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The trick is to guess the intended number first. Start in science. Focus “By division remainders” |
The text of the work is posted without images and formulas. "The subject of mathematics is so serious that it is useful to seize the opportunity to make it a little entertaining" B. Pascal When we first met in a mathematics lesson, the teacher promised to guess the date of birth of each student in our class if we quickly and correctly performed the arithmetic operations she suggested. First, we had to multiply our birthday by 2, add 5 to the resulting number, multiply the resulting result by 50 and, finally, add the number of the month of our birth to the resulting number. After we told the resulting number to the teacher, she, as promised, guessed our date of birth and was mistaken only when we ourselves were to blame for the incorrect calculations. I really liked this trick. I also became interested in what lies at the heart of this trick. It was then that I decided that I would definitely research the issue of mathematical tricks, find out their secrets, make a selection of tricks and surprise and entertain my friends and acquaintances by demonstrating mathematical tricks in mathematics lessons, extracurricular activities and even at home parties. I read in Internet sources that mathematical tricks do not receive special attention from either mathematicians or magicians. The first consider them simple fun, the second consider them too boring. But, in my opinion, this is not at all true. Mathematical tricks have a deep meaning. Mathematical tricks are experiments based on mathematical knowledge, on the properties of figures and numbers, presented in an extravagant form. To understand the essence of this or that experiment means to understand a small, but very important mathematical pattern. A person's ability to guess numbers conceived by others seems amazing to the uninitiated. But if we learn the secrets of tricks, we will be able not only to show them, but also to come up with our own new tricks. And the secret of the trick becomes clear when we write down the proposed actions in the form of a mathematical expression, transforming which we obtain the secret of guessing. In my work, I want to prove that mathematical tricks help develop memory, intelligence, the ability to think logically, improve mental calculation skills and, finally, simply increase students’ interest in mathematics, which should improve the quality of their knowledge. Goal of the work: explore math tricks. Tasks: Study the literature on the topic under study. Demonstrate a few tricks. Explain them in terms of mathematics. Attract the attention of classmates to study mathematics. Subject of study: math tricks Object of study:"secrets" of mathematical tricks Research methods: study and analysis of literature on entertaining mathematics, independent modeling of mathematical tricks. Practical significance: The material can be used in mathematics lessons and extracurricular activities, at mathematical evenings and holidays, and during mathematical competitions. Chapter 1. History of the emergence of mathematical tricks. Focus- a skillful trick based on deception of vision, attention with the help of a deft and quick technique, movement (Ozhegov’s dictionary) The history of mathematical tricks. The first document that mentions the art of illusion is an ancient Egyptian papyrus. It contains legends dating back to 2900 BC, the era of the reign of Pharaoh Cheops. Initially, magic tricks were used by sorcerers and healers. The priests of Babylon and Egypt created a huge number of unique tricks using excellent knowledge of mathematics, physics, astronomy and chemistry. The list of miracles performed by the priests can include: thunderclaps, lightning flashes, temple doors opening by themselves, statues of gods suddenly appearing from underground, the sounding musical instruments themselves, voices. In Ancient Greece, the harmonious development of personality was unimaginable without games. And the games of the ancients were not only sports. Our ancestors knew chess and checkers, and they were no strangers to puzzles and riddles. Scientists, thinkers, and teachers have always been familiar with such games. They created them. Since ancient times, the puzzles of Pythagoras and Archimedes, the Russian naval commander S.O. Makarov and the American S. Loyd have been known. We find the first mention of mathematical tricks in the book of the Russian mathematician Leonty Filippovich Magnitsky, published in 1703. We all know the great Russian poet M.Yu. Lermontov, but not everyone knows that he was a great lover of mathematics, he was especially attracted to mathematical tricks, of which he knew a great variety, and he invented some of them himself. The enormous cognitive and educational value of intellectual games was repeatedly pointed out by K.D. Ushinsky, A.S. Makarenko, A.V. Lunacharsky. Among those who were interested in them were K.E. Tsiolkovsky, K.S. Stanislavsky, I.G. Erenburg and many other outstanding people. I would especially like to mention the American mathematician, magician, journalist, writer and popularizer of science Martin Gardner. He