The Korean king's magic square: a brilliant algorithm in a k-drama (plus geomagic squares)
A double feature on magic squares featuring Bachet's algorithm embedded in the Korean historical drama series Tree with deep roots and the Lee Sallow's geomagic squares.
00:00 Intro
02:52 Part 1: The king's magic squares
09:40 Proof
18:22 The order 5 and 7 magic squares
19:17 Part 2: Geometric magic square
30:59 Thanks!
The Korean historical drama Tree with deep roots is available here www.viki.com/tv/1585c-tree-wi... All the magic square action takes place in episodes 1 and 2.
Episode 1: the king's study with magic squares at 33:17 again at 42:00 (father "simplifies" magic squares)
Episode 2: lunchbox action starts around 46:00 then again at 58:39 (the AHA moment)
Lee Sallows's book: Geometric magic squares, Dover (2013)
His website: www.leesallows.com
His comprehensive online gallery of stunning geomagic squares: www.geomagicsquares.com/
Nice write-up about the 33x33 magic square in Tree with deep roots: tinyurl.com/mszxrf2w
Wiki page on Claude Gaspar Bachet de Méziriac: en.wikipedia.org/wiki/Claude_...
Eduard Lucas 3x3 magic square equation: en.wikipedia.org/wiki/Magic_s...
An app by Ilm Narayana that demonstrates the king's method for magic squares up to order 33 (thank you very much Ilm for accepting my coding challenge:) editor.p5js.org/ilmnarayana/f...
Bachet's magic square algorithm write-up / bachets-magic-square (this article by Humanicus features the same proof that I present in this video)
Here is a magic square from an old Chinese manuscript en.wikipedia.org/wiki/Magic_s... Among other things they are writing from top to bottom which would also have been done in Korea at the time. In fact, in the drama the tech support ladies can be seen writing from top to bottom. So that's a bit of a blooper when it comes to the writing on the tiles. A few other minor issues are discussed in the comments. What's also interesting here that they went for a 33x33 magic square and 33x33=1089. 1089 is a four-digit number and all the tiles are only labelled with three numerals. How did they write 1089 and still make sense? :) Why not use a 31x31 magic square? 31x31=961.
For the second method for building odd order magic squares check out this link: en.wikipedia.org/wiki/Magic_s...
Some bugs:
- at 18:48, there is no green circle on the 2nd row 3rd column square.
- at 23:00, I should have said: take any 3 or more of the numbers that add to 15, then the corresponding pieces combine into the 4x4 with the bite (important because, for example, 7 and 8 don't work).
- at 28:20 one of the pentominoes is a hexomino :)
Today's music: Ardie Son - Counterparts
Today's t-shirt: 31415... Cannot remember where I found this t-shirt.
Enjoy!
Burkard
You know it's going to be a great video when you're only one minute in and you've already seen three excellent results.
You know that it is going to be a great video when it is Mathologer, you have just started the first second of the video, and you've already seen three comments pointing out how exceptional the video is. I am confident that I am going to agree with your assessment.
You know it's going to be a great video when it's posted by Mathologer.
you knoiw its going to be a great video when its mathologer
You know it's going to be a good video when the intellectuals watching the channel make idempotent statements just to emphasise how great the video is. No judgment, @neonblack211 :)
KZhead needs a super like button for this level of content. Bravo Mathologer!
Oh, that's what an empty lunch box means. I just thought my mom was a little absent-minded. But this makes so much more sense.
Empty lunchbox = you don't need any food = you're a dead man
I hope she wasn't too disappointed and decided better of it.
💀
either absent-minded or just a socialist either way I think Karl Marx would approve
Oh no…
Dear Mathologer. I can see how much work goes into these video's, but please never stop doing them. When I was a child I remember asking myself why we didn't have a TV channel that just showed educational programs. You (and a handful of others) make youtube what it can and should be. Thank you.
There’s a reason that the all-time mathematical greats like Euler, Ramanujan, Laplace, and Fermat were fascinated by magic squares and other patterns! It’s the knack and habit of recognizing those underlying structures which led them to some of the greatest insights and advances in mathematical history. THIS is why it’s so important to teach our kids more than rote memorization of numbers, facts, tables, and theories; we need to teach them how to see them as patterns which lead to other patterns within even further patterns. People with the gift of innate intuition about patterns are the people who change the world.
