Subtask 1

For this subtask, since ~R = 1~ and ~C = 1~, we can try and make only the first row and column a palindrome. One way to do this is to fill the first row and column with the letter a and fill the rest of the grid with the letter b.

Subtask 2

Because there are so few cases, it suffices to solve this subtask on paper and hardcode the answers. To reduce the amount of casework, notice that if we have a solution for ~(R, C)~, we can get a solution for ~(C, R)~ by flipping the grid.

Another possible approach is to write a brute force since there are only four important possibilities for each cell.

Subtask 3

This subtask is intended to aid students in thinking about the problem.

Subtask 4

The solutions to subtasks 1 and 2 should give some intuition here. From subtask 1, we propose the following general solution:

Fill the first ~R~ rows and the first ~C~ columns with the letter a and fill the rest of the grid with the letter b.

We notice that this fails when ~R = 0~, ~C = 0~, ~R = N~ or ~C = M~. We can handle these in pairs since we can flip the grid.

If ~R = 0~, we can fill the first ~M-1~ columns with the letter a, and the last column with the letter b, and then increment the last ~M-C~ characters of the last row. For instance, for the input 4 4 0 2 we can output:

aaab
aaab
aaab
aabc

If ~R = N~, every row must be a palindrome. Because of this, starting from a full grid of a characters, we can easily reduce the number of palindromic columns by two at a time by making matching columns of the first row b characters. For instance, for the input 4 4 4 2, we can output:

baab
aaaa
aaaa
aaaa

However, this fails if ~C~ is not the same parity as ~M~, that is, if we cannot get ~C~ by subtracting ~2~ an integer number of times from ~M~.

In this case, it depends: if ~M~ is odd, we can use the middle column. For instance, for the input 5 5 5 2:

babab
aaaaa
aaaaa
aaaaa
aaaaa

However, the input 4 4 4 1 is impossible since there is no middle column.

We can handle ~C = 0~ and ~C = M~ symmetrically by flipping the inputs, processing, and then flipping the output.