was born on October 21, 1914. Graduated from the Faculty of Mathematics of the University of Chicago. Founder (mid-50s), author and presenter (until 1983) of the “Mathematical Games” column of the Scientific American magazine (“In the World of Science”). Gardner interprets entertaining as a synonym for fascinating, interesting to learn, but alien to idle entertainment. Gardner's works include philosophical essays, essays on the history of mathematics, mathematical tricks and "comics", popular science sketches, science fiction stories, and intelligence problems. Gardner's articles and books on entertaining mathematics gained particular popularity. Seven books by Martin Gardner have been published in our country, which captivate the reader and encourage independent research. “Gardner’s” style is characterized by intelligibility, brightness and persuasiveness of presentation, brilliance and paradoxicality of thought, novelty and depth of scientific ideas. Among our compatriots I would like to mention the name of Ya. I. Perelman. Yakov Isidorovich Perelman did not make any scientific discoveries, did not invent anything in the field of technology. He did not have any academic titles or degrees. But he was devoted to science and for forty-three years he brought people the joy of communicating with science. It is with his books that the journey into the fascinating world of mathematics, physics, and astronomy begins. And it was his books that helped me write this work. Ignatiev E.I., Kordemsky B.A. made their enormous contribution to the popularization of mathematics. and many other Russian scientists, teachers, methodologists. Mathematical tricks are interesting precisely because each trick is based on mathematical laws. Their meaning is to guess the numbers conceived by the audience. Millions of people in all parts of the world are addicted to mathematical tricks. And this is not surprising. “Mental gymnastics” is useful at any age. And tricks train memory, sharpen intelligence, develop perseverance, the ability to think logically, analyze and compare. Chapter 2. Mathematical tricks Focus “Guess the intended number.” Let's ask any student to think of a number. Then the student must multiply this number by 2, add 8 to the result, divide the result by 2 and take away the intended number. As a result, the magician boldly calls the number 4. The solution to the trick: The viewer thought of the number 7 1) 7●2 = 14 2) 14 + 8 = 22 3) 22/2 = 11 4) 11 - 7 = 4 The number X is guessed. 2) X●2 2) X●2 + 8 3) (X●2 + 8)/2 4) (X●2 + 8)/2 - X = X + 4 - X = 4 We got 4 regardless of the originally guessed number Focus “Magic table”. You see a table in which numbers from 1 to 31 are written in a special way in five columns. I invite those present to think of any number from this table and indicate in which columns of the table this number is located. After that I will tell you the number you have in mind. The solution to the trick: This table is compiled as follows: each column corresponds to a certain number, after calculating the sum of which the magician guesses the number you have chosen For example: You thought of the number 27. This number is in the 1st, 2nd, 4th and 5th columns. It is enough to add the numbers located in the first row of the table in the corresponding columns, and we will get the intended number. (1+2+8+16=27). Focus “Favorite number”. Each of those present thinks of their favorite number. I suggest he multiply the number 15873 by his favorite number multiplied by 7. The solution to the trick: 1) 15873 * 7 = 111111. Thus, multiplying 15873 by 7 and by the favorite number, we get a number written only by the favorite number. For example, favorite number is 5 1) 15873 *(7*5) 2) 15873 *35 = 555555. 4. Focus “Guess the intended day of the week.” Let's number all the days of the week: Monday is the first, Tuesday is the second, etc. Let someone think of any day of the week. I suggest you the following actions: multiply the number of the planned day by 2, add 5 to the product, multiply the resulting amount by 5, add 0 to the resulting number at the end, and report the result to the magician. The solution to the trick: Let's say Thursday is planned, that is, day 4. Let's do the following: ((4×2+5)*5)*10 = 650, 650 - 250 = 400. The number of hundreds shows the hidden day of the week. By the way, the trick that our teacher showed us at the beginning of the school year for guessing the date of birth has the same secret. Let the day of my birth (and this is a single or double digit number) X, and the number of the month of my birth at then we have: (2 · X+ 5) · 50 + at= 100 · X + 250 + u. If you now subtract 250 from the result, you get a three or four-digit number, the last two digits of which indicate the month number, and the first one or two digits indicate the birthday. 5. Focus “Familiar numbers” After this, the magician immediately calls out the intended numbers. The solution to the trick: 6. Focus 2. Ask a friend to write a number from 100 to 999. The only condition! The difference between the first and last digits must be greater than one. For example, the number 346 is suitable, since 6 - 3 = 3, and 3 is greater than 1. But the number 344 is not suitable, since 4 - 3 = 1. 