What are magic squares?

0:47 Welcome to Maths Is Not Boring by Kidopoly.com. I'm Mira! And I'm Finn. If you haven't already, tap follow wherever you're listening so you never miss an episode.

0:59 Hey Mira, what strange mathematical wonders are we exploring today?

1:04 I want you to imagine a perfectly ordinary-looking grid of numbers, Finn. Just a simple square box drawn on a piece of paper, divided up into smaller squares inside.

1:15 Okay, I'm picturing a square grid. Maybe something like a tic-tac-toe board with three rows going across and three columns going down.

1:24 Exactly! That is the perfect shape to picture. But in this specific grid, the numbers are all different and never repeated. They are playing an invisible, completely secret game right under your nose.

1:37 A secret game? What do you mean by that? Are they hiding something?

1:42 Every single row of numbers, from left to right, adds up to the exact same total. And it does not stop there. Every single vertical column, from the top of the box all the way to the bottom, adds up to that exact same total too.

1:57 Wait, hold on. Even the diagonals? The lines going from corner to corner right through the middle.

2:03 Yes, every row, every column, and both main diagonals add up to the exact same number. They are called magic squares, and mathematicians, artists, and puzzle lovers have been completely obsessed with them for thousands of years.

2:20 So it's essentially just a box of numbers, but they all perfectly balance out. That sounds incredibly difficult to build.

2:27 It is brilliant. The most famous early example of one uses the simple numbers from one to nine, and the story of how human beings first noticed it is actually an ancient Chinese legend.

2:41 Ooh, I really love a good legend. What happened? Where did it come from?

2:46 Long, long ago, in the time of the great Emperor Yu, there was a terrible, dangerous flood along the Luo River.

2:55 Oh wow! Did the Emperor find a way to stop the water from destroying everything?

3:00 Well, according to the ancient story, as the people were trying to figure out how to calm the rushing river, a giant turtle climbed right out of the water and stood on the riverbank.

3:12 A giant turtle just walking onto the shore out of nowhere.

3:17 Exactly. And when the people looked very closely at the turtle's shell, they saw an intricate pattern of coloured dots on its back. Dots? You mean like the dots you would see on a pair of dice? Yes, just like that. The dots were carefully arranged in the little three by three square. And when they counted the dots in any row or any column or any diagonal, the number always added up to exactly 15.

3:46 No way. You're telling me the turtle just happened to have perfectly balanced math hiding on its back.

3:52 That is exactly how the legend goes. It's told as a wonderful story of discovery, and that tiny 3x3 grid of dots is known all over the world as the Lo Shoe Magic Square.

4:06 Okay, so that covers the 3x3 square with the numbers 1 to 9. But did anyone ever try to make bigger ones?

4:14 They absolutely did. Let's jump forward in history to the year 1514 and travel to the bustling city of Nuremberg in Germany.

4:24 Nuremberg in 1514. Who are we going to meet there?

4:29 A brilliant and incredibly famous Renaissance artist named Albrecht Dürer. He made a masterpiece engraving, which is a highly detailed picture printed from a carved metal plate.

4:42 What was the picture of? I'm trying to picture it in my head.

4:46 It is called Melancholia the First, and it is absolutely packed with strange, mysterious objects. There is a winged angel looking really gloomy, sitting with her head in her hand. There's a sleeping greyhound dog, an hourglass, and a whole bunch of heavy carpenter's tools scattered on the floor.

5:08 That sounds like a very crowded, messy room.

5:11 It really is. But if you look closely at the stone wall right behind the angel, high up in the corner of the picture, there is a square grid carved deeply into the stone. A magic square hidden in the artwork? Yes, and it is a slightly bigger one. It is four boxes across and four boxes down. It uses every single number from one all the way up to 16 exactly once.

5:39 Whoa! So what is the secret balancing total for this bigger one?

5:44 34. Every single row of four numbers adds up to 34. Every single column of four numbers adds up to 34. And both long diagonals total exactly 34.

5:58 Wait, wait, wait. Just getting the rows and the columns and the diagonals to match perfectly is already incredibly hard to do with 16 numbers.

