What is a tangram?

0:47 Welcome to Maths Is Not Boring by kidopoly.com. I'm Mira! And I'm Finn. A big hello to seven-year-old Rey Anch in St. Louis, Missouri, who love sharing cool facts and stories from the podcasts on car rides.

1:02 And a huge hello to eight-year-old Willa and six-year-old Zelda.

1:06 If you haven't already, tap follow wherever you're listening so you never miss an episode.

1:12 Hey Mira, before we start today's puzzle, I was thinking about that episode of our other show, Philosophies Not Boring. Oh, the one called How Did Babies Count? Yes. It makes you wonder if our minds are just naturally born with a built-in sense for amounts, even before anyone teaches us anything. You can go and play it right after this one.

1:34 Search for Philosophies Not Boring wherever you listen. But speaking of noticing amounts and shapes without having to count them, today we are looking at something you can hold in your hands. Hold in our hands? Yes. Picture seven little flat pieces of wood resting on a table. If you slide them around, you can make a picture of a cat with pointy ears, or a sailing boat, or a person running fast.

2:00 I've seen these. You can make thousands of different shapes out of just those same seven pieces. It's like magic.

2:08 You really can. But here is the genuinely strange part, Finn. No matter what picture you make, every single one of them covers the exact same amount of space on the table.

2:19 Wait. A giant tall boat and a tiny crouched cat cover the exact same amount of space? That can't be right.

2:29 The boat looks huge. It is completely right. Because you are using the exact same seven pieces to make them every single time.

2:42 Right, a tangram. I have played with these. They usually start out as a perfect solid square, all neatly packed into a box. But it is chopped up into smaller pieces.

2:53 Exactly. If you cut a square into exactly seven pieces, you get five triangles, one square, and a slanted shape called a parallelogram.

3:03 Oh yeah, a parallelogram is a four-sided shape with perfectly parallel opposite sides. Like a regular rectangle that someone pushed over slightly, so it's leaning to one side.

3:14 Yep. These seven pieces form a puzzle that originally came from China. And over 200 years ago, people across the world went completely wild for them.

3:26 Like a massive craze. Like when everyone suddenly has to have the same toy. A huge one.

3:32 In 1815, an American sea captain named M. Donaldson was given books of tangram puzzles in China, and he brought them all the way home on his ship, which was called the Trader.

3:44 That is a really long voyage just to bring back a puzzle.

3:48 True, but it started something incredible. By 1817 and 1818, there was a massive tangram craze sweeping all across America and Europe. People were utterly obsessed with finding new shapes.

4:03 I bet grown adults were staying up all night trying to build a perfect swan or a bridge out of little wooden shapes.

4:10 They actually were. It was so popular that the French Emperor Napoleon is said to have owned a beautifully carved ivory tangram set while he was exiled on the remote island of Saint Helena.

4:22 The Emperor of France sitting on a lonely island, trying to figure out how to make a little wooden boat.

4:28 That is the story! And the quiet maths hiding inside those pieces is an extraordinary idea called conservation of area.

4:37 Okay, conservation of area. What does that actually mean when you break it down?

4:42 Well, area is just the mathematical word for how much flat space something covers. It is how we measure carpets for a room, or grass for gardens, or the size of a giant football pitch.

4:55 So if I buy a carpet, the area is just how much floor it hides. And a bigger area means it covers more floor.

5:02 Right! Now, imagine your perfect Tangram square sitting perfectly still on the table. The area of that square is exactly the amount of table it covers underneath it.

5:14 Makes total sense. It is just a square footprint on the table.

5:18 Now, scramble those seven pieces up, slide them around, build a picture of a tall house with a pointy roof and a little chimney on top. Okay, I've got my house. It looks great. A mathematician would look at your house and say, without even having to measure a single side of it, that your house has the exact same area as the original square.

5:42 Wait, but it is a completely different shape now. The square was blocky and compact, and my house is tall and pointy. It stretches way higher up the table.

5:52 It does look different, but you use the exact same seven pieces, right? You didn't sneak in any extra pieces from another box?

6:01 No, just the seven. And you didn't leave any pieces out, or stack them on top of each other so they overlap? No, they are just sitting flat next to each other on the table, touching at the edges.

6:12 Then the area is conserved. Conserved just means saved, or kept exactly the same. The amount of flat space those pieces cover cannot magically grow or shrink just because you move them around into a new pattern.

