# The Big Lock-Down Math-Off, Match 14

Welcome to the 14th 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.

## Colin Beveridge – Conway’s Circle, a proof without words

Colin blogs at flyingcoloursmaths.co.uk and tweets at @icecolbeveridge.

Given triangle ABC, extend sides $$a$$ and $$c$$ through B by a length equal to side $$b$$. Similarly, extend sides $$b$$ and $$c$$ through A and sides $$a$$ and $$b$$ through C. Then the ends of these sides lie on a circle (the Conway Circle) which is concentric with the incircle of ABC.

QEF.

## Alex Cutbill – The difference of two squares with a twist

Alex is a freelance maths education consultant. He tweets at @intersectarian.

The difference of two squares, $a^2-b^2$, can be factorised as $(a+b)(a-b)$.

This lovely little identity can be proved algebraically using the distributive and commutative laws:

$(a+b)(a-b) \equiv a(a-b) + b(a-b) \equiv a^2-ab+ba-b^2 \equiv a^2-ab+ab-b^2 \equiv a^2-b^2$

But, like many things, it’s much nicer to prove it with pictures. Here are the graphics from Wikipedia:

This geometric proof compares favourably with the original algebraic proof, not least because it goes in the right direction, from $a^2-b^2$ to $(a+b)(a-b)$. The algebraic proof feels a bit like factorising 28,554,961 by multiplying 6,899 and 4,139 together.

Anyway, let me come to the point of my pitch: you’re probably already sold on this geometric proof, having seen it long, long ago, but there’s another geometric proof which you may not have seen.

Take a while to get used to the perspective – it’s in 3D! – and then slowly increase the value of θ to 180 by moving the slider.

I think of both geometric proofs in terms of physical manipulatives. Demonstrating the first proof involves rearranging two blocks, carefully positioning them relative to each other. The second proof suggests a single manipulative, whose two halves are held together by the axle which provides the only affordance; all you can do is twist, and with a twist the identity is proved. To me, the second manipulative, and hence the second proof, is more elegant.

I can’t take credit for this proof; while I rediscovered it independently, I subsequently found it in the wild:

I’d be interested to hear from you if you’ve seen this proof elsewhere. I’d also be interested to hear from you if you can make me the twisty manipulative I describe above! (And so, the ulterior motive for sharing this pitch is revealed.) If, like me, your practical skills are practically non-existent, you might like to consider the following puzzle instead. It also comes with a twist!

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

Match 14: Colin Beveridge vs Alex Cutbill

• Colin with Conway's circle (63%, 31 Votes)
• Alex with the difference of two squares (37%, 18 Votes)

Total Voters: 49

This poll is closed.