What are geometric puzzles. Math puzzles for kids and adults. What is a geometric puzzle
Tangram - old oriental puzzle from the figures obtained by cutting the square into 7 parts in a special way: 2 large triangles, one medium, 2 small triangles, a square and a parallelogram. As a result of folding these parts with each other, flat figures are obtained, the contours of which resemble all kinds of objects, ranging from humans, animals and ending with tools and household items. These types of puzzles are often referred to as "geometric construction sets", "cardboard puzzles" or "cut puzzles".
With a tangram, a child will learn to analyze images, highlight geometric shapes in them, learn to visually break an entire object into parts, and vice versa - to compose a given model from elements, and most importantly - to think logically.
How to make a tangram
A tangram can be made from cardboard or paper by printing out a template and cutting along the lines. You can download and print the tangram square diagram by clicking on the picture and selecting "print" or "save picture as...".
It is possible without a template. We draw a diagonal in a square - we get 2 triangles. Cut one of them in half into 2 small triangles. We mark the middle on each side of the second large triangle. We cut off the middle triangle and the rest of the figures at these marks. There are other options for how to draw a tangram, but when you cut it into pieces, they will be exactly the same.
A more practical and durable tangram can be cut from a rigid office folder or a plastic DVD box. You can complicate your task a little by cutting out tangrams from pieces of different felt, overcasting them around the edges, or even from plywood or wood.
How to play tangram
Each figure of the game must be made up of seven parts of the tangram, and at the same time they must not overlap.
The easiest option for preschool children 4-5 years old is to assemble figures according to diagrams (answers) drawn into elements, like a mosaic. A little practice, and the child will learn to make figures according to the contour pattern and even invent their own figures according to the same principle.
Level one - download and print a colored tangram, so it will be easier to navigate the diagram.
Schemes and figures of the game tangram
V Lately tangram is often used by designers. The most successful use of tangram, perhaps, as furniture. There are tangram tables, and transformable upholstered furniture, and cabinet furniture. All furniture, built on the principle of tangram, is quite comfortable and functional. It can be modified depending on the mood and desire of the owner. How many different options and combinations can be made from triangular, square and quadrangular shelves. When buying such furniture, along with instructions, the buyer is given several sheets with pictures on various topics that can be folded from these shelves.In the living room you can hang shelves in the form of people, in the nursery you can put cats, hares and birds from the same shelves, and in the dining room or library - the picture can be on construction theme- houses, castles, temples.
Here is such a multifunctional tangram.
TANGRAM There is an opinion that the history of tangram has about 4000 years. However, this is a common misconception. The myth about this was created by S. Loyd. In 1903, by releasing the book The Eighth Book of Tang, in which he first published his beautiful version of the ancient origins of the game. The place where the game was invented is China. In China, the name Tangram is unknown, and the game is called Shi-Chao-Chu (seven ingenious figures). The date of creation can be determined around the 17th - 18th centuries. The first known ancient book on tangram is the Collection of Figures in Seven Parts (China 1803). It was published on rice paper.
Each of the seven books on tangrams has exactly one thousand figures. These books are now very rare. One of the books, printed in gold on parchment, was discovered in Peking by an English soldier who sold his find for £300 to a collector of Chinese antiquities, who kindly provided some of the most exquisite figurines to be reproduced in this book.
According to Loyd's legend, Tang was a legendary Chinese sage who was worshiped as a deity by his countrymen. He arranged the figures in his seven books according to the seven stages in the evolution of the earth. His tangrams begin with symbolic images of chaos and the principle of yin and yang. Then the simplest forms of life follow, as we move along the tree of evolution, figures of fish, birds, animals and humans appear. Along the way, in various places, images of what was created by man come across: tools, furniture, clothing and architectural structures. There are references to well-known Chinese proverbs in Loyd.
