A square puzzle

Long ago, there was a farmer whose land was in the shape of a square.
Each side of the square was exactly one hundred paces long.

One day, a tired, dusty man knocked on the farmer’s door and asked for
something to eat. The farmer, being a kind person, gave the man a nice
lunch.

After the man had eaten, he said, “Farmer, I am your king! As a reward
for your kindness in giving me food when you thought I was just a humble
stranger, you may double the size of your land. But when you have added
the new land, your farm must still be in the shape of a square I”

The farmer was overjoyed, for now he could plant twice as many crops. He
went out at once to measure his new land so that he could put a fence
around it. But he suddenly found that he had a problem.

At first, the doubling of his square of land seemed easy. Since each
side of the square was one hundred paces long, it seemed as if the sides
of the new square should be two hundred paces long—twice the length of
the old sides. But this didn’t work. His new square, with sides two
hundred paces long, wasn’t twice as big as the old square—it was four
times as big! In fact, it contained as much land as four squares with
sides one hundred paces long!

The farmer scratched his head. Finally, he went back to his house and
got out paper and pencil. He thought long and hard. Then he had an idea.
He would divide his square of land into four small squares. Then he
could simply add on four new squares the same size as the small ones.
This way he’d have exactly twice the amount of land that was in the old
square.

First, he tried adding all the small squares onto one side of his old
square. But this was no good. It gave him a rectangle, not a square.

Then he tried adding two small squares to one side of the old square and
two to another side. This almost worked. But there was a piece missing
out of one corner, so it wasn’t a perfect square. And if he filled in
the missing piece, he would more than double the size of his farm.

Again, the farmer scratched his head. Then, he had another idea. He
divided his old square into four parts by drawing diagonal lines between
the opposite corners. Now the square was divided into four triangles. If
he could add four more triangles of exactly the same size— and somehow
make a square—he would have solved the puzzle!