3. Suppose your friend has already chosen a number and written it down. Your task is to rewrite this number in reverse order (346, and you write 643). 4. Now subtract the smaller number from the larger number (643 - 346 = 297). 6. Add both numbers (297+792). The solution to the trick: 100a + 10b + c; a - c > 1. 100a + 10b + c - 100c - 10b - a = 99a - 99c = 99(a - c). a - c = 2.99 * 2 = 198.198 + 891 = 1089, a - c = 3.99 * 3 = 297.297 + 792 = 1089, a - c = 4.99 * 4 = 396.396 + 693 = 1089, a - c = 9.99 * 9 = 891.891 + 198 = 1089. 7. Focus A circle of comrades who are not privy to the mathematical secret of Scheherazade’s number can be amazed by the following trick. Let someone write on a piece of paper - secret from the magician - a three-digit number, then let him add the same number to it again. The result is a six-digit number consisting of three repeating digits. The magician invites the same comrade or his neighbor to divide - secretly from him - this number by 7: at the same time he warns that there will be no remainder. The result is passed on to another neighbor, who divides it by 11; there should be no remainder. The result obtained is passed on to the next neighbor, who is asked to divide the number by 13 (again without a remainder). The result of the third division is transmitted to the first comrade with the words: Here is the number you have in mind. The solution to the trick: This beautiful arithmetic trick, which gives the impression of magic to the uninitiated, can be explained very simply. Attaching it to a three-digit number itself means multiplying it by 1001 (Scheherazade’s number), that is, by the product 71113. It is clear that if you first multiply the intended number by 1001, and then divide it by 1001, then you will get it yourself. This focus can be changed. Suggest division by 7, then by 11, and then by the intended number. Then we can say with confidence that the result will be 13. 8. Trick “Guess the result of calculations without asking anything” Let's write some number between 1 and 50 on a piece of paper and hide it without showing the participants the trick. In turn, have each participant write whatever number he wishes, greater than 50 but greater than 100, and, without showing you, do the following: will add 99 - x to its number, where x is the number you wrote on a piece of paper (you will calculate this difference in your head and tell the participants of the trick the finished result); cross out the leftmost digit in the resulting sum and add the same digit to the remaining number; the resulting number will be subtracted from the number originally written down by him. As a result, all participants will get the same number, exactly the one you wrote down and hid. The solution to the trick: My number X , Where " X" more than 1 but less than 50. Intended number at , Where " y" greater than 50 but less than or equal to 100. y - (y + 99 - x - 100 + 1) = y - y - 99 + x + 100 - 1 = x. 9. Focus modeled by myself. Guessing the house and apartment number of a participant in the trick. Add 8 to the house number, multiply the result by 8, multiply the result by 125, add the apartment number to the result. Tell me how much you got, and I will tell you your house number and apartment number. The secret of the trick: (X + 8) * 8 * 125 + Y - 8000 = 1000X + 8000 + Y - 8000 = 1000X + Y. The last one, two, three digits are the apartment number, the first 1 - 2 digits are the house number. Conclusions. Previously, I did not understand the significance of mathematical tricks because I knew little about them. I learned that the secret to solving many tricks is equations. While doing research, I became convinced that mathematical tricks are interesting to schoolchildren. Thanks to my work, I increased my knowledge and also realized that magic tricks sharpen the ability to think logically, analyze and compare. In addition, I realized that my current knowledge is not enough to understand the nature of many of the tricks I encountered while researching the topic. This applies to knowledge of algebra and geometry. Therefore, I will continue to study math tricks in future classes. Conclusion There is an interesting parable. “Once upon a time there was an old man who, when he died, left 19 camels to his three sons. He bequeathed half 1/2 to his eldest son, a fourth to his middle son, and a fifth to his youngest. Unable to find a solution on their own (after all, the problem in “whole camels” has no solution), the brothers turned to the sage. O wisest one! - said the elder brother, - my father left us 19 camels and ordered us to divide them among ourselves: the eldest - half, the middle - a quarter, the youngest - a fifth, but 19 is not divisible by 2, 4, or five. Can you, O venerable one, help our grief, for we want to fulfill the will of our father? “There is nothing simpler,” the sage answered them. - Take my camel and go home. The brothers of the house easily divided 20 camels in half, into 4 and into 5. The eldest brother received 10 camels, the middle one 5, and the youngest 4 camels. At the same time, one camel (10 + 4 + 5 = 19) remained extra. The brothers returned to the sage and complained: Oh, sage, again we did not fulfill the