PENTAMINO Pentomino (from other Greek pevta five, and dominoes) - polyominoes of five identical squares, that is, flat figures, each of which consists of five identical squares connected by the sides (“rook's move”). The same word sometimes called a puzzle, in which such figures are required to fit into a rectangle or other shapes. There is only one type of domino, two types of trominoes, and five types of tetraminos. In pentomino, the number of different figures increases immediately to twelve. There are 35 different varieties of Hexamino and 108 varieties of Heptomino.
Obviously, it is impossible to cover an 8x8 chessboard with only trominoes (if only because the number 64 is not divisible by 3). Is it possible to cover the same board with twenty-one straight trominoes and one monomino? By ingeniously coloring the squares that make up the tromino bones in three different colors, Golomb showed that this is possible only when the monomino covers one of the shaded squares. On the other hand, it can be proved by full mathematical induction that twenty-one trominoes and one monomino can completely cover the chessboard, no matter where the monomino is.
It turns out that the board can be covered with sixteen identical tetraminoes of any type, except for the zigzag one. Zigzag tetraminoes cannot even be laid so as to cover at least a strip at the edge of the board. If the board is painted with multi-colored stripes, then it can be proved that 15 L-shaped tetraminoes and one square tetramino cannot form coverings. By coloring the board with zigzag stripes, we will prove that a square tetromino plus any combination of straight and zigzag tetrominoes cannot cover the entire board either. When looking at pentominoes, the question involuntarily arises: can these 12 pieces and one square tetramino be used to fold an ordinary 8x8 chessboard? The first solution to this problem appeared in 1907. It belonged to Henry Dudeney. In Dudeney's solution, the square tetramino is adjacent to the side of the board.
Principles of the game 1. Play in such a way that there is always room for an even number of "dice" (if you play together). 2. You find it difficult to analyze the created position, try to complicate it as much as possible so that the enemy finds himself in an even more difficult position than you.
STOMACHION It was invented 2200 years ago by the ancient Greek thinker and mathematician Archimedes (BC), but the problem was solved only in 2003 by the American mathematician Bill Cutler. Using a specially designed computer program he knew everything possible solutions, which, without taking into account the rotation of the square and its mirror reflections, there are 536, and including all options
Archimedes tried to establish how many variants of new configurations of a square can exist with its 14 constituent parts. This question is solved by combinatorics, which turned into an independent discipline only in the 19th century. We do not know whether Archimedes succeeded in solving his own problem. Netz stumbled upon it by accident while copying old Archimedean works from 10th-century parchment sheets, which are considered the last copies of the original records. But from the parchment, which is stored in the Walters Art Museum in Baltimore, the monks scraped off the old letters and rewrote the document in a new way. As Netz was transcribing parchment one morning, he received a gift from the post office, a children's game based on the stomachion model. The researcher immediately noticed the similarity of the contents of the package with a drawing on parchment depicting a square cut. He came up with the idea that Archimedes created not a game for children, but the basics of combinatorics. Eureka!
Conclusion: These games have a thousand-year history. The penchant for geometric riddles is characteristic of people different eras and nationalities. These puzzles are interesting for people of any age, but first of all they are of great benefit to children, as they stimulate figurative, spatial and creative thinking, develop memory, logic and imagination.
To solve the puzzles collected in this chapter, knowledge of the full course of geometry is not required. Those who are familiar with only a modest circle of initial geometric information are able to cope with them. The two dozen problems proposed here will help the reader to ascertain whether he really possesses the geometric knowledge that he considers mastered. True knowledge of geometry consists not only in the ability to enumerate the properties of figures, but also in the art of managing them in practice to solve real problems. What good is a gun to a man who can't shoot?
Let the reader check how many well-aimed hits he will have out of 24 shots on geometric targets.
72. Cart.
Why does the front axle of the cart wear out more and catch fire more often than the rear?
73. In a magnifying glass.
An angle of 1 1/2° is viewed through a loupe magnifying 4 times. What size will the angle appear (Fig. 66)?
