Painted Purple
Three faces of a $3 \times 3$ cube are painted red, and the other three are painted blue. How many of the 27 smaller cubes have at least one red and at least one blue face?
Age
14 to 16
Challenge level
Problem
A wooden cube has three of its faces painted red and the other three of its faces painted blue, so that opposite faces have different colours.
It is then cut into $27$ identical smaller cubes.
How many of these new cubes have at least one face of each colour?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
If opposite faces are painted in different colours, three red faces will meet at one corner, and three blue faces will meet at the diagonally opposite corner.
The three cubes that are adjacent to these two corner cubes will also be painted in just one colour.
The six cubes at the centre of each face will also be painted in just one colour.
The cube at the very centre will not be painted.
The remaining $12$ cubes will have at least one face painted in each colour
(27 - 2 - 2x3 - 6 - 1 = 12)