Every cell holds 0 to 3 grains. A cell with 4 topples: one grain to each neighbour. Grains that fall off the edge are gone.
0 1 2 3 grains
Before I drew it, I guessed:
✓ the same under every turn and flip
✓ no two empty cells side by side
✓ adding it changes nothing
✗ 2.0 to 2.3 grains per cell (it passes 2.3 once the square is wide)
✓ a plain square at the centre (about 0.42 of the width)
½ detail along the diagonals: plain near the centre, then a staircase of smaller and smaller squares runs into each corner
What it said that I didn't ask: single grains are the rarest, and they get rarer as the square grows (not at every step). Tap 32, then 256.