74. Carpentry level.
You are familiar, of course, with a carpenter's level with a gas bubble (Fig. 67), extending towards the 01 mark, when the base of the level has an inclination. The greater this slope, the more the bubble moves away from the middle mark. The reason for the movement of the bubble is that, being lighter than the liquid in which it is located, it floats up. But if the tube were straight, the bubble, at the slightest inclination, would run off to the very end of the tube, that is, to its highest part. Such a level, as it is easy to understand, would be very inconvenient in practice. Therefore, the level tube is taken bent, as shown in Fig. 67. With a horizontal base of this level, the bubble, occupying the highest point of the tube, is located at its middle; if the level is inclined, then the highest point of the tube is no longer its middle, but some point adjacent to it, and the bubble moves away from the mark to another place in the tube.
The question of the problem is to determine how many millimeters the bubble will move away from the mark if the level is inclined by half a degree, and the radius of the tube bending arc is 1 m.
75. The number of faces.
Here is a question that no doubt will seem to many too naive or, on the contrary, too clever:
How many edges does a hexagonal pencil have?
Before looking at the answer, think carefully about the problem.
76. Lunar sickle.
The figure of the crescent moon (Fig. 68) needs to be divided into 6 parts, drawing only 2 straight lines.
How to do it?
77. From 12 matches.
From 12 matches, you can make a figure of a cross (Fig. 69), the area of \u200b\u200bwhich is equal to 5 "match" squares.
Change the location of the matches so that the outline of the figure covers an area equal to only 4 "match" squares.
It is not possible to use the services of measuring devices.
78. From 8 matches.
From 8 matches you can make quite a variety of closed figures. Some of them are shown in Fig. 70; their areas are, of course, different. The task is to make a figure from 8 matches that covers the largest area.
79. The path of the fly.
On the inner wall of a glass cylindrical jar one can see a drop of honey three centimeters from the upper edge of the vessel. And on the outer wall, at a point diametrically opposite, a fly settled (Fig. 71).
Show the fly the shortest path it can take to reach the honey drop.
Can height 20 cm; diameter - 10 cm.
Do not rely on the fly to find the shortest path on its own and thus make it easier for you to solve the problem: for this it would need to have a geometric knowledge that is too extensive for a fly's head.
80. Find a plug.
Before you is a board (Fig. 72) with three holes: square, triangular and round. Can there be one plug of this shape to cover all these holes?
81. Second plug.
If you have coped with the previous task, then perhaps you will be able to find a plug for such holes as shown in fig. 73?
82. Third plug.
Finally, another problem of the same kind: is there one plug for the three holes in fig. 74?
83. Pass a nickel.
Stock up on two coins of modern coinage: 5 kopeck and 2 kopeck. On a piece of paper, make a circle exactly equal to the circumference of a 2-kopeck coin, and carefully cut it out.
Do you think a nickel will fit through this hole? There is no catch here: the problem is truly geometric.
84. Tower height.
There is a landmark in your city - a high tower, the height of which, however, you do not know. You also have a photograph of the tower on a postcard. How can this picture help you know the height of the tower?
85. similar figures.
This problem is intended for those who know what geometric similarity is. You need to answer the following two questions:
86. Wire shadow.
How far does a sunny day stretch in space full shadow, rejected by a telegraph wire, the diameter of which is 4 mm?
87. Brick.
Building bricks weigh 4 kg. How much does a toy brick made of the same material weigh, all dimensions of which are 4 times smaller?
88. Giant and dwarf.
Approximately how many times is a giant 2 meters tall heavier than a dwarf 1 meter tall?
89. Two watermelons.
Two watermelons of different sizes are sold at the collective farm market. One is a quarter wider than the other, and it costs 1 1/2 times more. Which one is better to buy?
90. Two melons.
Selling two melons of the same variety. One circumference is 60, the other is 50 cm. The first is one and a half times more expensive than the second. What is the best melon to buy?
91. Cherries.
Cherry pulp surrounds the stone with a layer of the same thickness as the stone itself. We will assume that both the cherry and the stone are in the form of balls. Can you figure out in your mind how many times the volume of the juicy part of the cherry is greater than the volume of the stone?
