Season 11 Episode 1

Last year we talked about magic squares, but there's lots more to say! No need to watch last year's episode with Ayliean first, but if you like this one then you might like that one too.

Watch on YouTube

Further Reading

Previous Ayliean episodes

For the previous episode on magic squares, see Season 10 Episode 5.

Ayliean was also on the show last year to talk about Morse code. TOOT! Season10 Episode 0.

You can find more Ayliean content on YouTube and on TikTok.

If you like fractal time-lapses, you might like to start with this TikTok.

Basic Magic

I’d like to say more about what I call the "linearity of magic", using Ayliean’s description of 3x3 magic squares. If you remember, there was a point where Ayliean talked about the difference between the number in a corner and the number in the middle, using the word see-saw to describe the balance between that pair and the number in the opposite corner.

We can express this with algebra; let $A$ be the number in the middle, and let’s call the number in the top-left corner $A+B$. Then because the sum of the numbers in a diagonal is three times the number in the middle (see the previous episode for a proof!), the number in the opposite corner is $A-B$.

Similarly, let’s call the number in the top-right corner $A+C$. Then we can fill in the other numbers in the magic square to get
$$\left(\begin{matrix}A+B & A-B-C & A+C\\A-B+C & A & A+B-C\\A-C & A+B+C & A-B\end{matrix}\right)$$

It’s worth working that out for yourself on a scrap of paper to convince yourself that everything else in the magic square really is determined from these three numbers.

What I’d like to do next is to separate out the bits that depend on $A$ and $B$ and $C$, re-writing our magic square as
$$\left(\begin{matrix}A & A & A \\A & A & A \\A & A & A\end{matrix}\right)\quad + \quad \left(\begin{matrix}B & -B & 0 \\-B & 0 & B \\0 & B & -B\end{matrix}\right)\quad +\quad \left(\begin{matrix}0 & -C & C \\C & 0 & -C \\-C & C & 0\end{matrix}\right).$$

Now, I'm being a bit silly here by just writing + to mean "add up the corresponding elements of these magic squares". But not that silly.

If you like, there are three basic 3x3 magic squares
$$\left(\begin{matrix}1 & 1 & 1 \\1 & 1 & 1 \\1 & 1 & 1\end{matrix}\right)\quad\text{and}\quad\left(\begin{matrix}1 & -1 & 0 \\-1 & 0 & 1 \\0 & 1 & -1\end{matrix}\right)\quad\text{and}\quad\left(\begin{matrix}0 & -1 & 1 \\1 & 0 & -1 \\-1 & 1 & 0\end{matrix}\right).$$

Then all other 3x3 magic squares come from a combination of those, multiplying each by some constant, and then adding them together.

To me, this feels a little bit like discovering that light is made of primary colours, or molecules are made from a small set of atoms, or DNA contains just four bases!

There’s a concept in mathematics called a basis for a vector space, and that’s precisely what’s happening here for the magic squares. You will learn more about this if you take Mathematics at university, probably in a course called something like Linear Algebra.

Sums to a cube

Ayliean showed us that the sum of consecutive odd numbers gives a square number, and we saw a proof that uses a diagram of a square. What happens in higher dimensions?

Once you’ve seen a square being built layer-by-layer (with a corner bit), perhaps the next thing to think about is a cube made out of layers somehow? With edge bits? I’ll let you fill in the details. I’m being intentionally vague! It should not be clear from that description precisely what we’re adding together. We know that the answer is going to be $n^3$... but what’s the question?

Doubly alpha-magic squares

There's some code here for the doubly alpha-magic squares, the ones where you count the number of letters in the words that spell out the number.

We got a bit side-tracked by talking about the difference between a British billion ($10^{12}$). Since the livestream I’ve looked it up and it turns out that this definition has been obsolete for the last 50 years (I now understand that one Slido message about 1974!). A British billion is now $10^9$, same as everywhere else.

Users in slido also taught me about the Indian counting system including lakh for $10^5$ and crore for $10^7$, and Chinese numerals which include special characters for $10^4$ and $10^8$. Putting these together, perhaps we could say that the more languages you speak, the easier it is to describe large numbers!

Thanks for the helpful Slido comments, and keep them coming.

Other cool stuff

Ayliean heard someone mention Rubik’s cubes and wants you to know that there is a special “sub cube” recently added to the Maths Gear website. It's just like Rubik’s cube, except the corners and the edges each have rotational symmetry. That sounds more complicated than it is! To solve this cube, you just need to get the pieces into the right places, and you don't need to worry about their orientations. Technically, the symmetries of a the sub cube form a sub-group of the symmetries of a normal cube. Maybe one day we’ll explore that on OOMC.

Pythagorean triples

Blackpenredpen has a video with a method to find all Pythagorean triples: YouTube | finding ALL pythagorean triples (solutions to a^2+b^2=c^2)

Keith Conrad at the University of Connecticut has a different method which converts the problem of finding integer (whole-number) solutions to $a^2+b^2=c^2$ to the related problem of finding rational (fraction) solutions to $x^2+y^2=1$, points on the unit circle. There’s a really elegant method to find these involving a straight line with rational gradient, and I’m including it here because the method works for certain other equations. It’s more advanced than A-level or equivalent, but if you skip to page 4 then you might get a sense of what the idea is. PDF here.

Ayliean’s plots come from this blog (which uses a third way to generate Pythagorean triples!) and from the Wikipedia page, where the faint lines on the plot are related to the variables in Euclid’s formula for Pythagorean triples. The Wikipedia page contains a little widget for you to calculate your own Pythagorean triple using Euclid’s formula (I haven’t seen a similar widget anywhere else on Wikipedia, so I think that this is pretty neat!).