Beginning with a 3cm by 3cm by 3cm cube, a 1cm by 1cm by 1cm cube is cut from one corner and a 2cm by 2cm by 2cm cube is cut from the opposite corner, as shown.
In cm, what is the surface area of the resulting solid?
, 2014
Pick one
Solution
Assume the top face of the cube is coloured green.
Since the front face of the cube shares an edge with the top face, it cannot be coloured green. Thus, we need at least two colours.
Thus, we assume that the front face is coloured blue, as shown in Figure 1.
Since the right face shares an edge with the top face and with the front face, it cannot be coloured green or blue. Thus, we need at least three colours.
Thus, we assume that the right face is coloured red, as shown in Figure 2.
We have shown that at least 3 colours are needed. In fact, the cube can be coloured with exactly 3 colours by colouring the left face red, the back face blue, and the bottom face green (Figure 3).
In this way, the cube is coloured with exactly 3 colours and no two faces that share an edge are the same colour.
Therefore, 3 is the smallest number of colours needed to paint a cube so that no two faces that share an edge are the same colour.
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