What's also interesting is that a LOT of people who are into maths look down on things like magic squares not realising how important these little things are in the grand scheme of things :)
my gift to the world is pointing out the OBVIOUS associations between the proof the fake Pitagoras used for the right triangle theorem, the 3x3 Lo Shu magic square, the 5x5 Rotas Sator palindrome and the 12,000+ year old 2D CHIRAL swastika ... but the ignorant do not want to acknowledge this insight due to WW2 crimes committed against humanity ... the truth is genius, hidden in plain sight and a bitter pill to swallow ....
@J Lund wtf
@@Mathologerhi mathologer I love your videos thank you for making such great content
This is why 바둑/圍棋/囲碁 is important to teach to al children.
So this is why sudoku named one of their optional rule as magic square.
18:28 mentions the general rule of filling odd magic square we were taught in China in primary school. The rules was written as so: 一居上行正中央，依次斜填切莫忘 上出框时向下放，右出框时向左放 排重便在下格填，右上排重一个样 Translation: 1. put 1 in the middle of the first row 2. fill next consecutive numbers diagonally 3. when going out on the top, put into the bottom row 4. when going out on the right, put into the left column 5. if the square is occupied, put into the square below 6. if going diagonally on the top right, the same rules applies
Very nice. The Chinese text looks like it may rhyme. Does it?
@@Mathologer It does rhyme! and it is deliberately written in the form of seven-word-poem (much like the solution poem to the Chinese remainder theorem). On this same topic, there's another, much more ancient (and famous) "poem" on just the 3x3 magic square: 九宫之义，法以灵龟，二四为肩，六八为足，左三右七，戴九履一，五居中央 Translation: """ The way to fill a 9-square palace, is to imagine a turtle (back): 2 and 4 as the shoulder, 6 and 8 as feet, 3 on the left, 7 on the right, 9 as hat, 1 as shoe, and 5 in the middle """ This text came from an ancient manuscript, that says these numbers/pattern comes on a turtle, and it's a sign of miracle. I guess that's part of what you said in the video: some people think magic square is truly magical.
@@DarknessGu1deMe That's great. Thank you very much for sharing this with me :)
@@DarknessGu1deMe There is a much simpler version by 杨辉(1127~1279) 九子斜排，上下对易，左右相更，四维挺出。 Which translates roughly to: 1. arrange the 9 numbers "along the diagonal direction" (3 X 3 tilted square) 2. switch top and bottom 3. switch left and right 4. "stick out" the 4 corners.
@@DarknessGu1deMe I think the poem ever showed up in 1994 chinese tv series "Legend of the Condor Heroes" when the main character was trying to solve the 3x3 magic square puzzle to open up a door to a secret place. The female character solved the puzzle whilst citing the poem after she saw a turtle nearby. I remember she put 5 in the middle square as her last move to complete the puzzle. cmiiw.
I can't believe you mentioned _Tree With Deep Roots._ Knowing next to nothing about Korean or Korean, I stumbled onto that drama around 2011 and was intrigued by the story of Hangul woven into it. I learned to read it and eventually became somewhat proficient in Korean, all because of a random K-drama! Funny how those things work.
I liked the drama but not the way it ended :)
@@Mathologer oh, yeah, I forgot the ending. By around the midpoint or so it began to get a bit ridiculous, I thought, but I did watch it till the end. (And it _did_ get me into Hangul and Korean quite unexpectedly.) _Edit:_ The original comment should read “Korea or Korean.” Oops!
8:24 The magic constant is sum of all numbers in the square divided by the number of rows. For those looking for a concise formula: it should be (n²(n²+1)/2)/n or simply n(n²+1)/2. Plugging in 33 will give us the magic constant of 17985. 18:48 Go up-right one space. If you exit the board, wrap around to the other side. If you run into a space already filled in, drop straight down one space instead.
That's it :)
You're right on. You know the sum of each row, but how do you figure out the individual numbers? This video showed an interesting technique.
I was using another way of reaching the solution. Is there a reason to add every single number together? Since we have an odd number of pieces intuitively the middle number multiplied by the number of rows is going to be the only answer to the question pertaining the sum. This is immediately obvious because if a number larger than this was the answer then there necessarily exists another column/row which is lacking. Average value is (n^2+1)/2 and we can multiply by n, reaching the same conclusion. I love your algebra though.
I think it's called the Siamese method for solving magic square in that fashion. cmiiw.
