The magician drew 2/kloc-0 cards from a deck of playing cards and said to an audience, "Please remember one card". The audience saw it and remembered one of them. The magician washed the cards and then divided them on the table. As shown in the figure, put the first card in the position of 1, the second card in the position of 2, ..., and the last card in the position of 2 1, with the cards facing up. Divide them into three groups with 7 people in each group. At this time, ask the audience which group of cards they remember. When the audience says that they are in a certain group, the magician puts away the three groups of cards separately, and keeps the order of the cards in the group unchanged when putting them away. Then fold the three sets of cards and hold them in your hand. When folding, secretly put the group that the audience confirmed had dictation cards in the middle layer. The magician stopped shuffling and immediately began the second card division. The division is the same as above, the first one is placed at 1, and the third one is placed at 2. Then ask the audience. Which group is stuck now? When the audience tells which group they belong to, the magician gathers and folds again as before. Then divide the third card. After dividing, ask the audience which group they remember. When the audience points out their group. At this time, the magician did not hesitate to draw a card from the group, which happened to be the one memorized by the audience and won applause.

sequence

one

Zu Di

two

Zu Di

three

group

1

four

seven

10

13

16

19 2

five

eight

1 1

14

17

20 3

six

nine

12

15

18

2 1

After seeing this magic trick, I became interested in uncovering its mystery. After experiments and careful consideration, I finally found the shadow of mathematics. In fact, after the first deck, the cards memorized by the audience, such as A cards, may appear anywhere in any group. However, after the second division, the position of card A on the map can only be one of the numbers 8~ 14, because the seven cards on the number 8~ 14 were originally put in the middle layer by a magician on purpose. Now no matter which new group A is assigned to, it is only one of the three cards in the middle of the new group, that is, the third, fourth and fifth cards in the group. After three points, the position of a card on the map can only be one of 10, 1 1 2. The reason is that the three cards in these three positions are the three cards in the middle of the group where A is located, before folding. Now, since the positions 10, 1 1 and 12 are among the three groups respectively, it is absolutely correct for the magician to draw the cards in the middle of the group as long as the audience tells which group A belongs to. The secret of magic is to put the group containing A card in the middle layer every time you fold cards, without attracting the attention of the audience.