We’re going to talk about magic squares of squares, so first thing we’re going to do is talk about what a magic square is. We’re going to start with a three by three grid: 8, 1, 6; 3, 5, 7; 4, 9, 2. This is a three by three array of numbers and what’s incredible here is that if you add the numbers down the columns (e.g. 8 + 3 + 4) you’re going to get 15, no matter what column you add down. The same goes for the rows, and the two diagonals - all of them add up to 15. This is a perfect, pure magic square and is also the earliest known recorded magic square, usually called the Lo Shu. Records of it date back to about 2200 BC, and the Chinese tradition has it that the emperor saw this magic square on the back of a turtle.
Is a magic square still a magic square if any number is repeated or is that kind of imperfect? I would argue that those are imperfect magic squares, and in fact that’s what makes it hard to find magic squares if you allow repetitions. You can easily find magic squares if you just repeat all the numbers, like taking your favorite number minus 19. ins in this case the condition that all the entries be square numbers then this is a magic square and so in a way it’s like a very very close to being a magic square of squares
The Parker Square is an attempt at a magic square of squares, where every single entry is a square number. However, one of the diagonals does not add up to the same number as the other entries, making it a semi-magic square. The Bremner Square is another attempt, however two entries are not squares. The Salos Square is a third attempt, and it is almost a magic square, with all the entries being distinct. However, one of the diagonals does not add up to the same number as the other entries. ature and then we looked at other slices and we found that it had finitely many of these rational or elliptic curves
This actually has to do with Pythagorean triples, but that’s a story for a different day. What’s important here is that if $T$ is an integer or some rational number, then these expressions are also going to be integers or fractions. More generally, there’s heaps of rational numbers on the real line, and so I just picked some rational number $T$ and I do this construction and it gives me a point on my curve that has a rational coordinate. So rational curves have this incredible property that the moment you have one point, you can use that one point to construct lots and lots and lots of other points.
Then there’s elliptic curves, which sort of have this funny kind of shape usually defined by qubit polynomials. Elliptic curves have this incredible thing that you can add points on them. So if I take $P$ and $Q$, I can call this $P + Q$ and this sort of crazy little construction happens to satisfy things like associativity. This allows you to also create new points out of old points. If I have an old point, I can add it to itself for example and then create a new point, and then add that point to itself and create a new point.
Then there’s general type curves, which are more complicated, but one thing that’s amazing is that they always have finitely many points with coordinates that are rational. This is a spectacular theorem from the early 80s, proved by Faltings and awarded the Fields Medal.
So when we have curves, they sort of come in three different flavors. Two of those flavors have lots and lots of points, and one flavor does not. What happens with the surface that we were calling the Parker surface is that, as a consequence of work done by me and a couple of co-authors, we managed to show that on this surface there are only finitely many curves of these two types. This is something that surprised us. Our surface is full of curves made of curves, and if you’re looking for points on that give you a magic square of squares, one thing you might try to do is instead of looking at the whole surface, just look at a curve inside that surface and maybe you get lucky and maybe your curve is one of these two types and then you can use what we know about these two types to construct magic squares of squares.
The Parker surface has 368 of these rational or elliptic curves. This means that it only needs one, but it has more than one. The way we’ve constructed these is that we looked at slices like this - what happens if I take $X_1$ equal to $X_2$? That gives me some kind of curvature, and then we looked at other slices and found that it had finitely many of these rational or elliptic curves. so
It’s kind of crazy that there’s a 4x4 magic square, but not a 3x3. You can find a 5x5 and a 6x6 on the internet, and it’s not surprising from a geometry point of view. If you take an NxN grid with N at least 4, you can produce one of these magic squares. It would be really cool to prove that you can get an NxN for any N at least 4. It’s possible that there could be an NxN square where there isn’t a solution - that would be amazing! There’s some really interesting mathematics going on behind the scenes, and this square that Euler came up with is important in the proof that every integer is a sum of four squares.