will of our father! This camel is superfluous.” “Not superfluous,” answered the sage, “this is my camel.” Return him and go home.” “There are no unsolvable problems. There is always a way out” (folk wisdom) Mathematical tricks are varied. In many mathematical tricks, numbers are veiled by objects related to numbers. They develop skills in rapid mental calculation, calculation skills, because... you can guess small and large numbers, awaken the imagination, surprise, fascinate, develop the creative principles of the individual, artistic abilities, stimulate the need for creative self-expression. Mathematical tricks promote concentration. The magic of magic can wake up the sleepy, stir up the lazy, and make the slow-witted think. After all, without unraveling the secret of the trick, it is impossible to understand and appreciate all its charm. And the secret of focus most often has a mathematical nature. Literature Perelman, Ya.I. Interesting arithmetic. Numbers and tricks / Ya.I.Perelman. - M.: OLMA Media Group, 2013 Perelman, Ya.I. “Living Mathematics”, D.: VAP, 1994 Kordemsky, B.A. Mathematical savvy. - M.: Science. Ch. ed. physics and mathematics lit., 1991 Ignatiev E.I. In the kingdom of ingenuity - M.: Science. Ch. ed. physics and mathematics lit., 1984 M. Gardner “Mathematical miracles and mysteries” - Moscow: “Nauka”, 1988 Application Focus 1: “Familiar numbers” Write down the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in sequence on a piece of paper. Ask one of the students to add in their mind any three numbers following one another. And the result is to be named. For example, he will choose 5, 6 and 7. In this case, the sum will be 18. After this, I immediately call out the intended numbers. The secret of the trick: To do this trick you just need a little intelligence. When they call the sum (5+6+7) = 18, divide it in your head by 3. In our case, you get 6. This is the desired average figure. The number in front of it is 5, and after it is 7. The whole effect of this trick is in the lightning-fast response. Focus 2 1. Write the number 1089 on a piece of paper and put it aside temporarily (without showing it to anyone). 2. Ask a friend to write a number from 100 to 999. The only condition! The difference between the first and last digits must be greater than one. For example, the number 346 is suitable, since 6-3=3, and 3 is greater than 1. But the number 344, for example, is not suitable, since 4-3=1. It's clear? If not quite, read first)) 3. Suppose your friend has already chosen a number and written it down. Your task is to rewrite this number in reverse order (346, and you write 643). Ready? 4. Now subtract the smaller number from the larger number (643-346=297). 5. Now write down the resulting answer in reverse order (it was 297, it will become 792). 6. Add both numbers (297+792). 7. Voila! Show me your piece of paper with the magic number 1089. You knew in advance what the answer would be! Indeed, 297+792=1089! Hocus Pocus!!! The most interesting thing is that this algorithm always works! |
For lovers of mathematical tricks, I am posting a new selection! There are some pretty interesting options. Enjoy! :) Focus “Phenomenal memory”. To perform this trick, you need to prepare many cards, put its number on each of them (a two-digit number) and write down a seven-digit number using a special algorithm. The “magician” distributes cards to the participants and announces that he has memorized the numbers written on each card. Any participant names the number of the roll, and the magician, after thinking a little, says what number is written on this card. The solution to this trick is simple: to name a number, the “magician” does the following: adds the number 5 to the card number, turns over the digits of the resulting two-digit number, then each next digit is obtained by adding the last two; if a two-digit number is obtained, then the units digit is taken. For example: the card number is 46. We add 5, we get 51, rearrange the numbers - we get 15, we add the numbers, the next one is 6, then 5+6=11, i.e. take 1, then 6+1=7, then the numbers 8, 5. Number on the card: 1561785. Focus “Guess the intended number.” The magician invites one of the students to write any three-digit number on a piece of paper. Then add the same number to it again. The result will be a six-digit number. Pass the piece of paper to your neighbor, let him divide this number by 7. Pass the piece of paper further, let the next student divide the resulting number by 11. Pass the result further, let the next student divide the resulting number by 13. Then pass the piece of paper to the “magician”. He can name the number he has in mind. The solution to the trick: When we assigned the same number to a three-digit number, we thereby multiplied it by 1001, and then, dividing it successively by 7, 11, 13, we divided it by 1001, that is, we obtained the intended three-digit number. Focus “Magic table”. There is a table on the board or screen in which numbers from 1 to 31 are written in a well-known manner in five columns. The magician invites those present to think of any number from this table and indicate in which columns of the table this number is located. After that, he calls