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How many hypercubes can be adjacent?

2D grid of squares, one square is surrounded by 8 others. 3D grid of cubes, one cube is surrounded by 26 others.

What is this number for 4D? Is there a general formula?

The tricky thing is diagonals. In 2D there are only pure corner diagonals. In 3D there are edge diagonals and pure corner diagonals. I suspect that in 4D, some new kind of diagonal will be there. But I want to include all classes of diagonals in my definition.

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