Q. A solid cube is painted yellow on all its faces. The cube is then cut into 60 smaller but equal pieces by making the minimum number of cuts.

Explanation : 

1. Let the cuts be made by slicing the cube with planes parallel to its three pairs of faces.
• Suppose we make a₁ cuts along the x–direction, a₂ cuts along y, and a₃ cuts along z.
• Then the total number of small pieces is
(a₁ + 1)·(a₂ + 1)·(a₃ + 1) = 60.
• The total number of cuts is
N = a₁ + a₂ + a₃,
and we want N as small as possible.

2. Factor 60 into three positive integers whose sum is minimal.
Possible triples (a₁ + 1, a₂ + 1, a₃ + 1) and their sums:
1×6×10 → sum = 17
2×5×6 → sum = 13
3×4×5 → sum = 12 ← minimal
Thus take (a₁+1, a₂+1, a₃+1) = (3, 4, 5).
So a₁=2, a₂=3, a₃=4, giving N = 2+3+4 = 9 cuts.
This proves statement I.

3. Count the pieces not painted on any face.
A piece lies entirely interior if it is not on any outer layer.
Along each axis we must avoid the two boundary layers, so interior count =
(a₁+1 − 2)·(a₂+1 − 2)·(a₃+1 − 2)
= (3−2)·(4−2)·(5−2)
= 1·2·3
= 6.
This proves statement II.

Hence both statements I and II are correct.