### Picture Story

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

### There and Back

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

### Doesn't Add Up

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

# Cuboid Perimeters

##### Age 14 to 16 ShortChallenge Level

Answer: $35\text{cm}^3$

Label the widths $a, b, c$

This perimeter is equal to $a+b+a+b=2a+2b$

This perimeter is equal to $2a+2c$

Similarly, the third perimeter will be equal to  $2b+2c$

$$2a+2b=12\Rightarrow a+b=6\\ 2a+2c=16\Rightarrow a+c=8\\ 2b+2c=24\Rightarrow b + c = 12$$

Solving by elimination
$\quad\qquad a+b=\ 6\\ \underline{+\ \quad\quad a+c=\ 8\ \ }\\ \quad 2a+b+c=14$

$\quad 2a+b+c=14\\ \underline{-\ \quad\quad b+c=12\ \ }\\ \quad \qquad\quad \ 2a=\ 2$

$\therefore a=1$
So $b=5$, $c=7$, volume $=1\times5\times7=35\text{cm}^3$

Solving by substitution
a+b=6\Rightarrow b=6-a\\ a+c=8\Rightarrow c=8-a\\ \begin{align}b + c = 12\Rightarrow &(6-a)+(8-a)=12\\ \Rightarrow &14-2a=12\\ \Rightarrow &a=1\end{align}

So $b=5$, $c=7$, volume $=1\times5\times7=35\text{cm}^3$

Adding all of the equations together
$\quad\qquad a+b=\ 6\\ \qquad\quad a+c=\ 8\\ \underline{+\ \quad\quad b+c=12\ \ }\\ \ \ \ 2a+2b+2c=26$

$\therefore a+b+c=13$

$c=13-6=7$
$b=13-8=5$
$a=13-12=1$
So the volume is $1\times5\times7=35\text{cm}^3$.

You can find more short problems, arranged by curriculum topic, in our short problems collection.