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Make Magic Squares like a Renaissance Artist!

Albrecht Dürer ranks as one of my favorite artists, and wouldn't you know it, Renaissance artists were obsessed with math - probably moreso than me, or any of us reading (unless Terence Tao has found his way on my neocities :þ).

I would like to focus in on this with one of his most famous works, that being Melencolia I

(above image is sourced from Wikimedia)

Specifically, in the background, you can see a 4x4 square of numbers - these are:

when adding any row or column of these numbers, it will always add up to the same number, which is 34 in this case. We call squares of numbers with this property "magic squares," In addition, when all of the diagonals also add up to that same number, we call it "pandiagonal." Further still, the number "opposite" any other number in the Melencolia square adds up to 17, which makes it a "symmetric" magic square.

So, we have a single square of numbers satisfying loads of cool, unique properties - is that why am I making a post about it? No! I'm talking about it because it's NOT unique!

Tons of other magic squares are pandiagonal and/or symmetric - I've done so by playing around with Euler's Method (not to be confused with Euler's Method in calculus, or the billion other things named after Leonhard Euler). What I wanted, however, was to generalize a specific property of the Melencolia square I find very unique and satisfying.

My favorite property of the Melencolia square is one I have failed to mention up until now: it has a very unique pattern to how numbers are placed. I kind of struggle to describe it in words, so I hope this makes sense (I may rewrite this later if it's too confusing)

1. Place a 1 in the bottom right (or whatever corner you like, if you're willing to flip the directions in this guide)
2. Place numbers increasing by 1 going left to right BUT cross to the opposite row on the first move and then every other move after that. (in this case, we go from bottom row to top row for 1 to 2. We have only 3 moves, so we stay in the top for 2 to 3 and the go back down for 3 to 4)
3. There's a number of heuristics to "continue" the pattern, but they should do the same thing. I like to repeat the process by placing the largest number in the opposite corner to where the 1 is placed, then repeat the above process but going in the opposite direction and counting down. This completes the bottom and top rows. Then, with the last number i wrote down in a given row, I put the number that would follow it on the same column, but as far away as possible vertically (look at the distance between 13 and 12 or 5 and 4 on the Melencolia square, for instance.) Then I repeat step 2 with the remaining space.

ONE MUST NOTE!!!: This only works for magic squares whose side lengths are divisible by 4 - if its odd, you're left with a single middle row and the instructions fall apart - if it's even but not divisible by four, you're left with a situation where the the 4 numbers in the middle have to add up to n^2+1 with both the number opposite them AND the number directly above/below them, which cannot be done if we're using unique numbers in each column (which we kind of have to, otherwise we could just put in a magic square full of nothing but the number 1 and call it a day)

So, as an example, I composed a magic square of side length 12 following that same sort of pattern as the Melencolia square, since you should do the 8x8 by yourself, as its the easiest to do using this method besides the Melancolia square itself: