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# Euclid the Crazy Cat
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"Oh, no!" I groaned, staring at the sheet in front of me. I had completely forgotten about the summer math worksheet until it resurfaced like a ghost from the bottom of my schoolbag. It was due on the first day of the new semester, which was... tomorrow.
There were two pages of questions, and the first one was:
<div class="formula"> 1 + 2 + 3 + ... + 99 + 100 = ___. </div>
I tried to recall something called arithmetic sequence we had learned before the summer break.
[[But, I have forgotten everything!!->2-euclid]]
[[Wait, I know the answer! ->1-finish]]config.footer.right: "{back link} {restart link} [[Exit|https://www.mathland67.com/]]"
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I panicked. How I wished that someone would come and help me.
Someone tapped me on the shoulder. I turned around and saw Euclid.
Euclid was the new cat in the neighborhood. Nobody knew where he came from. One day, he just showed up, as if he had landed from outer space. He liked to visit our house becasue we often gave him food and water. Dad called him Euclid becasue he liked curling up in all kinds of geometric shapes—triangles, squares, circles...
I had no idea how Euclid had gotten into the house, but there he was, balancing precariously on my shoulder and staring curiously at my worksheet.
[["Hey, Euclid, want to do some math problems?" I asked.->3-summation]]
[["Go away, Euclid," I said. "I'm trying to focus."->2-finish]]
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Euclid jumped off my shoulder and stopped in front of the his food bowl.
Of course. Cats were more interetsted in food than math. But, wait...
Euclid hold up one piece of kibble with his paw and placed it on the ground. Then he added two more pieces in the column next to it, followed by three more in the next column, and four more in the fourth column.
Then, he looked up at me and smiled.
[["What're you doing?" I asked in surprise. Then it hit me. Euclid was trying to show me something.->4-summation]]
[["Don't play with food, Euclid," I said. "Now I have to clean it up for you."->3-clean]]config.footer.right: "{back link} {restart link} [[Exit|https://www.mathland67.com/]]"
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Euclid went back to work, bulding another triangle of the same shape and size—only this time, it was up side down. Together, the two triangles formed a rectangle with four rows and five columns.
[["Oh, I see!" I exclaimed excitedly, the kibble striking me like a lightening bolt.->5-summation]]
[["I don't get it," I said, shaking my head.->4-finish]]
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I sat down at my desk and begin sketching. Yes! Euclid's diagram could be used to add up any arithmetic sequence.
To add up the numbers from one to one hundred, I simply made two giant kibble triangles and fit them together to form a rectangle of 100 rows and 101 columns. Then, with multiplication, I could easily calculate the total pieces of kibble.
[["The answer is 10,100," I shouted excitedly.->5-summation-wrong]]
[["The answer is 5050," I shouted excitedly.->6-summation]]
[["The answer is 20,200," I shouted excitedly.->5-summation-wrong]]
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{embed image: './6-summation.png'}
Eulid held up his thumb.
"Yes," I high-fived him and turned back to my worksheet.
Wait, does a cat have thumbs? But before I could look into it, Euclid had already gone back to his usual sitting position, staring stony-faced at the second question on my worksheet:
<div class="formula"> 1 + 3 + 5 + ... + 99 = ___. </div>
Hmmm...the sum of odd numbers. I guess we can use the same approach.
I sketched another pairs of triangles, only this time, instead of 1, 2, 3..., the numbers went 1, 3, 5...
The number of rows is the first number plus the last number, which is 100. The number of columns is how many numbers we are adding up. The distance between 1 and 99 is 98, and the step size is 2, so there are 98 ÷ 2 + 1 = 50 columns.
[["The answer is 2,500," I said, showing Euclid my answer.->7-summation]]
[["The answer is 5,000," I said, showing Euclid my answer.->6-wrong]]
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Euclid smiled slyly. He jumped to the floor, cleaned away the rectangle, and placed a single piece of kibble on the ground.
"Okay, that's one," I said.
Euclid added three more pieces, but instread lining up as the socond column, he bended and placed them in an L-shape around the first piece, forming a two-by-two square.
"Interesting," I said. "You added three, but in a different way."
Then, in the third round, Euclid added five pieces, again in an L-shape, making a three-by-three square.
"Aha, I see where this is going," I said. The sum of odd numbers are square numbers! Euclid had made it perfectly clear without meowing a word.
[["The sum of all odd numbers from 1 to 99 is 2500." ->8-summation]]
[["The sum of all odd numbers from 1 to 99 is 2601." ->7-wrong]]
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Armed with the cat food drawing, I solved the two more summation problems with ease:
<div class="formula">
1 + 8 + 16 + 24 + ... + 96 = ___
4 + 12 + 20 + 28 + ... + 100 = ___
</div>
Both answers are square numbers. Apparently, you can decompose a square in many differnt ways.
[["The answers are 625 and 676," I said, showing my answers to Euclid. "Am I right?"->11-equation]]
[["The answers are 676 and 729," I said, showing my answers to Euclid. "Am I right?"->8-wrong]]
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Euclid nodded solmnly.
"Yippee!" I cheered. "Next question."
<div class="formula">
{embed image: './11-eq.png'}
</div>
Hmmm... It must have something to do with the distributive property. But what is distributive property? I turned to Euclid for help.
"What do you think?"
Euclid picked up a pencil from my desk and began drawing on my worksheet.
"Hey, stop!" I shouted. "Don't doodle on my worksheet."
[[But, wait... can a cat draw with a pencil?->12-equation]]config.footer.right: "{back link} {restart link} [[Exit|https://www.mathland67.com/]]"
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Euclid sketched a big square and divided it into four parts with a horizontal and a vertical line.
"I see. The side of the big square is x+y. Therefore, the area of the square is (x+y)<sup>2</sup>."