6:07 Oh Finn, Albrecht Dura did not stop there. This is the part where the math gets truly wild and mind-bending.

6:14 How could it possibly get any wilder than every line matching perfectly?

6:19 If you take the four numbers right in the dead centre of the square, they total 34.

6:25 The middle block? Just sitting in the centre? No way!

6:29 Way! And what about the four numbers sitting way out in the very corners of the huge square? They also total exactly 34!

6:38 That cannot be real! How did he even figure out how to arrange the numbers like that without a supercomputer?

6:45 It is an absolute masterpiece of numbers. Even the four smaller quadrants, meaning the top left four boxes, the top right four boxes, and the bottom ones, all independently add up to 34.

6:59 It is like the number 34, is trapped inside the box, bouncing around and showing up everywhere you look.

7:06 And the absolute best part is Dura's secret signature. Look closely at the two numbers sitting side by side, right in the middle of the very bottom row.

7:17 Okay, I'm looking. What are those two numbers?

7:20 15 and 14. The year he made the engraving was 1514. And he managed to hide that exact year right inside the perfectly balanced math.

7:32 That is so incredibly cool! It actually reminds me a lot of those number puzzles my grandpa does in the back of the newspaper every Sunday. Sudoku!

7:42 You have a fantastic eye for patterns. Sudoku puzzles are actually like cousins of magic squares. They are part of the same mathematical family.

7:51 Cousins? So they aren't exactly the same thing, but they are related. Exactly.

7:57 Sudoku actually descends from something called Latin squares. Those were heavily studied by a famously brilliant mathematician named Leonhard Euler.

8:07 Ah, I think I get it. Because in a Sudoku puzzle, you just have to make sure you never repeat a number in a line or a box. But you don't actually have to add them up to a specific secret total.

8:19 You nailed it, but they both share that exact same wonderful, satisfying feeling of everything fitting together perfectly without any pieces left over.

8:30 I bet people who love puzzles could sit and stare at these magical grids all day long.

8:35 They absolutely did. In fact, one of the famous founding fathers of the United States, Benjamin Franklin, used to make massive magic squares himself. Benjamin Franklin?

8:47 You mean the famous inventor with the kite and the lightning experiments?

8:51 The very same man. He worked as a clerk in the Pennsylvania Assembly, and when the political meetings got really long and boring, he would invent huge 8x8 magic squares, using giant numbers, just to pass the time.

9:06 Now that is what I call one seriously intense doodle to do during a meeting.

9:10 Alright, Finn, it is finally quiz time. Let's see what you remember from today's journey.

9:16 Okay, I'm taking a deep breath. I am ready. Let's see if I was paying close enough attention.

9:23 First question. What is the real mathematical name for these grids of perfectly balanced numbers? Next one. Question three. What specific year did the artist Albrecht Jura hide at the very bottom of his famous square? Here's number four. What secret number does every row, column, and diagonal add up to in Jura's masterpiece?

10:09 Last question. Which famous American inventor made giant magic squares when he was bored during long meetings? Great job! Let's see how you did on those. The first answer is they are mathematically called magic squares. For question two, the ancient legend says it was a giant turtle. Question three. Jura cleverly hid the year 1514 side by side in the bottom row.

10:43 For number four, the secret total hiding everywhere in Jura Square is exactly 34. And the last answer is, it was the inventor Benjamin Franklin.

10:55 You know, I really don't think I am ever going to look at a simple grid of numbers the exact same way again. It feels like there are secret, invisible patterns hiding right in front of us every day.

11:07 That is exactly how it feels when you see it. A simple, ordinary box of numbers on a page. But when you look inside, it is a perfectly balanced, beautiful puzzle holding the world together.

11:20 If Albrecht Durer's secret Hidden 1514 blew your mind today, please leave us a five-star review. Just scroll down on your app and tap the stars.

11:30 Oh, and we have sister shows too! History, science, and philosophy is not boring.

11:35 Go give those a listen and a follow as well. Want your name on the show? Send us an email at hello at kidopoly.com.

11:43 Thank you so much for exploring with us. See you next time on Maths Is Not Boring. Bye everyone.

11:49 Keep looking for those patterns!