What is the least possible number of cuts required to cut a cube into 64 identical pieces?

The correct option is 9.

To understand why 9 cuts are the least possible number of cuts, let us look at how cuts divide a cube into smaller, identical pieces.

A cube is a three-dimensional object. To cut it into smaller identical pieces, we make cuts along three mutually perpendicular directions: parallel to the X-axis (lengthwise), parallel to the Y-axis (widthwise), and parallel to the Z-axis (heightwise).

Let:
_x be the number of cuts made along the X-axis,
_y be the number of cuts made along the Y-axis, and
_z be the number of cuts made along the Z-axis.

Making _x parallel cuts in one direction divides the cube into (_x + 1) slices in that direction. Therefore, the total number of pieces, $N$, obtained by making cuts along all three directions is given by the product of the slices in each direction:

$N=(n_x+1)\times (n_y+1)\times (n_z+1)$

We are given that the cube must be cut into 64 identical pieces. Therefore, we set:

$(n_x+1)\times (n_y+1)\times (n_z+1)=64$

To minimize the total number of cuts, which is $n_x+n_y+n_z$, we need the factors $(n_x+1)$, $(n_y+1)$, and $(n_z+1)$ to be as close to each other as possible. This is because, for a constant product, the sum of terms is minimized when the terms are equal or close to equal.

Since 64 is a perfect cube, we can write it as:

$64=4\times 4\times 4$

By equating the factors, we get:
_x + 1 = 4 ⇒ _x = 3
_y + 1 = 4 ⇒ _y = 3
_z + 1 = 4 ⇒ _z = 3

Now, we calculate the total number of cuts required:

$\text{Total cuts}=n_x+n_y+n_z=3+3+3=9$

Thus, the minimum or least possible number of cuts required to partition the cube into 64 identical pieces is 9.