Mathematical card tricks and their secrets. Math tricks with cards

21 cards is a classic card trick with which many famous wizards started. His distinctive feature- simplicity of execution, which has absolutely no effect on the final effect. Moreover, in order to perform this trick, you do not need any incredible sleight of hand - just basic skills and a good memory to remember all the steps - there are quite a lot of them.

So what will you need for this trick?

• New, high-quality deck of cards;
• Your hands;
• Instructions and a couple of workouts;
• Spectators.

The “21 cards” trick itself is performed by two methods - a standard one, with the simplest ending, and a more complex one, which is suitable for those who like to show card tricks in the form of a show or want to attract spectators to direct participation in the trick. Let's look at both.

First serve.

Simply count out 10 cards and show that the next card is the one the spectator chose. You can add a little mentality by asking the viewer to follow the cards, and saying that you felt “something” on his card. And it doesn’t matter how you flip through the cards - face up or face down.

Second serve.

Deal all the cards one at a time into a not-so-neat pile. At the same time, silently count out the 11th of them, remembering where it lies in this pile. After performing magical movements, take out the spectator's card.

Hello everyone!

Today we are visiting mathematical tricks with cards with a deck of 52 cards. Today's focus is a mathematical one. It's damn easy to do, so let's not waste time chatting and let's get started.

What does magic look like in the eyes of the viewer?

It's simple, guys!

There are two almost equal packs on the table. Now you ask the spectator to take any pack that he likes best and count the number of cards in it, but do not tell you.

When the procedure is completed, you ask the spectator to add up the digits of the number in his head and remember the card using this number.

This is the most difficult part of this trick, but the viewer can do it. When everything is ready, you randomly say any number and count this number from the spectator's pile. And then Tada-da-da-dammmmm…..the spectator’s card appears.

The Secret of Math Trick

The secret is simply catastrophically simple. So…

1) Remove Jokers from the deck, otherwise the trick will not work.

2) Shuffle the deck and give it to the spectator so that he divides it into two equal parts. Of course, it’s unlikely that it will be equal, but it will be almost equal.

3) Now tell the spectator to choose any packet and count the number of cards in it.

4) Did you manage? Great! Now he needs to get in his mind the sum of the digits of this figure... In my example, I got 23 cards in a pile. So I add 2+3=5. You will get your own number.

5) Let’s say he got 7 or 4 or some other number. He must count out the given number of cards from the bottom of “his” pile and remember the last one. For example: I got 5. I count 5 cards from the bottom and remember the fifth. In my example, this is the King of the Cross.

6) When he remembers, let him put “his” pile on the remaining one. You can spend a couple if you know how.

7) And now again your exit - you need to know that his card on top of the deck will be the 19th (in all cases if the deck is divided approximately equally). You can relate this number in some way to the viewer. For example, if he is 19 years old or apartment 19. In general, whatever you want, then the effect will be cooler. Now count out 19 cards. The last one will be the spectator's card.

And finally, a selection of simple tricks

I think that this easy mathematical trick will not cause serious problems. Yes, I don’t think it will cause any problems.

Playing cards have some specific properties that can be used when composing tricks of a mathematical nature. We will indicate five such properties.

• 1. Cards can be considered simply as identical objects that are convenient to count; the images on them do not play any role.
  You could just as easily use pebbles, matches or pieces of paper.
• 2. Cards can be assigned numerical values ​​from 1 to 13 depending on what is depicted on their front side (with Jack, Queen and King
  are taken as 11, 12 and 13 respectively) 1).
• 3. They can be divided into four suits or into black and red cards.
• 4. Each card has a front and back side.
• 5. The cards are compact and identical in size. This allows you to arrange them in different ways, grouping them in rows or making piles, which are immediately
  can be easily upset by simply mixing the cards.

With so many possibilities, card tricks must have been around a long time ago, and mathematical card tricks can be considered to be certainly as old as the game of cards itself.

Apparently, the earliest discussion of card tricks by a mathematician is found in Claude Gaspard Bachet's entertaining book "Problemes plaisants et delestables", published in France in 1612. Subsequently, references to card tricks appeared in many books devoted to mathematical entertainment.

