Hexeneinmaleins (Witch's Multiplication Table)

The princess and I have completed the administrative process to re-enter our country. After that, our car drove slowly towards the station to board the Alleinbahn. The princess told Sisy and the other three guards to drive slowly because they were not sure about the next plan.

Of course, we did not say honestly that we slowed down the car because we were waiting for the reaction of the Jonquil Magic Ball. Before entering the immigration building, we saw that there was no visible change from the ball. After walking a few hundred meters, one kilometer, and two kilometers from the Guldendorf Immigration and Customs Office, the round thing still did not move.

"What's going on? I hope there is no damage to the ball," I said with worry.

The princess, who had a similar expression to mine, said, "Yes. Hopefully, nothing happens to this little one. Hopefully, he doesn't get sick." She said that while rubbing the Jonquil Magic Ball that she was now holding again.

We agreed to wait for the ball's reaction until we arrived at the station in the city of Guldendorf. During the trip, we filled our time by chatting about other topics. From that conversation, the princess gave me information that I had just heard.

"You could say that Mr. Warm was the Big Five member that I knew best before he became a former Big Five member. Now, out of the five, the one I know best is Mrs. Zeusleben," the princess told me.

I, who guessed that it probably started from business, asked, "What kind of business do the Princess and Mrs. Zeusleben run that made you two closer?"

The princess answered, "Not business, but a hobby."

"Hobby?" I asked.

The princess said, "Yes. Recently, I have acquired a new hobby, which is bathing in warm water. Because of that, I became close to Mrs. Zeusleben, who has had that hobby for a long time."

"Ooohhh..." I commented briefly.

The princess added her explanation: "I've bathed together with Mrs. Zeusleben in a hot spring outdoor bath in a villa in the Brocken Mountains. I've seen the big wing tattoo on her back. I've talked a lot with her. Therefore, I can say that 'The Angel' is the member of the Big Five who is closest to me right now".

"Ooohhh..." I commented briefly again.

Suddenly, at that moment, our conversation had to stop. However, we were actually happy because the one who stopped our conversation was Jonquil Magic Ball. The round thing suddenly showed a hologram and made its voice again.

"LOCATION DETECTED. THANK YOU FOR BRINGING ME BACK TO DEUTSCHTONIA, Vogel and Wolke.”

“Now, I will give you a long talk.”

“You as one and one must think of the Hexeneinmaleins from the literary work of Goethe. Interpret it into a matrix of numbers where one element is both a real number and an imaginary number. This will be the first answer.”

“Change the middle and last columns of the matrix to kana according to the kana order. Just as Mr. Warm ignored the Five and was no longer vocal, here ignore its first column, which contains five vocal words. This will be the second answer.”

“Write both answers on my hologram."

Hearing the long-winded instructions from Jonquil Magic Ball, I immediately said, "It's my turn."

As a history postgraduate, I knew what was meant by the Hexeneinmaleins from the literary work of Goethe. Goethe must have meant a literary figure whose full name was Johann Wolfgang von Goethe. Hexeneinmaleins is one of the things that appears in one of Goethe's most famous works, entitled "Faust".

"Hexeneinmaleins" is a term in German that means "witch multiplication" or "witch woman multiplication." Hexeneinmaleins is a rhyme that appears in the Witches' Kitchen scene in the novel Faust. I still remember the 13 lines of Hexeneinmaleins lyrics.

Du musst verstehen! ("You must understand!)
Aus Eins mach' Zehn (Make one into ten).
Und Zwei lass gehn (and let two go).
Und Drei mach' gleich, (and make three equal,)
So bist Du Reich. (and you'll be rich.)
Verlier' die Vier! (Lose the four!)
Aus Fünf und Sechs, (Make five and six,)
So sagt die Hex' (so the witch says,)
Mach' Sieben und Acht (make seven and eight).
So ist's vollbracht (and it's done).
Und Neun ist Eins (and nine is one).
Und Zehn ist keins. (and ten is none.)
Das ist das Hexen-Einmal-Eins! (That's the witch's multiplication table!)

There are several interpretations if we want to change Hexeneinmalein's word to become a matrix of numbers. Different interpretations will produce different results. For the interpretation that Jonquil Magic Ball expects this time, he has given the keyword—or you could say the key number.

Jonquil Magic Ball said, "One of its elements is a real number as well as an imaginary number." The number in question is zero (0). Zero can be a symbol of the absence of real and imaginary parts in complex numbers; this is because 0 can be an expression of 0 + 01 in the complex plane. Zero is located at the origin, which represents the real and imaginary axes in the complex plane.

In Hexeneinmalein's words, the lyrics “Lose the four!” can be translated as eliminate the fourth digit, or in other words, the fourth digit becomes 0.

But if there is a fourth digit, how many digits are there in the matrix being searched for? I can interpret from the eleventh and twelfth lines. “And nine is one. And ten is none.”

Ten is none, I can interpret as there is no tenth digit, meaning that this matrix has nine elements or nine matrix members.

Nine is one. I can interpret that the nine matrix digits that exist form a unity that has certain characteristics.

We will interpret the lyrics of the second line, “Make one into ten,” as a command to change the first element of the matrix to a value of 10. Furthermore, from the lyrics of the fourth and third lines, “and let two go, and make three equal” is a command to leave the second and third elements, respectively, with values ​​of 2 and 3.

Furthermore, as discussed previously, “Lose the four!” can be interpreted as a command to change the fourth element to a value of 0. Then the next three lines of lyrics, "Make five and six, so the witch says, make seven and eight," can be interpreted to exchange the values ​​of elements 5 and 6 with the values ​​of elements 7 and 8. From the sequence "5,6,7,8" to "7,8,5,6."

We have obtained eight matrix element numbers, namely 10, 2, 3, 0, 7, 8, 5, and 6. Only one number is missing. To find the one missing number, we can again observe that the nine matrix numbers form a unity that has certain characteristics.

A certain characteristic that we can capture is that this matrix has the same number of rows and the same number of columns, namely three. In addition, the matrix numbers, when entered into a 3 x 3 matrix, can form a semi-magic square.

A semi-magic square is a magic square or mathematical magic box where the number of numbers in each column and each row is the same, but the number of numbers diagonally is not the same. With these calculations, we can fill in the last, which is the number 4.

I filled in the first answer by writing the numbers:

10 2 3
  0 7 8
  5 6 ​​4