92. Model of the Eiffel Tower.
The Eiffel Tower in Paris, 300 meters high, is made entirely of iron, of which about 8,000,000 kg went into it. I wish to order an exact iron model of the famous tower weighing only 1 kg.
What height will she be? Above glass or below?
93. Two pots.
There are two copper pans of the same shape and with walls of the same thickness. The first is 8 times more spacious than the second.
How much heavier is it?
94. In the cold
An adult and a child are standing in the cold, both dressed alike. Which one is colder?
State educational institution Tula region"Tula special (correctional) boarding school for students, pupils with disabilities"
Distance Education Center
Related presentation
Games to recreate from geometric shapes figurative and plot images.
- Puzzles are toys for all time. Before the advent of computers and the rapid development board games, one of the main entertainments for most people was the Tangram puzzle game. Nowadays, a lot of people are addicted to puzzles. They are loved not only by children, but also by adults. The game helps to develop logical thinking, geometric intuition. This is a way of distraction from everyday problems and is aimed at the development of various thought processes - comparison, generalization, establishing a sequence, determining the relationship "whole" - "part". All these skills are necessary for future mathematicians.
- Now puzzles are sold in different versions - wooden, paper, and plastic.
- On the plane, you need to lay out any shapes that you can think of or you can use a sample. In this case, you can not overlap the parts so that they overlap each other. And also you need to use all the details.
TANGRAM GAME
Tangram ( whale. 七巧板 , pinyin qī qiǎo bǎn, lit. "seven boards of mastery") - puzzle , consisting of seven flat figures , which are added in a certain way to get another, more complex, figure (depicting a person, animal, household item, letter or number, etc.). The figure to be obtained is usually given in the form silhouette or outer loop. When solving the puzzle, two conditions must be met: first, all seven tangram figures must be used, and second, the figures must not overlap.
How many big triangles?
How many small triangles?
How many middle triangles?
How many triangles are there and how big are they?
Two large two small and one medium
- Each assembled figure must include all seven elements.
- When drawing up figures, the elements should not overlap each other.
- Elements of figures should adjoin one another.
- You need to start by finding the place of the largest triangle.
- As a result of the game, a planar silhouette image is obtained. It is conditional, schematic, but the image is easily guessed by the main characteristic features of the object: its structure, proportional ratio of parts and shape.
- Tangram - very ancient game- puzzle. She appeared in China more
- 4000 years ago. Exists whole line versions and hypotheses of the emergence of the game "Tangram".
- Legend one.
Broken tiles.
More than 4000 thousand years ago, a porcelain tile fell out of the hands of one person and broke into seven pieces. Frustrated, he hurriedly tried to fold it, but each time he received more and more interesting images. This lesson turned out to be so fascinating that subsequently a square made up of seven geometric figures was called the Board of Wisdom.
Shi Chao Chu
- The second legend: three wise men came up with "Shi-Chao-Chu".
The appearance of this chinese puzzle associated with a beautiful legend.
Almost two and a half thousand years ago, the long-awaited son and heir was born to the elderly emperor of China. Years passed. The boy grew up healthy and quick-witted beyond his years. One thing bothered the old emperor: his son, the future ruler of a vast country, did not want to study. It gave the boy more pleasure to play with toys all day long. The emperor summoned three wise men to himself, one of whom was known as a mathematician, the other became famous as an artist, and the third was a famous philosopher, and ordered them to come up with a game, amused by which, his son would comprehend the beginnings of mathematics, learn to look at the world with the watchful eyes of an artist, would become patient, like a true philosopher, and would understand that often complex things are made up of simple things. Three wise men came up with "Shi-Chao-Chu" - a square cut into seven parts.
Seven Books of Tan
- Legend Three: The Seven Books of Tan.
“In the notes of the late Professor Challenor, which fell into the hands of the author,” Loyd argued, “there is evidence that seven books on tangrams, each of which has exactly a thousand figures, were compiled in China more than 4000 years ago. These books have now become so rare that in the forty years that Professor Challenor spent in China, he only once managed to see the first edition of the first of seven volumes.