Conjecture: the number at the center of an odd magic square is always the middle number in the list, i.e. the average number in each cell i.e. (n^2+1)/2 (which is an integer as n is odd). Can you prove it? If this is false, can you find a counter-example? Also, given n odd, how many magic squares do exist (with the equivalence relation given by the trivial horizontal/vertical reflections and 90° rotations)?
The proof sketch was brillianty displayed. That "aha" moment you experience when it finally sinks in is priceless, almost addictive.
Love the Pi Tshirt, Nice Visualisation of Pi.
Neat proof of an amazing result! I'll have to watch that show because it sounds like the best use of math in a film / tv show ever. Also, nice pi shirt!
It's not a T-shirt, it's a Π-shirt! (Is there something encoded in blue vs white dots?)
@@ahcuah9526 It looks as if every third digit is in white; I'm unaware of any significance in that sequence of numbers, but with Mathologer I just *never* know!
The level of video and animation is amazing. This is a huge and talented work! СПАСИБО БОЛЬШОЕ 👌
And turns out, there is a 3D equivalent for magic squares called a magic cube. Going beyond 3D, we have magic hypercube. How can I never hear of this before? I love that the well-known classic problem has a lot of different variations to tinker about. Especially when each has a unique approach to the problem. The wonder of math always amazes me.
Reminds me of wordsquares that are cubed. For instance "CUBE" can be squared like this: CUBE UGLY BLUE EYES But then each of those other three words can be made into a wordsquare as well, which is then considered to have cubed the word "cube": UGLY GLUE LULL YELP BLUE L### U### E### etc.! (I'm sorry, I have forgotten the rest...)
@@FLScrabbler , I've tried working out the rest of this cube from what you've provided, but I am unsatisfied with the set up. The cube properties applied to the used words duplicates "lull" as stemming from the L in "ugly" and the L in "blue", and then duplicates "yelp" as stemming from the Y in "ugly" and the Y in "eyes" (I don't know if it is possible, but I would think a cube with unique words at every step would be more interesting). Then that sticks you with a 'ul' to start a word with, which is a tough fill. Here is the final result I came to: CUBE UGLY BLUE EYES UGLY GLUE LULL YELP BLUE LULL ULTS ELSE EYES YELP ELSE SPED
@@SgtSupaman Very nice! Well done..!
because if you try to make a magic cube it doesn't work.
@@SgtSupaman Very nice. Also; if you allow proper names, then one possibility would be to replace ”ULTS” with ”ULAM”, and ”ELSE” with ”ELMO”. 😌👍🏻
My favorite fact about the Dürer magic square: it was completed in the year 1514, and in the middle of the bottom row you can see it says 15 14. The only similar example I know of is the Basel problem, posed in 1644 whose solution is pi^2/6 = 1.644... However, sources differ on whether it was first posed in 1644 or 1650.
Actually the Durer square has a couple of other magic properties. In particular the four 2x2 blocks in the corners and the one in the middle also all add to the magic constant. Sadly these extra properties are not replicated by the geomagic counterpart :)
@@Mathologer This property is the reason why Dürer was able to put the year in middle of the bottom row after generating the MS by inverting the diagonals: The two central columns were swapped without affecting the magic constant...
I love seeing Magic Squares used in Sudoku variants. Thanks for sharing.
In case you're wondering, the dots on his shirt are the digits of pi
Of course, a nice feature of the 3*5 magic rectangle is the fact that it quite well approximates the golden rectangle, considering the size of its denominator.
It really makes me glad to see more China-related or Eastern Asia-related videos here!
this channel is absolutely fantastic!
Wanted to generalize the formula for the magic sum. for an n x n magic cube, I noticed that the sum of the "columns" are just Σk (k=1..n) and the sum of the "rows" is nΣk (k =1..n) - n² (the equivalent of nΣk (k =0..n-1) . That means the sum of all of those unique values (and the magic sum of the square) = Σk (k=1..n) + nΣk (k =1..n) - n² = (n + 1) nΣk (k =1..n) - n² substituting n/2(n+1) for Σk (k =1..n) = n/2 (n+1)² - n² And some math autopilot = n/2 (n² + 2n + 1) - n² = n³/2 + n² + n/2 - n² magic sum of n order magic square = *n/2 (n² + 1)* S₃₃ = 33 * (33² + 1)/2 = 33 * (1089 + 1)/2 = 33 * 1090/2 = 33 * 545 = *1795* _Also note it will always be an integer because n is odd so n² is odd and (n² + 1) is even and divisible by 2._
I was proud of myself for decoding your shirt 🧐 Not everyone can be a genius
One of your finest videos in my opinion, I really enjoyed it!
