Divide the cross into figures of 5 cells. Cutting tasks.docx - cutting tasks

1. A square contains 16 cells. Divide the square into two equal parts so that the cut line runs along the sides of the cells. (The ways of cutting a square into two parts will be considered different if the parts of the square obtained with one method of cutting are not equal to the parts obtained with another method.) How many solutions does the problem have?
2. A 3x4 rectangle contains 12 cells. Find five ways to cut a rectangle into two equal parts so that the cut line goes along the sides of the cells (cutting methods are considered different if the parts obtained by one method of cutting are not equal to the parts obtained by another method).
3. The 3X5 rectangle contains 15 cells and the central cell has been removed. Find five ways to cut the remaining figure into two equal parts so that the cut line goes along the sides of the cells.
4. A 6x6 square is divided into 36 identical squares. Find five ways to cut a square into two equal parts so that the cut line goes along the sides of the squares. Note: the problem has more than 200 solutions.
5. Divide the 4x4 square into four equal parts so that the cut line goes along the sides of the cells. How many different ways of cutting can you find?
6. Divide the figure (Fig. 5) into three equal parts so that the cut line runs along the sides of the squares.

7. Divide the figure (Fig. 6) into four equal parts so that the cut line runs along the sides of the squares.

8. Divide the figure (Fig. 7) into four equal parts so that the cut lines go along the sides of the squares. Find as many solutions as possible.

9. Divide the 5x5 square with the central square cut out into four equal parts.

10. Cut the figures shown in Fig. 8 into two equal parts along the grid lines, and each part should have a circle.

11. The figures shown in Fig. 9 must be cut along the grid lines into four equal parts so that there is a circle in each part. How to do it?

12. Cut the figure shown in Fig. 10 along the grid lines into four equal parts and fold them into a square so that the circles and stars are symmetrical about all axes of symmetry of the square.

13. Cut this square (Fig. 11) along the sides of the cells so that all parts are the same size and shape and that each contains one circle and an asterisk.

14. Cut the 6×6 checkered paper square shown in Figure 12 into four identical pieces so that each of them contains three colored squares.

10. A square sheet of checkered paper is divided into smaller squares by segments running along the sides of the cells. Prove that the sum of the lengths of these segments is divisible by 4. (The length of the side of the cell is 1).

Solution: Let Q be a square sheet of paper, L(Q) be the sum of the lengths of those sides of the cells that lie inside it. Then L(Q) is divisible by 4, since all considered sides are divided into four sides, obtained from each other by rotations of 90 0 and 180 0 relative to the center of the square.

If the square Q is divided into squares Q 1 , …, Q n , then the sum of the lengths of the division segments is equal to

L (Q) - L (Q 1) - ... - L (Q n). It is clear that this number is divisible by 4, since the numbers L(Q), L(Q 1), ..., L(Q n) are divisible by 4.

4. Invariants

11. Given a chessboard. It is allowed to repaint in a different color all the cells of any horizontal or vertical at once. Can this result in a board with exactly one black cell?

Solution: Repainting a horizontal or vertical line containing k black and 8-k white cells will result in 8-k black and k white cells. Therefore, the number of black cells will change to (8-k)-k=8-2k, i.e. for an even number. Since the parity of the number of black cells is preserved, we cannot get one black cell from the original 32 black cells.

12. Given a chessboard. It is allowed to repaint in a different color all the cells located inside a 2 x 2 square at once. Can exactly one black cell remain on the board?

Solution: Recoloring a 2 x 2 square containing k black and 4-k white cells will result in 4-k black and k white cells. Therefore, the number of black cells will change to (4-k)-k=4-2k, i.e. for an even number. Since the parity of the number of black cells is preserved, we cannot get one black cell from the original 32 black cells.

13. Prove that a convex polygon cannot be cut into a finite number of non-convex quadrilaterals.

Solution: Suppose that a convex polygon M is cut into non-convex quadrilaterals M 1 ,…, M n . To each polygon N we assign a number f(N) equal to the difference between the sum of its interior angles less than 180 and the sum of angles complementing its angles to 360, greater than 180. Compare the numbers A=f(M) and B=f(M 1)+…+ f(M n). Consider for this all points that are the vertices of the quadrilaterals M 1 ..., M n . They can be divided into four types.

1. Vertices of the polygon M. These points contribute equally to A and B.

2. Points on the sides of the polygon M or M 1. The contribution of each such point to B on

180 more than in A.

