## My Analyze Method

This magic game contains two mathematical methodologies. In order to reveal the game's logic, I use the circle-list method and display the card faces on a table. You will easily see and understand the mathematical logic in the game.

Take a piece of paper and divide it into four equal-sized cards. Draw four different pictures on the cards and label the cards on the left and right sides with A, B, C, and D.

Wash the cards well. Fold a line in the middle.

Tear the cards into halves based on the fold line. Each half has four cards.

Put one half of four cards on top of another half. Now you have a deck of eight cards.

In order to better explain, keep all cards face up and place them clockwise in a circle. Use a piece of green-coloured paper to mark the tail. The next card, clockwise, is the head.

As an example, use the D card as the tail. The head is the next card, clockwise, A. After A, the sequence of cards is B, C,...

With the cards ready, let's start to explain, focusing on two mathematical methods, the logic of the game.

## Math One: Repliated Sequence

The first math used in the game is to arrange the cards in a replicated sequence.

As shown in the above picture, we can see that no matter where the start or head card is, the sequence of remaining cards follows a common rule: the first four cards are replicated in the second four cards in order. The following are some examples:

D as head, the order is D-A-B-C D-A-B-C

A as head, the order is A-B-C-D A-B-C-D

B as head, the order is B-C-D-A B-C-D-A

and more

No matter how you wash your cards at the beginning or how many times you place the head card at the bottom of the tail card, This sequence follows a similar replicated sequence.

Another very interesting feature is that taking the first three cards, the first half card, and the last one of the remaining five cards is the matched card, i.e., the original C card. For example, taking three cards from the head D card, the head and tail of the remaining cards are all from the C card.

Based on those math discoveries, the next step in the magic game is to take the first three cards out and place them between any remaining cards. For example, place the first three cards in any position of the organ arrows shown in the following picture.

Choose a position, as shown in the following picture, between B and C.

After inserting, the following is the case of eight cards in a sequence.

Now the head and tail are all C. Take the head C, the secret card, out, and place it aside.

The number of remaining cards is seven. Place the secret card in the middle of the circle.

Please note that the tail card is another C, a key card. At this moment, the other cards above the key are not important. Their sequence can be anything. As in the magic show, you can ask the audience to place those non-key cards anywhere, but keep the key at the end. For you, the magician, you should clearly remember that the last card is the matching card to the secret card.

## Math Two: Josephus Problem

Another math in the magic show is the famouse Josephus math problem.

This is an old story from an ancient war. The historian Josephus and a team of soldiers were in a trap and had no way to escape. To avoid being captured and killed, they chose to commit suicide by each other. Then they designed a method, either by luck or the hand of God. They stood in a circle, one starting to kill the next one clockwise, and the next remaining one repeated the kill until the last one killed himself. The last remaining one could, in fact, survive.

The story became a math problem. N soldiers in a circle list. Start at one clockwise to count, the odd number remains, and even one out. Repeat this process until there is only one left. The math problem is to find out the index position of the surviving soldier. The problem is a very interesting pure math and computer science problem.

Liu Qian's game was designed based on this math problem. For a game with five or six cards in a list, Liu organised several deceiving steps to find a way to move the last matching card to the surviving position so that he could make the perfect match. Coincidentally, the seven-charcter magic word can move the last card in the 5 or 6 card list to the survive position! Interstingly, I found out that the same step applies to a seven-card list.

Continue the game based on the result of the seven cards in the above picture. Throw the head-A card away. This is the result:

Read out aloud the seven-letter magic word one by one and move the head card to the bottom at the same time. The last card C is moved to the fifth position of a six-card list. This is exactly the surving position:

Based on the rule of odd-in and even-out in the Josephus problem, the other half C card is the last remaining card:

At last, the match was made!

## Conclusion

Based on those two math problems, Liu elegantly designed his very interesting magic game. The first math is a list of replicated sequences. He could repeatedly move two matched cards to the head and tail of the list. Then keep the first card as a secret, to-be-matched card. The second is the Josephus problem. He continued his deceiving steps to move the last card to the right position and finally found the matched card.

With my above analysis, you may further design your own and more interesting steps. You may use various numbers of cards, such as 2, 3, 5, or even more.

After this game, do you have any other interesting math problems? You may define another totally different but also very interesting gamic game. Share yours with us!

## References

- My previous blog: Liu Qian's Magic Card Show in TV Gala
- Wikipedia：Josephus problem
- A computer science class on the Josephine Problem. At 12:57, a table is displayed with the n-list and survive index solution.

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