the number you have in mind. The solution to the trick: For example, you thought of the number 27. This number is in the 1st, 2nd, 4th and 5th columns. It is enough to add the numbers located in the last row of the table in the corresponding columns, and we will get the intended number. (1+2+8+16=27). Trick “Guess the crossed out number” Let someone think of some multi-digit number, for example, the number 847. Invite him to find the sum of the digits of this number (8+4+7=19) and subtract it from the conceived number. It turns out: 847-19=828. including the one that comes out, let him cross out the number – it doesn’t matter which one – and tell you the rest. You will immediately tell him the crossed out number, although you do not know the intended number and did not see what was done with it. This is done very simply: you look for a number that, together with the sum of the numbers given to you, would form the nearest number that is divisible by 9 without a remainder. If, for example, in the number 828 the first digit (8) was crossed out and you were told the numbers 2 and 8, then, having added 2 + 8, you realize that the nearest number divisible by 9, i.e. 18, is not enough 8. This is the crossed out number. Why does this happen? Because if you subtract the sum of its digits from any number, you will be left with a number that is divisible by 9 without a remainder, in other words, one whose sum of digits is divisible by 9. In fact, let in the conceived number a be the hundreds digit, b be the hundreds digit tens, s – units digit. This means that the total number of units in this number is 100a+10b+s. Subtracting the sum of the digits (a+b+c) from this number, we get: 100a+10b+c-(a+b+c)=99a+9c=9(11a+c), i.e. a number divisible by 9. When performing a trick, it may happen that the sum of the numbers given to you is itself divisible by 9, for example 4 and 5. This shows that the crossed out number is either 0 or 9. Then you must answer: 0 or 9. Focus “Who has what card?” An assistant is needed to perform the trick. There are three cards with ratings on the table: “3”, “4”, “5”. Three people approach the table and each takes one of the cards and shows it to the “magician’s” assistant. The “magician” must guess who took what without looking. The assistant tells him: “Guess,” and the “magician” names who has which card. The solution to the trick: Let's consider the possible options. Cards can be arranged as follows: 3, 4, 5 4, 3, 5 5, 3, 4 3, 5, 4 4, 5, 3 5, 4, 3 Since the assistant sees which card each person took, he will help the “magician”. To do this, you need to remember 6 signals. Let's number six cases: First – 3, 4, 5 Second – 3, 5, 4 Third – 4, 3, 5 Fourth – 4, 5, 3 Fifth – 5, 3, 4 Sixth – 5, 4, 3 If the case is the first, then the assistant says: “Done!” If the case is the second, then: “Okay, done!” If it’s the third case, then: “Guess!” If it’s the fourth, then: “So, guess!” If it’s the fifth, then: “Guess!” If it’s the sixth, then: “So, guess!” Thus, if the option starts with the number 3, then “Ready!”, if with the number 4, then “Guess!”, if with the number 5, then “Guess!”, and students take the cards in turn. Focus “Who took what?” To perform this ingenious trick, you need to prepare three small things that fit in your pocket, for example, a pencil, a key and an eraser, and a plate with 24 nuts. The magician invites three students to hide a pencil, key or eraser in their pocket during his absence, and he will guess who took what. The guessing procedure is carried out as follows. Returning to the room after the things have been hidden in their pockets, the magician hands them nuts from a plate to keep. The first one is given one nut, the second one two, the third three. Then he leaves the room again, leaving the following instructions: everyone must take more nuts from the plate, namely: the owner of the pencil takes as many nuts as were handed to him; the owner of the key takes twice the number of nuts that were given to him; the owner of the eraser takes four times the number of nuts that were given to him. The remaining nuts remain on the plate. When all this is done, the “magician” enters the room, glances at the plate and announces who has what item in their pocket. The solution to the trick is as follows: each way of distributing things in the pockets corresponds to a certain number of remaining nuts. Let's designate the names of the participants in the focus - Vladimir, Alexander and Svyatoslav. Let's also denote things by letters: pencil - K, key - KL, eraser - L. How can three things be located between three participants? Six ways: There can be no other cases. Let's now see which remainders correspond to each of these cases:
