9514 1404 393
Answer:
18 cubes
Explanation:
We assume that the 45° angle is measured between the cut plane and the face of the 4×4×4 cube. For the purpose of figuring this out we have defined the cube to lie with its upper front corner at (x, y, z) = (0, 0, 0), and its diagonally opposite corner to lie at (x, y, z) = (4, 4, -4). The cut plane has equation ...
x/4 +y/4 -z/√8 -1 = 0
For a given small cube corner point (x, y, z), the diagonally opposite corner point will be (x +1, y+1, z-1). We have assumed that a small cube is cut if its diagonally opposite corner points lie on opposite sides of the cut plane. The side of the cut plane the corner lies on will be indicated by the sign of the sum ...
x/4 +y/4 -z/√8 -1
For the 64 corners of interest, we can use a spreadsheet to calculate whether the cube corner and its diagonal opposite lie on opposite sides of the cut plane.
We find that 9 cubes on the top layer are cut, 6 cubes on the second layer, and 3 cubes on the 3rd layer are cut, for a total of 18 cut cubes. The entire bottom layer is untouched by the cut plane, which exits the z-axis at -√8 ≈ -2.8, above the top of the bottom layer at z = -3.
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The attachment shows the spreadsheet output for the top layer of cubes. Even values ±4 indicate the entire cube is on one side of the cut plane. Even values ±2 indicate the cube straddles the cut plane (is cut).
Odd values (±1 or ±3) or 0 indicate a corner of the cube lies on the cut plane. This is only the case for the top plane of cubes, where the face diagonal is the line of intersection.
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Alternate solution
The second attachment shows the 4×4 array of cubes with the cut lines at each layer. There is no cut line between layers 3 and 4, because the cut plane exits the cube above that layer. The cut cubes on each layer are ones that lie between the cut line on the plane above and the cut line on the plane below. (Here, we're using "cut line" to mean the line of intersection between the cut plane and the plane bounding a cube layer.)