3. Interior points of the polygon at which the corners of the quadrilateral meet,

less than 180. The contribution of each such point to B is 360 more than to A.

4. Interior points of the polygon M at which the corners of the quadrilaterals converge, and one of them is greater than 180. Such points give zero contributions to A and B.

As a result, we get A<В. С другой стороны, А>0 and B=0. The inequality A > 0 is obvious, and to prove the equality B=0 it suffices to check that if N is a non-convex quadrilateral, then f(N)=0. Let the angles N be a>b>c>d. Any non-convex quadrilateral has exactly one angle greater than 180, so f(N)=b+c+d-(360-a)=a+b+c+d-360=0.

A contradiction is obtained, so a convex polygon cannot be cut into a finite number of non-convex quadrilaterals.

14. There is a chip in the center of each cell of the chessboard. The chips were rearranged so that the pairwise distances between them did not decrease. Prove that in reality the pairwise distances have not changed.

Solution: If at least one of the distances between the chips would increase, then the sum of all pairwise distances between the chips would also increase, but the sum of all pairwise distances between the chips does not change with any permutation.

15. The square field is divided into 100 identical square sections, 9 of which are overgrown with weeds. It is known that weeds in a year extend to those and only those plots in which at least two adjacent (i.e., having a common side) plots are already overgrown with weeds. Prove that the field will never be completely overgrown with weeds.

Solution: It is easy to check that the length of the boundary of the entire weedy area (or several areas) will not increase. At the initial moment, it does not exceed 4*9=36, therefore, at the final moment, it cannot be equal to 40.

Consequently, the field will never be completely overgrown with weeds.

16. A convex 2m-gon А 1 …А 2 m is given. A point P is taken inside it, not lying on any of the diagonals. Prove that point Р belongs to an even number of triangles with vertices at points А 1 ,…, А 2 m .

Solution: Diagonals break the polygon into several parts. We will call neighboring those of them that have a common side. It is clear that one can get from any interior point of the polygon to any other point, passing each time only from the neighboring part to the neighboring one. The part of the plane that lies outside the polygon can also be considered one of these parts. The number of triangles under consideration for the points of this part is equal to zero, so it suffices to prove that when passing from a neighboring part to a neighboring part, the parity of the number of triangles is preserved.

Let the common side of two neighboring parts lie on the diagonal (or side) PQ. Then to all the considered triangles, except for triangles with side PQ, both of these parts either belong or do not belong. Therefore, when moving from one part to another, the number of triangles changes by k 1 -k 2 , where k 1 is the number of polygon vertices lying on one side of PQ. Since k 1 +k 2 =2m-2, then the number k 1 -k 2 is even.

4. Auxiliary coloring in checkerboard pattern

17. There is a beetle in each square of the 5 x 5 board. At some point, all the beetles crawl onto adjacent (horizontally or vertically) cells. Does this necessarily leave an empty cell?

Solution: Since the total number of cells on a 5 x 5 chessboard is odd, there cannot be equal numbers of black and white cells. Let there be more black cells for definiteness. Then there are fewer beetles sitting on white cells than black cells. Therefore, at least one of the black cells remains empty, since only beetles sitting on white cells crawl onto the black cells.

19. Prove that a board of 10 x 10 squares cannot be cut into T-shaped figures consisting of four squares.

Solution: Suppose a board of 10 x 10 squares is divided into such figures. Each figure contains either 1 or 3 black cells, i.e. always an odd number. The figures themselves should be 100/4 = 25 pieces. Therefore, they contain an odd number of black cells, and there are 100/2=50 black cells in total. A contradiction has been obtained.

5. Problems about coloring

20. The plane is painted in two colors. Prove that there are two points of the same color, the distance between which is exactly 1.

Solution: Consider a regular triangle with side 1.

transcript

1 M. A. Ekimova, G. P. Kukin MTsNMO Moscow, 2002

2 UDC BBK E45 E45 Ekimova M. A., Kukin G. P. Cutting problems. M.: MTsNMO, p.: ill. Series: "Secrets of teaching mathematics". This book is the first book in the Secrets of Teaching Mathematics series, designed to present and summarize the accumulated experience in the field of mathematics education. This collection is one of the parts of the course "Developing logic in grades 5-7". To all the problems given in the book, solutions or instructions are given. The book is recommended for extracurricular work in mathematics. BBK ISBN c Kukin G. P., Ekimova M. A., c MTsNMO, 2002.