The first and perhaps the only philosopher who condescended to consider card tricks was the American Charles Peirce. In one of his articles, he admits that in 1860 he “concocted” several unusual card tricks based, using his terminology, on “cyclic arithmetic.” He describes two such tricks in detail under the name “first curiosity” and “second curiosity.”

"The First Curiosity" is based on Fermat's theorem. Just describing the method of demonstrating it required 13 pages, and an additional 52 pages were devoted to explaining its essence. And although Peirs reports the "constant interest and amazement of the audience" caused by his trick, the climactic effect of this trick seems so out of proportion to the complexity of the preparations that it is difficult to believe that the spectators were not plunged into sleep long before the window. expectations of his demonstration.

Here is an example of how, as a result of modifying the method of demonstrating one old trick, its entertainingness increased extraordinarily.

Sixteen cards are laid out on the table front side up in the form of a square with four cards in a row. Someone is asked to think of one card and tell the one showing which one. vertical she lies next to her. Then the cards are collected with the right hand in vertical rows and sequentially folded into the left hand. After this, the cards are again laid out in the form of a square sequentially along horizontals; thus, the cards that were originally laid out in the same vertical row now end up in the same horizontal row. The person showing needs to remember which of them now contains the intended card. Next, the viewer is asked to once again indicate in which vertical next to him he sees his card. It is clear that after this the shower can immediately indicate the intended card, which will lie at the intersection of the just named vertical row and the horizontal row in which, as is known, it should be located. The success of this trick, of course, depends on whether the spectator follows the procedure closely enough to recognize the essence of the matter.

Several simple tricks with cards, I recommend for those who love mathematics.

So, we read:

"...and if someday they forget about cards on earth, perhaps this will mean that God has forgiven us. A person finds peace, but will he be able to part with his ancient habits?"

This is not a quote, not a saying, although perhaps one of the “greats” will agree with us, however, we will take the liberty of using the above thought as an epigraph.

Cards indeed ancient game. They are said to have been invented in France in the Middle Ages to amuse some bored king. But most likely this is an invention of the Chinese, in whose books there is a mention of them. In Europe, cards have been known since the Crusades, and in Italy the game of cards existed already in 1379, as evidenced by the book of one artist. In Russia, cards appeared in the 17th century, and it must be said that, despite severe persecution and persecution, they took root quite quickly. “Game is the disgrace of living rooms,” we read in one of the old books, “the corruption of morals and the brake on enlightenment.” Whether one wins or loses, the game remains an equally shameful affair. This is the triumph of fools, because the game does not require any talent, intelligence, or education, you can’t come up with anything better game in order to drive worthy people away from the living room and in their place attract fools and rogues. The game drives out the spirit of fun and liveliness from society. The ancients, whom we admire and are constantly proud of, knew better than we how to take advantage of the pleasures delivered by the company of people gathered for a pleasant pastime.

Currently, it is said further, cards are especially favored in our society; Everyone plays: ladies, girls, and boys, preferring the green field to dancing. This, of course, is a very sad phenomenon, but what to do, “live with wolves, howl like a wolf...”

Today the attitude towards cards is different. Their world is very diverse. People use them to predict fate, and often quite successfully. They are sometimes seen as a kind of magical destiny, forcing one to treat them with trepidation and reverence. Maps can also serve well in the development of logic and intelligence, being an indispensable tool for explaining many mathematical questions and combinations.

Most card tricks are based on dexterity , or simply on “diversion of eyes” and deception of those present. But along with this, there are other “tricks” that boil down to various math problems, developing intelligence and counting.

TASK 1

GUESSING THE NUMBER OF POINTS ON THE CARDS AND THE CARDS THEMSELVES

Guess how many points are in the three cards someone takes?

From a full deck of 52 cards, let someone take three cards and keep them. To find out how many points are in these three cards, do this...

They ask the one who took three cards to add so many cards to each card he took so that, together with the points of each card taken, the result is 15 (Each of the figures is counted as 10). After this, the guesser can only take the remaining cards, count their number to himself, subtract 4 from the resulting number, and you will get the exact sum of the points of the 3 cards taken.

