# Continuity

Computer Science Level 5

Consider an $$N \times N$$ grid, initially completely unpopulated. We can mark squares in this grid in at least $$2^{(N \cdot N)}$$ ways, such as for a $$2 \times 2$$ grid:

Of these $$16$$ grids, precisely $$13$$ are continuous. That is, there exists a path, using edge adjacency (no corners/diagonals) from every black square to every other black square, and there is at least one black square.

Of all $$3 \times 3$$ grids, $$218$$ are continuous.

How many $$10 \times 10$$ grids are continuous?

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