Something different. Glad you liked it :)
Lovely video on the magic square by a gentleman who is NEVER SQUARE!!! Gary in dreamland. Have a nice dream!!🙂☁️☁️☁️⛅💫🌟🌟🌟🌟🌟👉☁️⏰☁️👈
It's always exciting when these come out!
As always: beautifully and clearly presented.
Your animations were amazing, I would have never been able to visualize this without your excellent animations. I was truly impressed, I learned a lot thank you!
visual thinker to the point of mathematical difficulties, so these proofs scratch an elusive itch, so satisfying to watch those shapes fit
Your videos make may days… everytime… thank you.
Thanks for crafting such a lovely video. The hours spent in photoshop yielded a beautiful result, and the delightful subject matter made the video a joy to watch.
This is honestly the best video on magic squares on KZhead & the best comprehensible video on the topic I've ever seen. SUPER! B)
Your videos are just beautiful ! ❤ CONGRATULATIONS ! 🎉 🎉🎉
I discovered a similar trick to the king's magic square by taking an x-Sudoku and adding 9 * (n-1) from a 3x3 magic square to each cell in the corresponding region. It works because Sudoku grids are Latin squares and Latin squares are just magic squares with repeating numbers so just like your example it's taking another pattern with consistent sums and adding them together to make each number unique. You can use a similar technique to iterate magic squares creating any square of a length of power 3 (or any length multiplied by another) and it even works for magic cubes, I've checked up to length 125 by iterating a 5x5x5 twice. This video got me thinking you could construct a 4x4 latin square using playing cards and add 0, 4, 8 or 12 to each suit to create a magic square. It works.
This might be the most mindblowing piece of math I have ever seen. I had problems focusing on the rest of the video because my mind was reeling from the extreme and simplistic beauty of this structure!
you are a genius and such a good teacher who is willing to share to the world, thankyou
Great, as always!
I was shaking my head with disbelief half the time while watching this video. Amazing
Mathologer never disapoints. I'm so glad to know this channel.
Q: What’s the only thing better than a Mathologer video? A: Another Mathologer video
여기서 한국 드라마를 보게 될 줄은 꿈에도 몰랐네요. Never expected to see K-drama in this channel!!
There are a couple of other K-dramas that would be worth covering mathswise. In particular, Melancholia has got some good stuff :) en.wikipedia.org/wiki/Melancholia_(TV_series)
Thank you a lot professor Burkard, you are truly enlightening my life
I loved playing with the cube puzzles like those when I was a child. The manipulation of the pieces, not just in hand, but also in my mind lead me to a greater understanding of mechanics, physics, and engineering. Thank you for this great video!
I don't know if it's going to come up here but I once spent some time looking at 4x4 Panmagic Squares. There are the usual rows, columns, and diagonals, but, for instance there are also all 2x2 sub-squares many more sub-patterns.
You'll probably enjoy this one www.futilitycloset.com
Thanks again for anther great video. As I said i watch these on a nice Sunday afternoon to relax. :)
Ok. Im am a huge fan of k-dramas. And now you gave me a double reason to watch this one.
Maybe also watch Melancholia. That one has a lot of nice math(s), too :)
@@Mathologer Thanks for the tip.
I'm blown away by the amount of work necessary to build such a video, besides the knowledge and the insight needed 🙉
*edit:* this was such a cool video For an n-by-n square, the sum is n(n²+1)/2 The entire square is 1+2+...+n*n = n²(n²+1)/2, then divide that up into n "slices."
That's it :)
I never expected I could see a Korean drama in your channel. :-)
Gonna be another banger!
This was amazing!
Vielen Dank für das tolle Video!
Amazing! Thank you!
Another great video to share with my DP HL students. Thank you!
Magic, as always! 👍🏼👌🏼👏🏼
My fav YT Channel. Amazing Video.
You here use some magic to put this video together so nicely! 😍
Exceptional storytelling!
Could you do a video on Galois fields and the fundamental theorem of Galois theory? I remember my math professor trying to explain to us the importance and beauty of Galois fields during my computer science studies about 30 years ago, but the poor prof did not have today's tools available and somehow the magic of Galois fields remained hidden for us students.