EXAMPLE: For example, let someone take a four, a seven and a nine. Then he must add 11 cards to the four, 8 cards to the seven, and 6 cards to the nine. There are 24 cards left from the deck. Subtracting four from 24, we find that the sum of the 3 cards taken should be equal to 20, which is true.

In one of Somerset Maugham's stories there is the following dialogue:
- Do you like card trick?
- I can’t stand it.
- Then I'll show you a trick.

After the third trick, the victim runs away under some pretext. Most card tricks, if performed not by a skilled professional, but by an amateur, are unbearably boring. But there are other card tricks that do not require any sleight of hand. It is they who are of interest from a mathematical point of view.

Consider, for example, the following trick. The audience and the magician sit at the table opposite each other. The magician takes a deck of cards, face down, and, turning twenty of them face down, passes the deck to the spectator. The spectator carefully shuffles the deck and the turned over cards are distributed randomly. Holding the deck under the table so that neither he nor the magician can see the cards, the spectators count out the top twenty cards and, without removing them from under the table, hand them over to the magician.

The magician takes the stack, but continues to hold it under the table so that he can see the cards. “Neither you nor I know,” he says, “how many upside-down cards there are among the 20 you gave me. However, it seems to me that there are fewer of them, among the 32 that you have left. Without looking at the cards, I will now turn over a few more cards and try to equalize the number of turned over cards in my part of the deck and in yours.”

The magician fiddles with the cards for a while, pretending that he is trying to determine the top and bottom sides of the cards by touch. Then he pulls his cards up, lays them out on the table and counts the ones turned over. There are exactly the same number of them as among the 32 cards that are in the spectator’s hands.

This wonderful trick is best explained using one of the oldest math puzzles. Imagine that there are two vessels in front of you: a liter of water is poured into one of them, and a liter of wine is poured into the other. One cubic centimeter of water taken from the first vessel is poured into a vessel with wine and mixed thoroughly. Then take one cubic centimeter of the mixture and pour it back into the vessel with water. What is more now: water in wine or wine in water? (We are neglecting the fact that usually a mixture of water and alcohol takes up a smaller volume of alcohol and water before mixing).

The answer is: there is exactly the same amount of wine in water as there is water in wine. The funny thing is that this problem contains too much irrelevant information. It is completely unnecessary to know how much liquid is in each vessel, how much of it is poured, and how many times the transfusion is repeated. It does not matter whether the liquids are thoroughly mixed. It is not even important whether the amount of liquid in the vessels is the same before transfusion. The only really important condition is that each vessel, at the end of all transfusions, contains exactly the same amount of liquid as was in it at first. This condition means that no matter how much wine we take from a vessel with wine, we will certainly have to replenish the resulting deficit with the same amount of water. (We can say this: the lack of wine in a vessel with wine is equal to the amount of wine in a vessel with water. - Approx. Ed.)

If the above reasoning seems incomprehensible to the reader, he can understand it with the help of a deck of cards. Let 26 cards laid out in a row on the table face up represent wine, and 26 cards laid out in a row face down represent water. No matter how many times you move cards from one row to another, if in the end there are again 26 cards in each row, then the number of cards lying face down in one row will exactly coincide with the number of cards in the other row lying face up.

Let's take a stack of 32 cards, face down, and a stack of 20 upside down cards and move cards from one stack to another any number of times, making sure that 20 cards remain in the smaller stack at all times. By turning over the smaller stack, you close open cards and, conversely, open cards that were previously closed. Therefore, after turning over in both piles open cards will be equal.

Now it’s probably clear to everyone how the trick with cards works. First, the magician turns over exactly 20 cards. When he receives a pile of 20 from a spectator, the number of unturned cards in it is equal to the number of turned over cards in the rest of the deck. Then, while pretending to turn over some new cards, the magician actually turns over the entire stack of 20 cards he received. As a result, there are as many reversed cards in this pile as there are among the 32 cards remaining in the spectator's possession. This trick is especially surprising to mathematicians, which is why they come up with very complex explanations.

