We want to cut a cube with an edge length of 4 units into 64 smaller cubes, each with an edge length of 1 unit. This can be done simply with 9 cuts if the pieces resulting from the cuts are not moved apart from each other. How many cuts can this be reduced to if the pieces can be rearranged appropriately after each cut?
Since the cube is convex, we can cut it into two parts with one cut, meaning that with each cut, we can at most double the number of pieces we have. This means that we definitely need 6 cuts, because after 5 cuts, we can get at most $2^{5}=32$ pieces. Six cuts are sufficient, as shown in the figure.

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Remark. The necessity of 6 cuts for the dissection can also be shown as follows. Consider a unit-edged small cube inside the larger cube, which touches all its neighboring small cubes with each of its faces. This small cube must be cut out. Since the cube is convex, only one of its faces can be separated with one cut, so 6 cuts are needed to remove all its neighbors. This also implies that even for dissecting a cube of edge length 3 into 27 pieces, 6 cuts are required, since there is also a "inner" small cube in that case.