Euclid let out an approving meow.
"And the area is divided into four parts: two squares with the area x<sup>2</sup> and y<sup>2</sup>, and <b>two</b> rectangles with the area xy."
Ah, there it was:
<div class="formula">
{embed image: './12-eq.png'}
</div>
A proof without word!
[[Next question.->13-equation]]
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<div class="formula">
Prove x<sup>2</sup> - y<sup>2</sup> = (x+y)(x-y)
</div>
"Let me try it this time," I said, excited about the new trick I had learned.
First, I sketched a square with side length x. Then, starting from its top right corner, I drew a smaller square of side length y inside it. Now, since x<sup>2</sup> is the area of the big square, and y<sup>2</sup> is that of the small one, x<sup>2</sup> - y<sup>2</sup> is the area of the remaining L-shape.
"Okay, now we just need to show that this area is (x+y)(x-y)," I said, shading the L-shape.
[[But how?->14-equation]]
[[Oh, I see. It's obvious. Next question ->15-inequality]] config.footer.right: "{back link} {restart link} [[Exit|https://www.mathland67.com/]]"
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From the pen stand, Euclid pulled a pair of scissors.
"Careful!" I shouted. He pointed the scissors at the L-shape.
"You mean cutting it?"
I added an imaginary cut line to the diagram, and immediately noticed that the two rectangles created by the cut had the same height x-y.
In my mind, I rotated the rectangle on the right and taped it next to the other. Now I had a long rectangle, whose width was... x+y.
That's it! x<sup>2</sup> - y<sup>2</sup> = (x+y)(x-y).
[[Time for the last problem.->15-inequality]]config.footer.right: "{back link} {restart link} [[Exit|https://www.mathland67.com/]]"
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<div class="formula">
Prove x<sup>2</sup> + y<sup>2</sup> >= 2xy.
</div>
Could Euclid's digram be used to prove inequalities, too? I sketched two squares side by side, with side lengths x and y. Obviously, x<sup>2</sup> + y<sup>2</sup> is the total area of the two squares. But where is xy in the diagram?
Euclid shook both his head and tail at the same time.
[["Oh, don't look at me like that," I said. "Let me try again."->16-inequality]]
[["Well, if you can do better, why don't you do it?" I said.->16-inequality2]]config.footer.right: "{back link} {restart link} [[Exit|https://www.mathland67.com/]]"
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I crossed out the diagram and started over. This time, I did the same thing we had done before: I drew a big square with side length x+y and divided it into four pieces with a horizontal and a vertical line.
In the top left was a square with side length x, and in the bottom right was a square with side length y.
"Now we have something representing xy," I said, pointing to the two rectangles. "We just need to prove that their combined area is always smaller than the combined area of the two squares."
[[Hmmm...that didn't seem easy.->17-inequality]]config.footer.right: "{back link} {restart link} [[Exit|https://www.mathland67.com/]]"
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{embed image: './16.png'}
Euclid took up his pencil and sketched a big square. Then, he divided it into four pieces with a horizontal and a vertical line.
"That's the same diagram you have drawn last time," I said.
In the top left was a square with side length x, and in the bottom right was a square with side length y.
"Now we have something representing xy," I said, pointing to the two rectangles. "We just need to prove that their combined area is always smaller than the combined area of the two squares."
[[Hmmm...that didn't seem easy.->17-inequality]]config.footer.right: "{back link} {restart link} [[Exit|https://www.mathland67.com/]]"
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I shaded two rectangles and moved them into the big square. But to my annoyance, they overlapped, forming a square that was double-counted.
Euclid leaned in and drew a double-headed arraw on my worksheet, linking the double-counted square to the square at the bottom right.
Oh, yes! Aren't they the same? Togehter, the L-shape and the square at the bottom right should have the same area as the two rectangles. Therefore, the remaining empty area inside the x<sup>2</sup> square is simply x<sup>2</sup> + y<sup>2</sup> - 2xy.
When x = y, this area shrinks to zero. Otherwise, it is always greater than zero.
[[I showed my drawing to Euclid. "What do you think?"->18-end]]Euclid shook his head, looking disappointed.
[["Wait! Let me try again."->5-summation]]Euclid shook his tail, looking disappointed.
[["Wait! Let me try again."->6-summation]]Euclid curled up into a circle, covering its eyes with its tail.
[["OK, OK...let me try again."->8-summation]]config.footer.right: "{back link} {restart link} [[Exit|https://www.mathland67.com/]]"
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Euclid nodded imperceptibly and jumped off my desk. He took his time eating all the kibbles on the ground. Then, standing on his hind legs, he twisted the doorknob and opened the door to backyard.
Wait... can cats turn doorknobs?
But before I could figured it out, Euclid had already slipped out of the house. He walked silently and jumped on the fence. Above him, stars scattered across the sky, forming all kinds of shapes: triangles, squares, circles...
Well, thank you, Euclid, for solving all my algebra problems without using any algebra!config.footer.right: "{back link} {restart link}"
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Memory came to me like flood. I finished the worksheet in ten minutes. It's time for some video games.config.footer.right: "{back link} {restart link}"
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Euclid shook his head, looking disappointed. He jumped off my shoulder and walked right through the back door.
Wait...how did he just go through the door.
But before I could figure it out, he had disappered into the night.config.footer.right: "{back link} {restart link}"
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Euclid let out a soft meow and ate up all the kibble on the ground.
"Okay, good cat," I said, turning back to my worksheet. I guess I'll just use my calculator to add these numbers up.config.footer.right: "{back link} {restart link}"
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I guess I'll just use my calculator to add these numbers up. It'll takes some time but I'll get it done.Euclid let out a disappointed meow and walked away.
[["Wait! Let me try again."->7-summation]]