6:27 But my brain really wants to say the tall house covers more space because it stretches further up. It feels like it takes up more of the world. I know.

6:38 That is exactly why this idea feels like magic when you first see it. Our eyes get tricked all the time by how long or wide something looks. We see something tall and think it must be bigger. But the mass of area doesn't care about the outside outline.

6:55 Oh, I get it. It only cares about the actual stuff inside the outline.

7:00 Yes, that is the true heart of conservation of area. If you cut a shape into pieces and rearrange them without losing any pieces or overlapping them, the total area stays exactly the same forever.

7:14 So if I make a running person and the arms are stretched way out to the sides and the legs are mid-stride, the area is still exactly the same as the boring old square.

7:26 Exactly. Every single one of the thousands of tangram shapes you can possibly make has the identical area. The space it takes up never changes.

7:38 That is really beautiful, actually. The outline changes completely, and the picture tells a whole new story. But the amount of space it owns in the world never changes. It is incredibly beautiful.

7:51 And mathematicians actually use this exact idea to figure out the area of really weird, complicated shapes in the real world.

8:00 How do they do that? You can't always just chop things up.

8:04 In their minds they can. If they look at a weird, wobbly shape, they can imagine chopping it up and rearranging the pieces into a simple rectangle. Because of conservation of area, they know the area hasn't changed at all.

8:19 So instead of trying to do complicated maths on a wobbly, impossible shape, they just slide the pieces around in their mind until it is easy to measure.

8:28 Exactly. They turn it into something simple that they already know how to measure. It is like a superpower for seeing shapes.

8:36 Okay, I have to ask. Who actually came up with this perfectly balanced puzzle? Did someone just accidentally drop a square tile on the floor and watch it shatter into seven perfect geometric pieces?

8:50 Well, there is a very famous myth about exactly that. A story claimed that a 4,000-year-old god called Tan invented the puzzle when he dropped a special tile and it broke.

9:02 A god called Tan? Wait, is that actually true? That sounds like a fairy tale.

9:09 It is a fairy tale. It was known as the Eighth Book of Tan, and it was a giant 19th-century hoax. People loved the story, but it is only a myth.

9:20 Oh, good. Because dropping a tile and getting perfect mathematical shapes sounded completely impossible.

9:27 Right. The true history comes from China, probably around the year 1800, and the puzzle pieces are very carefully designed. You definitely do not get them by smashing something on the ground.

9:40 You said it's five triangles, a square, and a parallelogram.

9:44 Yes, and those five triangles are all what we call right-angled isosceles triangles.

9:50 I know those. Meaning they have one perfectly square corner, like the corner of a room, and the two sides that make that corner are exactly the same length. Spot on.

10:00 There are two large ones, one medium one, and two small ones. Because they are so perfectly proportioned, they click together in endless, surprising ways.

10:11 And every single time they click together to make a new shape, the conservation of area is right there, quietly doing its job. Exactly.

10:19 It is a quiet rule hiding in plain sight. So, next time you see a floor made of tiles, or you try to fit all your sandwiches perfectly into a lunchbox, you are playing with the exact same rules of area. Alright, Finn, it is time for the quiz. Let's see how much you remember from our tangram journey.

10:38 Okay, I'm ready. I feel like I learned a lot, but let's see. First question.

10:43 What is the name of the seven-piece puzzle we talked about today? Next one. What year did the massive tangram craze sweep across America and Europe? Question 3. Which famous French emperor is said to have owned a tangram set while exiled on an island? Here's number four. What is the mathematical word for how much flat space something covers?

11:20 Last question. What do we call the slanted, pushed over rectangle shape in a tangram set? Great job! Let's go through the answers and see how you did. The first answer is the seven-piece puzzle is called a tangram. For question two, the huge tangram craze happened around 1817 to 1818. Number three, the answer is the French Emperor is said to be Napoleon.

11:52 Question four. The mathematical word for how much flat space something covers is area.

11:59 And the final answer is that slanted shape is called a parallelogram. I am seriously never going to look at puzzle pieces the same way again. Just knowing that a tiny square and a giant running person hold the exact same amount of space in the world is so cool.

12:16 I know. It makes you want to go cut up some paper right now and see what wild shapes you can build without changing the area.

12:24 Hey, if you enjoyed exploring Tangrams and the magic of area with us today, please leave us a five-star review.

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13:06 See you next time on Maths is not boring. Bye, everyone.