve row 4? Hint: a board's validity is independent of adjacent boards; the initial layout was only as a hint to the name of the game.

Hint 9:

Each invalid board can be solved with a brief visual inspection; having a red and green pen in hand simply clues the mechanism. Nothing to do with checking; this puzzle is about valid boards. 3 of the 8 boards in row 4 are pawns.

Answer 1

I don't know what it was about the last hint that finally got me to the solution but I have it now.

Here's how I moved forward from Stiv's partial solution:

In each of the grids in the bottom row, one square is marked as wrong (red). Since only one square can be wrong at any given time, there is only a single possibility both for the wrong square and for the chess piece (or pawn) represented by the X in each grid.

Here they are:

As you can see I also added the green squares indicating the letters of the chess piece. If we now take the letter at the red square, we get: APERKATE
Applying ROT2 to the K yields APERMATE
Adding the P from the BISHOP in the example

finally gives

PAPERMATE, a brand of pen and also a fitting end for a game of chess on paper.

Many thanks to Stiv for their initial work, without which my answer would have not been possible!

Answer 2

^{A partial answer to explain the 'Given...' section.}

There are several clues in this puzzle to tell us that we should be focussing not on tic-tac-toe, but on the game of...

CHESS. Notably, these include the reference to 'check' in the last line of the puzzle (as per Hint 1) and the references to 'critical' in Hints 1 and 2 (as in 'critical position'), Hint 5's 'X to play', and - of course - the reference to 'boards' (chessboards) in the first line of the puzzle.

One other important element (as emphasised in Hint 6) is the symmetry of sorts exhibited by the second row of boards. In particular, the 1st and 8th are reflections of each other, as are the 2nd and 7th, and the 3rd and 6th. This should be leading us to think about...

the arrangement of chess pieces at the start of a game...



Note here that in the bottom row the rook pieces occupy the 1st and 8th columns, the knight pieces occupy the 2nd and 7th, and the bishop pieces occupy the 3rd and 6th - exactly replicating the symmetry of the puzzle set-up.

In fact, thinking about this further, we can now explain the 'Given...' example in its entirety:

1. Firstly, by considering the white pieces laid out in the image in the spoiler above you can see that the number of green squares shaded in each board matches the number of letters in the corresponding piece's name:



2. Secondly, if we consider the 'X' characters to represent the corresponding pieces and imagine them occupying those spaces, the 'O' characters actually represent legal moves (within the frame of the 3x3 area):

The next question then is what do we make of the next set of boards? At the moment, I don't know.

The complete absence of 'X' characters in the board containing merely seven 'O' characters is perplexing, and suggests we're looking for something a little different this time around. It seems the 'Given...' part of the puzzle might merely have been to help us work out we are looking at chess. We now need to use that knowledge in a different way altogether...

Similarly confusing is the appearance of the red square - this is new. Five cells being shaded green might imply that the pieces of interest in the first board might be the five-letter QUEEN rather than the four-letter PAWN, as before. But does the additional red cell suggest we are looking for something applicable to both five-letter pieces and six-letter pieces (which would include the KNIGHT and BISHOP)? Does it perhaps suggest an action in chess, like a piece being captured, or a king put into check? I'm not sure.

I shall continue to think, but am happy for anyone with more insight into the nuances of this game to make use of my answer and make the next move!