According to Loyd's legend, Tang was a legendary Chinese sage who was worshiped as a deity by his countrymen. He arranged the figures in his seven books according to the seven stages in the evolution of the earth. His tangrams begin with symbolic representations of chaos and the yin and yang principle. Then the simplest forms of life follow, as we move along the tree of evolution, figures of fish, birds, animals and humans appear. Along the way, in various places, images of what has been created by man come across: tools, furniture, clothing and architectural structures. Loyd frequently cites the sayings of Confucius, the philosopher Shufutse, the commentator Li Huangzhang, and the fictional Professor Challenor. Li Huangzhang is mentioned because, according to legend, he knew all the figures from the seven books of Tang before he could speak. There are also references in Loyd to "famous" Chinese proverbs such as "Only a fool would undertake to write the eighth book of Tang."
- The first image of a tangram (1780) was found on a woodcut by the Japanese artist Utomaro, where two girls fold the figures. The name "tangram" originated in Europe, most likely from the word "tan" (which means "Chinese") and the root "gram" (translated from Greek "letter") At first, it was used not for entertainment, but for teaching geometry .
- The tangram may have originated from titles type of furniture that appeared in times empire soong ., and later the word turned into a set of wooden figures for the game.
- Writer and mathematician Lewis Carroll considered a tangram enthusiast. He kept a Chinese book with 323 problems.
- At Napoleon during his exile Saint Helena there was a tangram set and a book containing problems and solutions.
- Book Sam loida "The Eighth Book of Tang" ( English The Eighth Book Of Tan ), released in 1903 , contains a fictional history of the tangram, according to which this puzzle was invented 4,000 years ago by a deity named Tang. The book includes 700 problems, some of which are unsolvable.
- What figure did you make?
- "The charm of the tangram lies in the simplicity of the material and in its seeming unsuitability for creating figures with aesthetic appeal"
M. Gardner:
- Columbus egg is a catch phrase denoting an unexpectedly simple way out of a predicament.
- Once upon a time there was an Italian Girolamo Benzoni in the 16th century. He loved to travel. And once at a dinner with Cardinal Mendoza, he met Columbus. This story happened there. According to legend, when Columbus, during a dinner with Cardinal Mendoza, talked about how he discovered America, the century Italian Girolamo Benzoni said: “What could be easier than discovering new land? In response to this, Columbus offered him a simple task: how to put an egg on the table vertically? When none of those present could do this, Columbus took the egg, broke it from one end and put it on the table, showing that it really was simple. Seeing this, everyone protested, saying that they could do it too. To which Columbus replied: "The difference is, gentlemen, that you could do it, but I actually did it."
- The name "Columbian Egg" is very appropriate for the proposed puzzle. It also takes a long time to puzzle over how to assemble a picture from ten pieces of an egg, and the resulting image is usually very simple. This mysterious and exciting game belongs to the class of geometric constructors (tangrams). Playing with geometric constructors contributes to the development of ingenuity, spatial imagination, constructive thinking, combinatorial abilities.
This is an oval of 10 parts: among them 4 triangles (2 large and 2 small), 2 figures similar to a quadrilateral, one of the sides of which is rounded, 4 figures (large and small) resembling a triangle, but with a rounded one side .
- . It is best to make silhouettes of birds from the details of the Columbus Egg puzzle (54 different forms of birds are known), you can also make silhouettes of objects, people, animals.
- What figure did you make?
- Geometric puzzles are a wonderful tool for developing the mind, and puzzles can also be used to create an interior:
- http://www.lobzik.pri.ee/modules/news/article.php?storyid=645
- http://www.youtube.com/watch?v=JClq8XIuK6M
Cuby Gami ( Cubigami)
- http://yandex.ru/video/search?p=1&filmId=nMtgVgv_UXI&text=%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8 %D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5%20%D0%B3%D0%BE%D0%BB%D0%BE%D0%B2%D0%BE %D0%BB%D0%BE%D0%BC%D0%BA%D0%B8&_=1417688865574&safety=1
Puzzle is an educational game for any age, aimed at enhancing spatial perception and imagination.
