It is said that black cards and red cards are printed with different inks, one is water-based and the other is oily. Get out of here! /Really?

In the oil-water magic, the magician has 10 cards: 5 red cards and 5 black cards (it doesn't matter if the red cards and black cards are other unequal numbers). At first, the magician turned all the red cards upside down, while all the black cards were upside down, which upset the order of the cards. Then, Uncle turned the five cards in the even positions of 2, 4, 6, 8, 10 upside down (the so-called "flip shuffling method"), which means that among the five cards in the odd positions (called the first group), the red card is still facing up, the black card is facing up, and the five cards in the even positions (called the second group) are the opposite. The five cards in the first group and the five cards in the second group just cross.

Next, the magician and the audience carried out a special shuffling method (Hummer shuffling method), that is, a stack of cards was mixed with the front and back, and two adjacent cards (or multiples of two, such as four, six or eight cards) were taken out at a time, so that the audience could choose to keep the same or flip the cards. In fact, this shuffling method is equivalent to taking out one card from the first group of cards and the second group of cards for exchange, and changing the front and back at the same time (you can think carefully about why this is the case). Then a * * * has the following four possible situations: (as shown in the figure below)

We can find that according to this shuffling method, no matter how many times shuffling, the black cards in the first group always face up and the red cards always face up; In the second group, the red card always faces up, and the black card always faces up, unchanged. Finally, after shuffling the Hummer for many times, the magician divided the odd and even cards into two stacks, and turned all the stacks over, so that the left and right stacks of cards with the same color and opposite faces returned to the original state: all the red cards were facing in one direction, or they were facing up at the same time, or they were facing up at the same time, and all the black cards were facing up in the other direction at the same time, thus realizing oil-water separation.

In this magic trick, the magician shuffles the cards several times by cutting cards (putting a certain number of cards from top to bottom), but cutting cards has no effect on the final result, just moving all the first five cards in the original odd position to the even position, and all the five cards in the even position to the odd position.

When the secret is revealed, you will find that the so-called "separation of water and oil" is just another trick.

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