A cube of side 15 cm, painted on all its faces, is cut into small cubes of side 3 cm. How many of the small cubes have only one face painted?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 54
A small cube has only one face painted when it lies in the middle part of a face of the big cube, away from every edge; cubes on an edge or a corner have two or three painted faces.
Small cubes along each edge \(=\displaystyle\frac{15}{3}=5\)
On each face, leaving out the two edge rows each way: \((5-2)^2=9\) cubes have one face painted
The cube has 6 faces: \(6\times9=54\)
Tip: For a painted cube cut into \(n\times n\times n\) small cubes: 3 faces painted \(=8\) (corners), 2 faces \(=12(n-2)\), 1 face \(=6(n-2)^2\), no face \(=(n-2)^3\).