Bravissimo! Thanks again guys. I love the way that your exploration of truly basic fundamentals of geometric-numeric logic inspire new ideas about efficient coding, data structures, etc. Cheers etc. ~ M
The diagonal construction of magic squares and the geo magic squares were both superbly presented... Really interesting!
Thank you very much, glad you liked it :)
Fascinating. Thank you for the clarity of the demonstration & the inspiration that comes with it!
Glad you enjoyed it!
Like many others in Mathologer-land, this video has helped me with some elementary school students I tutor. They need exercise in simple arithmetic, and need even more a window into how exciting and powerful math can be. As for me, I am now 4 episodes into Tree With Deep Roots. I got goosebumps watching the king solve the 33x33 square. Now I'm reading all this 15th century Korean political history. What a gas!
outstanding, thank you
What is interesting is that you can also turn any magic square into a new one by adding a non-zero integer constant to every square (the summations will change by the side length times the non-zero integer constant). Additionally there is probably some way to generate a new magic square by taking the modulus of every square (need to be careful about creating repeating numbers with certain values for the modulus. Edit: This actually might not be possible. I don’t have an example that it would work without causing a duplicate value).
Yep. There are a couple of other transformations that turn magic squares into new magic squares. Have a look here for a summary: en.wikipedia.org/wiki/Magic_square#Transformations_that_preserve_the_magic_property
@@Mathologer Since I made the comment I was investigating the group structure of flipping the square across the different axes and until I saw the wiki article you provided I completely missed that the group was isomorphic to D8 and could be simplified from 4 operations (a flip across the 4 different axes) to 2 (90 degree turn clockwise and a flip across one of the axes). Thank you for sharing the wiki article.
I saw this done as a trick by a teacher, who generated magic squares to a set numbers on demand, in reality they were just doing some maths tricks in their head. Very impressive today but as a child it was like genuine sorcery.
Brilliant. I will never forget how to make magic squares. Love it. I don't know if it will come in handy but I love it.
Fantastic video - loved it! 😍 I learned a method of constructing odd order magic squares when I was in Jr High School. It was much later that I became intrigued with even order magic squares. I finally managed to crack that by dividing the problem into even-odd (2, 6, 10, 14, etc) and even-even (4, 8, 12, 16, etc) cases with a different technique for each.
Great video. I really loved learning about Bachet's algorithm and geomagic squares. I was also reminded of orthogonal latin squares when you described Bachlet's algorithm.
I was actually thinking of mentioning orthogonal Latin squares but in the end decided against it :)
Another very enjoyable video to learn from!
I followed a lot of math youtuber for years. Mathologer is really the only one that consistently blows my mind.
Mission accomplished :)
I think a part 2 would be great. Geomagic squares. Self tiling, where all of the pieces combine to make larger versions of a single pieces and rearrange the pieces and get any of the other pieces. And....... Geomagic squares that are fractal. With any row combining to make the target shape but also all of the pieces combning to make the same shape but bigger.
Yes, a lot more variations are possible. Maybe have a look at Lee Sallows's book first, or his online gallery of geomagic squares (link in the description)
@@Mathologer The book is already on my list to get. The online gallery is amazing
Great video. Magic squares are classic puzzles. The insights I help my students understand for 3x3 magic squares is the sum is triple the central number, and all lines through the centre are arithmetic sequences. This can be seen in the general solution to the 3x3 magic square (with a, b and c), which we derive if the student is up for it. This makes solving 3x3 easy with very little information required - I think any 3 values can be used except if they are one vertex and the opposite two edges, or all three values in a line through the centre (assuming any solutions exist). In those cases, the problem is underdetermined.
Homeworks : 1) Considering a general nxn square, it is simplest to add the main diagonal, whose points stay in that diagonal. It becomes the middlemost row post transformation. Their points are (i,n-i) (1 at (0,0)) and the values are n+(n-1)*i for i from 0 to n-1. Adding, we get n^2 + n(n-1)^2 /2 = (n^3 +n)/2. For a 33x33, we get 17985.
I got 17688 ;-;
The way I went: there are 33*33=1089 tiles on the board. The sum of all tiles 1 to 1089 is the 1089th triangular number: 1089*1090/2=593,505. Divide *that* by the 33 tiles in any row, and you get 17,985.
@@jursamaj That's how I did it also. Very simple and straightforward 🙂
Very interesting topic!!
Great video!
