Continuity

Consider an N×NN \times N grid, initially completely unpopulated. We can mark squares in this grid in at least 2(NN)2^{(N \cdot N)} ways, such as for a 2×22 \times 2 grid:

Of these 1616 grids, precisely 1313 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 3×33 \times 3 grids, 218218 are continuous.


How many 10×1010 \times 10 grids are continuous?

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