One and three. "That adds up to four which is a perfect square so I can replace it with another perfect square. I pick nine."

"What is a perfect square?"

Tokens. "If you can put this many tokens into a square then the number is a square." Four tokens are a two by two square, like so. Nine tokens are a three by three square, like so.

"I don't think you get to pick your own square. I pick one."

"Awwww." He moves one.

Lintalai cackles.

Her parents beam at her. Aitim rolls an eleven. "That's good as it is."

Lintalai attempts to make an eleven square. She cannot even make a rectangle. "I think something is wrong with eleven."

"Eleven is not a square. The next square after nine is sixteen, which is higher than dice go."

"It's just a line."

"Eleven is prime, that's the word for numbers that won't rectangle."

"Oh." She is distracted from the game seeking rectangles of other numbers.

They have such a smart baby. They watch her adoringly.

One is a small square. Or a small line? Two is a line. Three is a line. Four is a square. Five is a line. "Gosh there are a lot of numbers that won't rectangle." Six rectangles.

"Most big numbers rectangle. Lots of small numbers are prime."

"Ohhh." Seven is a line. Eight is a rectangle. Nine is a square. Ten is a rectangle.

Such a clever baby! Their clever baby!

She runs out of tokens when she discovers that twenty is a rectangle.

"If you would like we can order some more online so you can check bigger numbers."

"What is online?"

"We use our pocket everythings to send instructions to the token store and then they send us tokens."

"I could just use blocks, I have lots of those."

"You could!"

She goes out to the courtyard and sacrifices her block city to prime numbers until she is bored with them and then reads How To Build Very Tall for block city tips.

A long time ago, everybody built single houses. They looked like this.

But lots of people wanted to live in the same place. How do they do that? Well, you can make the houses closer together, like this.

And you can put stairs in the house, and have a second floor and a basement, like this.

But even more people than that wanted to live in one space. What to do? 

Well, you can just put more houses on top of the houses, right? It turns out this is hard. When you build taller, buildings are heavy. This means they press down on the levels below them. If they are not carefully built, they might tilt, like this:

Or even worse, they might crumble, like this!

To learn how to build very tall, we had to learn how to build a base that could take lots and lots of weight.

Here is how buildings are built today. First big trucks dig a foundation. Look at how big and deep the foundation must be.

In some places they can dig all the way down to the solid bedrock under the ground, and attach the supports for the building to that. But in some places, there is no bedrock to be found! Then they pour concrete. It takes lots and lots of concrete to create enough support for a tall building. Concrete is a liquid, but it will turn solid once it is poured. It will turn into rock, and then we have the bedrock for our skyscraper.

Next, we need strong metal supports to be the spine of our skyscraper. Researchers worked for a long time to figure out what was the best way to do supports. Here are some different building spines. Which one do you like?

In some places, there are extra challenges! For example, in a city that has earthquakes, we need to build our spine so that it can shake with a quake. That way, even if a quake happens right under it, people will be safe. Here is an earthquake-proof building spine. 

Tall buildings will also have lots of wind blowing at them. They need to be able to sway in the wind without breaking or getting overbalanced. This is called 'sway'. Some sway is good - it means your building will not take all the force of the wind. But if you have very much sway, people in the building will notice, and they hate that! Engineers work very carefully to make sure that their buildings can sway a little - but not too much.

Now you have a very tall building - good job! But you have another hard problem - how do you get people to live in it. When we had small buildings, they had stairs. But no one can climb a hundred flights of stairs for work every day! 

So we invented the elevator. Early elevators were a box on a pulley, like this: 

They had a shaft in the building and carried people up and down, so they did not have to walk.

But as buildings got taller, there was a problem.You can only have one elevator in an elevator shaft - if you had two, they would collide! What if a hundred people wanted to go to work at the same time? Then we'd need lots of elevator shafts to hold them, right? Imagine if we tried to make enough elevators to move everyone quickly. We'd get a building like this - half of the floor space is elevators! What a waste!

So our engineers got to work again. They invented elevators that work like this. They go up one shaft, and down the other, in a big circle, and you can have a hundred of them at a time. When people need to get off the elevator, they get off the conveyor to let them out, so they don't slow down the people behind them. 

Now how much of our building has to be elevators?

That's right, not much!

But we still can't build as tall as we might like. Why not? Well, rock can only hold on to so much weight. If we built too heavy, even rock might not do! Also, it takes a long time for an elevator to reach the top of a hundred-story building. A two-hundred story building is pretty inconvenient! Building gets more expensive as buildings get taller, so we don't build as tall as we can everywhere. As we learn more things, maybe buildings will get even taller!