Highest n where an equal number in all cells is (im)possible

Question

Inspired by Board with all 2020s :

Zeroes are written in all cells of a n×n board. We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it.

• Is there a highest n for which an equal positive number can be reached in all cells simultaneously?
• Is there a highest n for which an equal positive number can not be reached in all cells simultaneously?

Note: It is possible for n = 1,2,4 and 5. It is not possible for n=3 and n=6

My LP solver tells me under 100 it is solvable for

n = 1,2,4,5,8,9,10,14,15,19,20,22,24,25,29,32,34,39,44,59,64,71,76,77,82,84,94,97 (I do not see a pattern)

Obviously at least 1 of the answers is no. But are there an infinite number of solvable and an infinite number on unsolvable sizes, or has one of the types a finite number of sizes? (I don't know myself)*

Hint: impossibility for specific cases can be proven mathematically:

- If a balanced matrix with only positive increment values exists, then a fully symmetric balanced matrix with only positive increment values can be constructed from it by adding mirror images. Hence: If no fully symmetric balanced matrix with only positive increment values exists, the case is infeasible

- Looking at increments for fully symmetric n = 3: - Corner total: T = 2*side+corner - Side total: T = 2*corner+centre+side - Centre total: T = centre+4*side Eliminating side and corner from these equations yields centre = -Total/7 -> infeasible

- I applied the same technique to prove n=6 is infeasible

It seems likely that if, with size, the number of equations increases, the chance of a negative value increases. However, there might a a pattern or redundant equation might appear, making (some, or all) high n cases feasible.

Answer 1

PARTIAL ANSWER:

Let us say a matrix is balanced if any cell plus its neighbours adds up to the same constant. This implies it is possible for n = 1,2,4,5 because each number tells us how many times we need to apply the +1 operation. With n = 3,6, we run into trouble because we need to apply the +1 operation a negative number of times. Hence the balanced matrix is useless. Note this does not prove the impossibility of n=3 or 6 since there might be a different balanced matrix which does work. But it gives us reason to suspect it's not possible. I believe what is required is a systematic method of generating balanced matrices for all n.

NOTE: Credit belongs to WhatsUp for finding the balanced matrix for n = 5 in the original Board with all 2020s problem.

Answer 2

Silly answer:

If you make zero presses then all cells will be equal and remain at zero. This means that it works for any $n$. To avoid this problem you must state that at least one press must be made.