Chequered Cuboid
A large cuboid is made from cubes of equal size. What fraction of the surface area of the large cuboid is black?
Problem
A cuboid is made from cubes of equal size, coloured alternately white and black as shown.
What fraction of the surface area of the cuboid is coloured black?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Consider the different faces of this cuboid. The faces on the ends of the cuboid will look like these diagrams on the right. Between them they have $9$ black squares and $9$ white squares.
The other four faces all look like the diagram to the right. This contains $6$ white squares and $6$ black squares.
Therefore, between the four faces there are $6 \times 4 = 24$ white squares and $6 \times 4 = 24$ black squares.
Therefore, overall there are $9+24=33$ white squares and $9+24=33$ black squares, so $\frac{1}{2}$ of the squares are coloured black.