Chapter 1  -  Falling to the Mysterious Land

Chapter 2  -  The Witch’s Soup 

Chapter 3  -  Win the King’s War! 

Chapter 4  -  Mixing Diamond’s Color

Chapter 5  -  Back to Reality 

Content 

Chapter 1: Falling to the Mysterious Land 

One dark night came along with thunder roaring and lightning splitting the sky, raindrops falling like straight pearl necklaces. Far far away, one light was still on, and one little girl named Linnea was too scared to fall asleep and was begging for her mom’s company. 

“Okay, okay. Just one more bedtime story then be a good girl and go to sleep,” her mom said while tucking her into bed, “Once upon a time...” 

Once upon a time...

Linnea closed her eyes and felt herself sinking into the soft bed. Gradually, Linnea’s world became quiet, when she opened her eyes, she found that she was lying on a warm lawn with a fresh smell of grass where the sunlight burst through the clouds and shone down through the tree leaves. 

“Where am I?” Linnea thought, “Why am I sleeping in the middle of a forest?” But she could not remember anything about what happened and how come she’d be here.

As Linnea wandered around, she found out that the world here was strange: trees grew straight up into the sky, and birds were always flying in a straight line. The grass was either light or dark green in a perfectly square rectangle, making a crisscross pattern, and the whole land was like a huge chessboard! 

This is so weird.

I want to go home.

I know who could bring you home.

Who’s that?

Up here!

A talking raven!

Follow me, I know the most powerful person in the forest, and she can help you out!

Then, the raven flew off without hesitation.  Linnea was a little scared but had no choice but to follow it. 

Chapter 2: The Witch’s Soup 

Linnea followed the raven through the forest, past the swamp, and finally, she saw a shabby house with a wisp of smoke coming out of the chimney.

There you go, here is a young witch called Greta. She is the nicest person I’ve ever known, and every animal in this forest visits her to get help with their problems. Good luck!

Linnea thanked the Raven, and it flew away.  She walked through a garden, also like a chessboard with strange planets and herbs she couldn’t tell in each square. She carefully pushed open the half-hidden door and found the inside of the house was in luxurious dark decoration, “Hello?” Linnea asked in a trembling voice.

“Hello my dear, how can I help you?” 

Linnea saw a young lady wearing a witch’s hat and a gorgeous black short dress.

“You must be Greta!” Linnea screamed out with excitement, “Hi my name’s Linnea, a raven led me here, I’m lost and looking for a way home”  

“Oh, the Raven is my sweetest servant and he’s right to bring you to me. Come here darling, let’s make a magical soup that can guide you home!” Greta smiled and led Linnea to a large pot with boiling water. 

RECIPE

Ingredients needed: 

2 jars of blueberry jam. 

1 pot of magical potion,  and

1 special-coloured diamond 

Greta listed all the ingredients for the jam and potion:

"For the jam, we need 5 blueberries, 2 cups of flour, 3 cups of sugar, and 1 bat's wing for each jar we're making."

"For the potion, we need 1 cup of flour, 1 cup of sugar, 2 bat's wings, and 1 dragon scale."

"We can write the ingredients for each part in a box like this:

For 1 jar of jam: [5, 2, 3, 1, 0]

For the potion: [0, 1, 1, 2, 1]

Where each number in the box matches the amount of a specific ingredient we need for the recipe.  Do you see how the first number in each matches the number of blueberries we need?"

"We need 2 jars of  jam for our soup, so we need to double the ingredients in our recipe:

For 2 jars of jam: 2 x [5, 2, 3, 1, 0] = [10, 4, 6, 2, 0]

So for the entire soup, we need to add the recipes for the 2 jars of jam and the potion:

[10, 4, 6, 2, 0] + [0, 1, 1, 2, 1] = [10, 5, 7, 4, 1]

And now we have the recipe for the soup!"

Using her recipe, our clever Linnea quickly found all the ingredients she needed for the jars of jam and potion. 

“Well done my dear! You are such a smart girl!” Greta smiled amiably “but one ingredient is missing, darling, the special-colored diamond is not any colored rock in my magical garden. The diamond is owned by a king in another kingdom, can you persuade him to give it to you? You know, finding the way back home isn’t easy, it is not enough to have wisdom, but also courage. Can you do that, my sweetheart?” 

“Definitely!” Linnea responded in confidence. 

“Well said darling, come, let me help you to use the magic portal to send you to that kingdom. It is located at the edge of the forest, and it’s in the middle of a war with a neighboring country.”

  Greta started drawing the magic circle by murmuring the spell. 

Within a second, Linnea was standing in front of the gate of a kingdom’s castle. Linnea took a deep breath and talked to the gate guard: “ Let me visit the king, I’m Linnea from another world, please help me to report to the king that I have a way to help him win the war.” One guard went in and after a few minutes, the gate opened slowly. 

Chapter 3: Win the King’s War! 

A king’s servant walked out hurriedly and guided Linnea walking towards the highest castle. After what seemed like millions of stairs, Linnea finally arrived in a palace where the walls were covered with gold foil. An old man wearing a large crown decorated with countless shining diamonds sat on a throne at the end of the palace, looking anxious.

The king welcomed Linnea, “You said you can help me win the war? Meet my military counselor,” the king pointed to a man dressed in a knight’s armor, “let’s see what you can do. Look, if you help me win the war, I can give you anything you want. However, if you fail, I’m going to lock you up forever for wasting my time!” The king let Linnea begin her analysis. 

5

5

I see the battleground formed in a check pattern, I can use some battle array that the enemy cannot predict so that you can give them a surprise attack. Let me introduce different types of formation.

Formation #1

Symmetric Matrix

It is a square matrix that is equal to its transpose matrix.

5

1

Let

be our soldiers, and

be our enemies.

6

6

2

2

1

Formation #2

Transpose of a Matrix

It switches the number of soldiers in thethe row and column by its diagonal to produce a new formation. 

Suppose an original battle array (Matrix) is given as below, where all entries are the number of soldiers.

1

2

4

1   3

2   4

3

Formation #3

Inverse of a Matrix (is obtained by dividing the adjugate of the given matrix by the determinant of the given matrix.)

1   2   1   0

3   4   0   1

1   2   1   0

0   1  3/2 -1/2

1   0  -2   1

0   1  3/2 -1/2

1   2   1   0

3   4   0   1

1   2   1   0

0  -2  -3   1

1   0  -2   1    

0   1  3/2 -1/2

Write the augmented matrix

Find the pivot in the 1st column in the 1st row

Eliminate the 1st column

Make the pivot in the 2nd column by dividing the 2nd row by -2

Eliminate the 2nd column

There is the inverse matrix on the right