I love how Taoism and the brilliant mathematical patterns therein are just casually a core part of Korean culture. South Korea's flag is literally the Taijitu and Ba Gua. It's great.
loshu is never old!(just inscribed on a turtle shell🐢).Love this channel!
I was looking at MSs a month ago. This threw new aspects into the mix. 🙏I like the music 🎶
Fascinating!
My 3½-year-old son is fascinated by the Numberblocks series. I can’t wait to see what he’ll make of this video when he’s older.
How delightful! I'm gonna 3D print one of these for sure.
I'm a natural-study at Math. Have been my whole life. I could derive and integrate common polynomial and other typical calc1/2 problems mentally when I was like 14. School couldn't keep up, though, and the internet wasn't meaningfully around yet. As a result, I found it boring as I got older, and I moved into comp sci instead, working at a FAANG company living the easy life. But you really bring out the romance in math out of me. I genuinely never found any appeal in any other mathematician on KZhead (even though I respect them and what they know, they just don't resonate with me). But your work is really incredible -- had you been doing your thing 20 years ago, I would've gone into math for sure. Keep up these videos. They are really so good.
Very good you explained it in a way other people can understand. 👍
awesome video :) thank you Burkard!
Great video as always! 8:24 The sum of all tiles from an order-n magic square is S = 1+2+...+n^2 = n^2(n^2+1)/2. Thus the magic number should be S/n = n(n^2+1)/2. It's 17985 when n=33. 18:50 Extend the square to the whole plane by translating horizontally and vertically (similar to the flat torus). Start from the middle tile in the top row, go towards top-right diagonally 4 times, then go down 1 tile. Repeat. P.S: There are 5 different starting places of tile 1 for this method to work (to make sure that 11 through 15 are the five numbers on the "/" diagonal). Actually, it's better to start from tile 11 - we just need it to be on the "/" diagonal. BTW, are there any sum-and-product double magic squares? I vaguely remember having read about it before.
Excellent!
Lesson learnt. There's always more to magic squares. Was not expecting that!
Wow, really beautiful!
The geometrical videos are awesome
I was not expecting Sejong and k drama from this channel
There are actually quite a few k dramas with interesting maths built into them. I probably will cover Melancholia at some point: en.wikipedia.org/wiki/Melancholia_(TV_series)
When I first heard the idea of magic squares, my initial thought was "no way! that's way overconstrained!" (though I wouldn't have used that word.) I thought sure you could get all the rows, or all the columns, but not both and definitely not the diagonals too. Then you learn more about how they work and how to make them and it becomes "well, of course! how could it be any other way?"
Whoa, whoa, wait a second! There's another picture of Euler out there?? 😲
Mind=blown, thank you !
"Since this will probably be my only ever Mathologer video on magic squares" ... I am already waiting for the next video on magic squares (or maybe magic cubes?) 🤩🤣
I have a different perspective on "the king's method". When I went about proving the method, I looked at the form the columbs take when we unwind the magic square back to the lunch box formation. Under that perspective each columb corresponds to two diagonals with a combined leangth equal to that of the columb. Than the key to proving the method is to show that when shifting from a columb to its' adjacent, the sum over one of the corresponding diagonals increase by exactly as much as the sum over the other will decrease.
Nice :)
@@gregoryford2532 Sorry, I'm not a native English speaker...
Great, now I have so many questions I want answered. Shapes: All shapes must be constructed from an integer number of squares connecting to each other across full edges. N=side length C=the magic number. 1. For what N is there a constant shape that is a square? 2. Is there an N with a set of shapes that only add to a square? 3. If a square can be built, it must have a C with a square root equal to or larger than the triangular number of N. Is there an N where all squares are possible once the minimum square size is reached? 4. Earliest N with a set of consecutive integers that can build a square? 5. So many more.
The magic square construction at 18:45 was the method I was shown many years ago. You start with one in that location, then go up and to the right one space for the next number. You pretend the grid wraps on itself, the top going to the bottom and the right to the left extremes, and if the square it is headed to one that is already occupied you go down one. This construction works for all odd grids as far as I know
That's it. The problem I have with most texts on magic squares is that they hardly ever bother to prove anything :(
Pretty magical!!
I've been testing my understadning of the mousatche method using the 33x33 square. It's been a journey so far. Drat you for making me do something interesting! ;)
Beautiful 😊
Just mind blowing
An interesting idea : Maybe the pattern of fitting by translation at 7:17 also works well for fitting fractals in a rectangular viewport? 🤔