Many tricks involving guessing the number of cards are based on elementary mathematical principles. Here is one of best tricks this type. Turning your back to the audience, ask someone present to take any number of cards from 1 to 12 from the deck and, without naming the number of selected cards, hide them in their pocket. Then your assistant must count out from the top of the deck exactly as many cards as he has already hidden in his pocket, and remember the next card after the last one counted. When all this is done, you turn to face the audience and ask them to name someone's last name and first name, which would have at least 13 letters. Let's say, for example, someone named Benvenuto Cellini. Holding a deck of cards in your hands, you turn to the spectator, in whose pocket the cards he has selected are hidden, and say that he must, naming each letter in the name and surname of Benvenuto Cellini, lay out one card at a time on the table. Showing how to do this, you remove one card from your deck and, saying each letter out loud, lay the cards face down on the table. You then collect these cards and place them on top of the remaining cards in the deck.

You hand the entire deck to the spectator and ask him to put the cards that are in his pocket on top. Be sure to emphasize that you don't know how many cards he has in his pocket. And yet, despite adding an unknown number of cards to the deck, after the spectator spells “Benvenuto Cellini” and does everything you said, the top card in the deck will be the card he had in mind!

It's not hard to see what's going on here. Let x be the number of cards in the spectator’s pocket and, therefore, the number of cards lying in the deck on top of the card he intended, and y be the number of letters in the name and surname of the person named by the spectators. By showing how to spell a first and last name, you reverse the order of the cards, as a result of which the “depth of occurrence” of the noticed card becomes y - x. Adding x cards to the deck results in the intended card ending up in the (y - x + x)th place, counting from above. The quantities x and - x cancel each other, and the intended card, after being at the letters, will be on top.

The following trick is based on a more subtle use of the fact that the results of individual manipulations with cards can cancel each other out. The spectator chooses any three cards face down on the table without showing them to the magician. The remaining cards, carefully shuffled, are returned to the magician by the spectator. “All the cards in the deck will remain in their places,” says the magician. “I just took one card out of the deck. In color and value, it matches the one you will now choose.” With these words, he takes one card from the deck and, without opening it, puts it aside.

The remaining cards are handed to the spectator and he is asked to reveal the three cards that he previously laid out on the table. Let's say it was a nine, a queen and an ace. On each of the open cards, the spectator places cards from the deck face down, while counting out loud. Laying out cards on the nine, he counts from 10 to 15 (that is, he lays out six cards in total). The queen has a value equal to 12 (jack - 11, king - 13), therefore, when placing cards on it, the count must begin with 12. Since the count always ends at 15, the queen will be closed by three cards. On top of the ace (value - 1) you need to lay out 14 cards.

After the required number of cards have been laid out, the magician asks the spectator to add up the value of the three lower (open) cards and find a card in the deck whose number matches the resulting amount. In this example, the total is 22 (9 +12 +1), so the spectator takes out the twenty-second card. Finally, the magician reveals the card set aside at the very beginning of the trick. Both cards - the one just taken out for the audience and the one put aside by the magician a long time ago - match both in meaning and color!

How is this trick done? When choosing his card, the magician must look at the color and value of the fourth card from the bottom and put aside the card that matches it in color and value. The rest is done automatically. (Sometimes this card is among those bottom cards decks. As soon as the spectator runs out of cards, do not forget to ask him to open the next card.) I leave it to the reader to carry out a simple algebraic proof that the trick should always be performed without misfires. The simplicity with which the cards are shuffled makes them very convenient for demonstrating a number of probability theorems, many of which are quite surprising and well deserve to be called tricks. Imagine, for example, that two people each have a deck of 52 cards. One of them counts out loud from 1 to 52. For each count, both lay one card face down on the table. What is the probability that at some point two identical cards will be placed on the table at the same time?

Many people probably think that this probability is small, but in fact it is greater! The probability of a mismatch is 1 divided by the transcendental number e. (This is not exactly true, but the error is less than 1/10.) Since e is 2.718..., the probability of a match is approximately 17/27. If there is someone willing to bet that there will be no match, you have a pretty good chance of winning the bet. It is interesting to note that by laying out cards from two decks, we obtain an empirical method for finding the decimal expansion of the number e, similar to finding the expansion of the number pi by throwing the Buffon needle. How more cards we take, the closer to 1/e the probability of mismatch will be.

The text from the book was typed by Nikita Sklyarevsky