Add rules to Minesweeper until you can start with nothing

Question

Think of classic Minesweeper game, with the following list of restrictions on the placement of mines:

• C (Connected): All mines are connected via 8-way neighborhood (orthogonal and diagonal).
• Q (Quad): Every 2x2 subgrid contains at least one mine.
• T (Triplet): No three mines are adjacent on a line (row, column, or diagonal). Think of Tic-Tac-Toe.
• O (Outside): All mines are connected to the outside via 4-way neighborhood (orthogonal), and all non-mine cells are connected via 4-way neighborhood.

Here are some examples under each restriction: (X are mines, and . are cells without mines)

C	Q	T	O
XXX.. ...X. .XX.. ..X.. .X.XX	.X..X ..X.. .X..X .X.X. .XX.X	X...X X..X. .X..X XX... ...XX	.XX.X ....X .X... XXXX. X....

Now, find the unique combination of two restrictions that allow you to open a cell with 100% certainty on a large enough grid. What kind of cell(s) can you open, and what is the minimum size of such a grid?

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Note: The answer to this puzzle spoils how to get a specific achievement in 14 Minesweeper Variants (2022) by Artless Games.

Answer 1

A grid that satisfies the condition is

a 7x7 grid with Triplet and Outside restrictions.

We can click on

the center cell.

Indeed, if it was a mine, then we would need to connect it to the outside border. To progress towards the border from the center mine, we must do some sort of S shape in any direction:



As we can see, we end up being stuck because of the triplet rule. Hence, the center cell cannot be a mine, so it's safe to click on it!