Cubic Masterpiece
Weekly Problem 49 - 2014
A blue cube is cut into 27 smaller cubes of equal size. What fraction of the total surface area of these cubes is blue?
A blue cube is cut into 27 smaller cubes of equal size. What fraction of the total surface area of these cubes is blue?
Age
11 to 14
Challenge level
Problem
A solid wooden cube is painted blue on the outside. The cube is then cut into 27 smaller cubes of equal size.
What fraction of the total surface area of these new cubes is blue?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Let the smaller cubes have side length $1\,\mathrm{unit}$. So the original cube had side of length $3\,\mathrm{units}$ and as a cube has six faces it had a surface area of
$$6\times(3\,\mathrm{units}\times 3\,\mathrm{units})=54\,\mathrm{units}^2\;,$$
all of which was painted blue.
The total surface are of the 27 small cubes is
$$27\times 6\,\mathrm{units}^2 = 162\,\mathrm{units}^2\;.$$
So the required fraction is
$$\frac{54\,\mathrm{units}^2}{162\,\mathrm{units}^2} = \frac{1}{3}\;.$$