GATE CS 2024 Set 2 — Question 9

To cut a cube into $n$ pieces with minimum cuts, we should distribute the cuts as evenly as possible across the three dimensions (x, y, z).
Let $x, y, z$ be the number of cuts along the three axes. The total number of pieces is given by $(x+1)(y+1)(z+1)$.We want $(x+1)(y+1)(z+1) = 8$.
To minimize the total number of cuts $k = x+y+z$, the factors $(x+1), (y+1), (z+1)$ should be as close to each other as possible.
Since $8 = 2 \times 2 \times 2$, we can set:
$x+1 = 2 \implies x = 1$
$y+1 = 2 \implies y = 1$
$z+1 = 2 \implies z = 1$Total cuts $k = 1 + 1 + 1 = 3$.
These 3 mutually perpendicular cuts passing through the center of the cube will divide it into 8 smaller cubes of equal size and shape.