# Problem of the Week Problem C and Solution I Want More Cubes

## Problem

Rashid has a wooden cube with a side length of $$10$$ cm. He makes three cuts parallel to the faces of the cube in order to create $$8$$ identical smaller cubes, as shown.

What is the difference between the surface area of the original cube and the total surface area of the $$8$$ smaller cubes?

## Solution

Solution 1

Each face on the original cube has an area of $$10 \times 10=100~\text{cm}^2$$. Since there are $$6$$ faces on a cube, the surface area of the original cube is $$100 \times 6=600~\text{cm}^2$$.

Each of the smaller cubes has a side length of $$5$$ cm. So the surface area of each smaller cube is $$5 \times 5 \times 6 = 150~\text{cm}^2$$. There are $$8$$ smaller cubes, so the total surface area of the smaller cubes is $$8 \times 150=1200~\text{cm}^2$$.

Therefore, the difference in surface area is $$1200-600=600~\text{cm}^2$$.

Solution 2

Each cut increases the surface area by two $$10~\text{cm}\times 10~\text{cm}$$ squares, or $$2\times 10 \times 10=200~\text{cm}^2$$.

Since there are three cuts, the increase in surface area is $$3\times 200~\text{cm}^2=600~\text{cm}^2$$.