You see that the remainder of the nuts is different in all cases, therefore, knowing the remainder, it is easy to determine what the distribution of things is between the participants. The magician again - for the third time - leaves the room and looks into his notebook with the last sign (there is no need to remember it). Using the sign, he determines who has what item. For example, if there are 5 nuts left on the plate, then this means the case (KL, L, K), that is: Vladimir has the key, Alexander has the eraser, Svyatoslav has the pencil. 4th magician (I team) Focus “Favorite number”. Each of those present thinks of their favorite number. The magician invites him to multiply the number 15873 by his favorite number multiplied by 7. For example, if his favorite number is 5, then let him multiply by 35. The result will be a product written only with his favorite number. The second option is also possible: multiply the number 12345679 by your favorite number multiplied by 9, in our case this is the number 45. The explanation of this trick is quite simple: if you multiply 15873 by 7, you get 111111, and if you multiply 12345679 by 9, you get 111111111. Trick: “Guess the intended number without asking anything.” The magician offers students the following actions: The first student thinks of some two-digit number, the second one adds the same number to it on the right and left, the third one divides the resulting six-digit number by 7, the fourth one by 3, the fifth one by 13, the sixth one by 37 and passes on his answer to the person who has planned it. who sees that his number has returned to him. The secret of the trick: if you assign the same number to the right and left of any two-digit number, then the two-digit number will increase by 10101 times. The number 10101 is equal to the product of the numbers 3, 7, 13 and 37, so after division we get the intended number. Fan competition – “Fun Score”. A representative is invited from each team. There are two tables on the board, on which numbers from 1 to 25 are marked in disarray. At the leader’s signal, students must find all the numbers on the table in order; whoever does it faster wins. Focus “Number in an envelope” The magician writes the number 1089 on a piece of paper, puts the piece of paper in an envelope and seals it. Invites someone, having given him this envelope, to write on it a three-digit number such that the extreme digits in it are different and differ from each other by more than 1. Let him then swap the extreme digits and subtract the smaller one from the larger three-digit number . As a result, let him rearrange the extreme digits again and add the resulting three-digit number to the difference of the first two. When he receives the amount, the magician invites him to open the envelope. There he will find a piece of paper with the number 1089, which is what he got. Focus “Guessing the day, month and year of birth” The magician asks students to perform the following actions: “Multiply the number of the month in which you were born by 100, then add your birthday, multiply the result by 2, add 2 to the resulting number, multiply the result by 5, add 1 to the resulting number, add 1 to the resulting number 0, add 1 more to the resulting number and finally add the number of your years. After that, tell me what number you got.” Now the “magician” has to subtract 111 from the named number, and then divide the remainder into three sides from right to left, two digits each. The middle two digits indicate birthday, the first two or one – month number, and the last two digits are number of years, knowing the number of years, the magician determines the year of birth. Focus “Guess the intended day of the week.” Let's number all the days of the week: Monday is the first, Tuesday is the second, etc. Let someone think of any day of the week. The magician offers him the following actions: multiply the number of the planned day by 2, add 5 to the product, multiply the resulting amount by 5, add 0 to the resulting number at the end, and report the result to the magician. From this number he subtracts 250 and the number of hundreds will be the number of the planned day. Solution to the trick: let’s say it’s planned to be Thursday, that is, day 4. Let's perform the following steps: ((4*2+5)*5)*10=650, 650 – 250=400. Focus “Guess the age”. The magician invites one of the students to multiply the number of their years by 10, then multiply any single-digit number by 9, subtract the second from the first product and report the resulting difference. In this number, the “magician” must add the units digit with the tens digit to get the number of years. |
Circle any number you wish. Cross out all the numbers that are in the same column and on the same row as the circled number. Circle any of the uncrossed numbers and cross out the numbers that are on the same row and in the same column. Circle any of the remaining numbers and cross out those numbers that are on the same row and in the same column. Finally, circle the only remaining number. Add up the numbers circled. Nowyou can call them amount. You got 34.
Secret focus.
Why does the drawn matrix “force” you to always choose four numbers that add up to 34? The secret is simple and elegant. Above each column we write the numbers 1, 2, 3, 4, and to the left of each line - the numbers 0, 4, 8, 12:
1 2 3 4
These eight numbers are calledgenerators matrices. In each cell we will enter a number equal to the sum of two generators located at the row and column at the intersection of which the cell is located. As a result, we get a matrix whose cells are numbered in order from 1 to 16, and their sum is equal to the sum of the generators.
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