# The Big Lock-Down Math-Off, Match 6

Welcome to the sixth match in this year’s Big Math-Off. Take a look at the two interesting bits of maths below, and vote for your favourite.

You can still submit pitches, and anyone can enter: instructions are in the announcement post.

Here are today’s two pitches.

## Zoe Griffiths – What’s wrong with Geistes Blitz (Ghost Blitz)?

Zoe Griffiths is a maths communicator for Think Maths. You can find her on Twitter at @zoelgriffiths.

Here is a zip file containing the cards and instructions for play for ‘Shape Match’:

Note, my observations are based on one edition of Geistes Blitz only.

## Emma Bell – A beautiful cuboid puzzle

Emma Bell is Maths Enhancement Manager at Grimsby Institute. On Twitter she is @El_Timbre.

Calculate the volume of the cuboid, given that:

\begin{align} pq &= 10 \\
qr &= 18 \\
&\text{and} \\
pr &= 20 \end{align}

I love this “puzzle” because of the journey it takes you on as you solve it…

Have a play with it, and come back to hear my thoughts!

The first path I took was to see if I could use the given products to find the values of each of the separate dimensions.

\begin{array}{ccc}
pq=10 & qr=18 & pr=20 \0.5em] \color{red}{q=\frac{10}{p}} & \color{blue}{r=\frac{18}{q}} & \color{blue}{r=\frac{20}{p}} \\[0.5em] \color{green}{p=\frac{10}{q}} & \color{red}{q=\frac{18}{r}} & \color{green}{p=\frac{20}{r}} \end{array} This looked promising! I had pairs of equations which I could group together, and perhaps the dimensions would fall out nicely? \begin{align} \color{green}{\frac{10}{q}} &= \color{green}{\frac{20}{r}} \\[1em] \color{red}{\frac{10}{p}} &= \color{red}{\frac{18}{r}} \\[1em] \color{blue}{\frac{18}{q}} &= \color{blue}{\frac{20}{p}} \end{align} With some substitution and manipulation, I got there… \[ \color{green}{ \begin{align} \frac{10}{p} &= \frac{18}{r} \\ \frac{10}{p} &= \frac{18}{\left(\frac{20}{p}\right)} \\ \left( \frac{20}{p} \right) \left(\frac{10}{p} \right) &= 18 \\ \frac{200}{p^2} &= 18 \\ 200 = 18p^2 \\ p^2 = \frac{100}{9} \\ p = \frac{10}{3} \end{align} \\ \text{(+ve only!)} }
\color{red}{ \begin{align} \frac{10}{q} &= \frac{20}{r} \\ \frac{10}{q} &= \frac{20}{\left(\frac{18}{q}\right)} \\ \left( \frac{18}{q} \right) \left( \frac{10}{q} \right) &= 20 \\ \frac{180}{q^2} &= 20 \\ 180 &= 20q^2 \\ q^2 = 9 \\ q &= 3 \end{align} \\ \text{(+ve only!)} }
\color{blue}{ \begin{align} \frac{10}{p} &= \frac{18}{r} \\ \frac{10}{\left(\frac{20}{r}\right)} &= \frac{18}{r} \\ 10 &= \left(\frac{18}{r}\right)\left(\frac{20}{r}\right) \\ 10 &= \frac{360}{r^2} \\ 10r^2 &= 360 \\ r^3 &= 36 \\ r &= 6 \end{align} \\ \text{(+ve only!)} }

\begin{align}
\text{Volume} &= p q r \\
&= \color{green}{\left(\frac{10}{3}\right)} \times \color{red}{3} \times \color{blue}{6} \\
&= 60 \text{ cubic units}
\end{align}

However, I wasn’t mathematically satisfied… yes, I’d reached a solution, but it didn’t seem elegant enough. It was too mechanical, and dare I say it, BORING…

I started again (and I’ll admit, it was handy to know where I was going!)

\begin{align}
(pq)(qr)(pr) &= 10 \times 18 \times 20 \\
p^2q^2r^2 &= 3600 \\
(pqr)^2 &= 60^2 \\
pqr &= 60
\end{align}

Now THAT’S beautiful!

So, which bit of maths made you say “Aha!” the loudest? Vote:

Match 6: Zoe Griffiths vs Emma Bell

Emma with cuboid volumes
Zoe with Geistes Blitz

Total Voters: 59

This poll is closed.