Unlocking the secrets of Magic Square puzzles

Magic Square puzzles are some of my favourite recreational math puzzles. Fill up a 3x3 square with the numbers 1 through 9 each used once so every row, column, and main diagonals add up to the same number. More generally, it can be a nxn square with numbers 1 through n^2, and there are actually tonnes of variants of these. In this video we will prove that there is precisely 1 possible 3x3 magic square (up to reflections and rotations). We'll do this by computing the Magic Number, aka the fixed sum each row must add up to, then figuring out the center must always sum to 5, and finally show how we get a single possible square. What's a bit crazy is that 4x4 has 880 possibilities, 5x5 has over 275 million possibilities, and for 6x6 it is so large we've never computed the exact number!

0:00 What is a Magic Puzzle?
0:45 Try these Magic Puzzles!
1:29 Ad hoc solving
2:27 Our 3x3 theorem
3:04 Gauss' counting trick
4:14 Sum formula
5:55 Central Square
7:09 Proving the theorem
9:25 Bigger magic squares

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[MSI-is]

My grandfather told me how to play this when i was young. I realized its importance when i was taught about it in 7th grade. He unfortunately passed away a year ago. R.I.P ❤💘💝

[jessstuart]

Constraint Programming is one of those topics I wish was higher in my learning priority queue.

[allenanderson]

Algorithm for solving any magic square with an odd number of squares on each side.

1. Place the numbers from 1 to n^2 in order.
2. Place 1 in the top middle square.
3. If the square up one and one to the right is open, next number goes there.
4. If the square up one and one to the right is outside of the magic square, then wrap around to the opposite side.
5. If no square is available, next number is placed one square down.

[jeanadams]

Got the first one quite easily, exactly the way he did after I hit play again. Need to take longer with the second one. I am 81 and have lost the ability to keep track of multiple steps mentally. I start out well, but will suddenly forget where I was going with it. lol It is still fun to try.

[geraldsnodd]

...A failed attempt at the magic square gives you a PARKER SQUARE.
😂
Jokes aside, I can't wait to learn through your content.
excited to study mathematics/ physics/ mathematics and computing subjects at university.
[I'll be done with my entrance exams by 3rd week of June :) ]
hope i get my desired university
Fingers crossed 🤞

[MaxCubing]

And so... We know how many possible combinations have 6x6 (and even bigger!) rubik's cubes, but we don't know how many combinations does it have a 6x6 magic square? We (humanity) know many things, but we still don't know and understand how many things in the universe!

[hifijohn]

Love the t-shirt, it reminds of a homer simpson quote "remember your hippopotamus oath"

[TheSilentLooters]

9:28 thanks man now i can write all 8 magic square by just memorizing one

[JacobIX]

Actually when I googled, I stumbled upon this:
JULY 2023:
"Prof. Hidetoshi Mino has counted the magic squares of order 6 to be 17, 753, 889, 189, 701, 384, 304 different 6x6 magic squares."
That's pretty new.. but yeah, now we know.
Also, the sum of all numbers in a 6x6 magic square is sigma 36 which is equal to 666, which is kinda cool lol.

[Deejaynerate]

In 2024, someone finally calculated how many 6x6 magic squares are possible, with the exact value being roughly 1.775*10^19. For context, the diameter of the Milky Way in kilometers is projected to be roughly 9*10^17. Crunching the numbers, a universe with a diameter of 1.775*10^19 km would be able to fit roughly 400 Milky Ways.

[coffeeconfessor]

So, there is only one permutation to the 3x3, but as the size increases, is there a recognizable pattern in the growth of the number of allowable permutations? I mean, only up to 5x5 has been solved, but it seems like there should be a mathematical pattern, or is it because only a very tiny number have had their permutation possibilities calculated that there isn't a direct pattern distinctly seen yet?

[StaticBlaster]

For the 4 * 4 magic square, it took me a while to figure out what the numbers were because I used the ad hoc method.

[matthewjohnson]

I already knew about Magic squares; CTC did a buzzle of more than 4 in a single puzzle.

[SimpleLivingHigherThinking]

Sir, I love the way you teach math in very simple and exciting way have started to love and enjoy math I feared a you so of

[mrkremps]

Well, this will make sudoku infinitely easier.

[BurstFlare]

Here's what I got for 0:45~1:03 before watching the rest of the video.

3x3

8, 3, 4
1, 5, 9
6, 7, 2


4x4

7, 13, 12, 2
10, 4, 5, 15
1, 11, 14, 8
16, 6, 3, 9

The video confirms that my 3x3 solution is correct, but what about my 4x4 solution?

[mouradbelkas]

Thank you, 3x 3 and 4x 4 are fairly easy. How about 5 x 5 and 6 x 6 . There is also other numbers than 1 to 9. Any consecutive 9 numbers (-5, - 4, -3 ...1, 2, 3, 4). The center number is the sum of the numbers divided by the number of cells

[alandmuhamad]

Im proud that I proved you can only have 8 solutions for the 3×3, on my own.

[vivekm.s]

First comment proffesor. 💓

It is also the ancient tradition of India. Which was also developed by Srinivasa Ramanujan

[LucilleSlate]

Really interesting!!! Thank you so much!