- What was interesting about the lesson?
- What was especially memorable?
- What composition would you prefer? Why?
- http http://www.golovolomok.net/component /
- http://yandex.ru/video/search?p=1&filmId=nMtgVgv_UXI&text=%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8 %D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5%20%D0%B3%D0%BE%D0%BB%D0%BE%D0%B2%D0%BE %D0%BB%D0%BE%D0%BC%D0%BA%D0%B8&_= 1417688865574&safety=1
- http:// www.youtube.com/watch?v=JClq8XIuK6M
- http:// www.lobzik.pri.ee/modules/news/article.php?storyid=645
- http://festival.1september.ru/articles/626772 /
- Animation:
- http://yandex.ru/images/search?img_url=http%3A%2F%2Fwww.mathpuzzle.com%2FInterlockingSpiralsAnimation.gif&uinfo=sw-1525-sh-858-ww-1506-wh-708-pd-0.89552241563797- wp-16x9_1366x768&_=1417717151222&p=2&viewport=wide&text=%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5 %D1%81%D0%BA%D0%B8%D0%B5%20%D1%84%D0%B8%D0%B3%D1%83%D1%80%D1%8B%20%D0%B0%D0 %BD%D0%B8%D0%BC%D0%B0%D1%86%D0%B8%D1%8F&pos=60&rpt=simage&pin=1
- http:// myweb.rollins.edu/jsiry/Deep_technology_tetrahedron.html
- http:// animating.ru/avatars/category_25.htm
Arithmetic puzzles
Exercise 1. Fill in the table with numbers from 1 to 4 so that adjacent cells contain different numbers. (Two cells are called adjacent if they have no common points, including vertices).
Grade
Task 2. Arrange the numbers from 1 to 10 in small circles so that the sums of the numbers in the four large circles are equal.
Grade: Each correct answer is worth 15 points.
Task 3. Is it possible to knock out 100 points with several shots?
Grade: Each correct answer is worth 15 points.
Task 4. Arrange the numbers from 1 to 12 so that the sum of the numbers in all selected areas is 26.
-0,02
Grade: Each correct answer is worth 15 points.
Task 5. Decipher the example. Same letters mean the same numbers, different letters mean different numbers.
Grade: Each correct answer is worth 15 points.
Geometric puzzles
Exercise 1. Cut the given figure into 3 parts and fold them into a square.
Solve the problem in several ways.
Grade: Each correct answer is worth 10 points.
Task 2. The points in the table are the vertices of the eight triangles. All of them have different areas from 0.5 to 4 unit squares. Restore the triangles, keeping in mind that they should not overlap or touch each other.
Grade: Each correct answer is worth 15 points.
Task 3. The 10x10 table is a plan of the forest, which shows the position of 20 trees. To each tree you need to tie your own tent (in a cage touching the cage of the tree with its side), in total exactly 20 tents. Cells with tents must not touch (even at an angle), and the numbers indicate the number of cells occupied by tents in the corresponding row or column.
Grade: Each correct answer is worth 15 points.
Task 4. For the task, a set of 12 2x2 squares is used, and four colors. In each square, one color is missing, two adjacent cells are painted with the second, and the third and fourth are applied to the remaining pair of cells, so that all possible combinations of colors are obtained.
Arrange the squares in a checkered grid without intersections, and calculate the area of the smallest figure of each color. The sum of the resulting four numbers will be your result. Try to make this amount as large as possible.
In the example, the only red area has an area of 12, the smallest yellow area is 3, the smallest green area is 2, and the smallest blue area is 1. The total is 18.
Grade: 30 points for best solution, 28 for next, 26 for next...
Task 5. Cut the 8X8 square into as many triangles as you can. All of them should be of different area and have vertices at the nodes of the grid. In the example, a 4X4 square is cut into five triangles. The area of each is indicated.
Grade: 30 points for best solution, 28 for next, 26 for next…
Exercise 1. Come up with a rebus on the topic of mathematical terms.