3 Introduction At present, the traditional view of the composition of subjects studied by schoolchildren is being revised and refined. Various new subjects are introduced into the school curriculum. One of these subjects is logic. The study of logic contributes to the understanding of the beauty and elegance of reasoning, the ability to reason, the creative development of the individual, the aesthetic education of a person. Every cultured person should be familiar with logical problems, puzzles, games that have been known for several centuries or even millennia in many countries of the world. The development of ingenuity, ingenuity and independence of thinking is necessary for any person if he wants to succeed and achieve harmony in life. Our experience shows that the systematic study of formal logic or fragments of mathematical logic should be postponed to the upper grades of secondary school. At the same time, it is necessary to develop logical thinking as early as possible. In fact, when studying school subjects, reasoning and proof appear only in the 7th grade (when the systematic geometry course begins). For many students, the abrupt transition (there was no reasoning became a lot of reasoning) is unbearably difficult. In the course of developing logic for grades 5-7, it is quite possible to teach schoolchildren to reason, prove, and find patterns. For example, when solving mathematical puzzles, one must not only guess (pick up) several answers, but also prove that a complete list of possible answers has been obtained. It's pretty good for a 5th grader. But in the process of teaching logic in grades 5-7 of secondary schools, teachers face certain difficulties: the lack of textbooks, didactic materials, manuals, and visual materials. All this has to be compiled, written and drawn by the teacher himself. One of the goals of this collection is to make it easier for the teacher to prepare and conduct classes. We will give some recommendations for conducting lessons before working with the collection.

4 4 Introduction It is desirable to start teaching logic to schoolchildren from the fifth grade, and maybe even earlier. Logic should be taught in a relaxed, almost improvisational style. This apparent lightness actually requires a lot of serious preparation from the teacher. It is unacceptable, for example, to proofread an interesting and entertaining problem from a thick handwritten notebook, as teachers sometimes do. We recommend that you conduct classes in a non-standard form. It is necessary to use as much visual material as possible in the lessons: various cards, pictures, sets of figures, illustrations for solving problems, diagrams. You should not deal with younger students on the same topic for a long time. When analyzing a topic, you should try to highlight the main logical milestones and achieve an understanding (and not memorization) of these points. It is necessary to constantly return to the material covered. This can be done at independent work, team competitions (during lessons), tests at the end of the quarter, oral and written olympiads, matboys (out of school hours). It is also necessary to use entertaining and comic tasks in the classroom, sometimes it is useful to change the direction of activity. This collection is one of the parts of the course "Developing logic in grades 5-7" "Problems for cutting". This part was tested at the lessons of logic in grades 5-7 of the lyceum school 74 in Omsk. Many scientists have been fond of cutting problems since ancient times. Solutions to many simple cutting problems were found by the ancient Greeks and Chinese, but the first systematic treatise on this topic was written by Abul-Vef, the famous Persian astronomer of the 10th century, who lived in Baghdad. Geometers seriously engaged in solving problems of cutting figures into the smallest number of parts and then composing one or another new figure from them only at the beginning of the 20th century. One of the founders of this fascinating branch of geometry was the famous puzzle compiler Henry

5 Introduction 5 E. Dudeni. A particularly large number of pre-existing figures cutting records were broken by an expert at the Australian Patent Office, Harry Lindgren. He is a leading figure cutter. Today, puzzle lovers are fond of solving cutting problems, primarily because there is no universal method for solving such problems, and everyone who takes on their solution can fully demonstrate their ingenuity, intuition and ability to think creatively. Since no deep knowledge of geometry is required here, amateurs can sometimes even outperform professional mathematicians. At the same time, slicing problems are not frivolous or useless, they are not far from serious mathematical problems. From cutting problems, the Boyai-Gervin theorem was born that any two equal-sized polygons are equally composed (the converse is obvious), and then Hilbert's third problem: is a similar statement true for polyhedra? Cutting tasks help schoolchildren to form geometric representations as early as possible on a variety of materials. When solving such problems, there is a feeling of beauty, law and order in nature. The collection "Problems for cutting" is divided into two sections. When solving problems from the first section, students will not need knowledge of the basics of planimetry, but they will need ingenuity, geometric imagination and fairly simple geometric information that is known to everyone. The second section is optional tasks. These included tasks, the solution of which will require knowledge of basic geometric information about figures, their properties and features, knowledge of some theorems. Each section is divided into paragraphs, in which we tried to combine tasks on one topic, and they, in turn, are divided into lessons containing each homogeneous tasks in order of increasing difficulty. The first section contains eight paragraphs. 1. Tasks on checkered paper. This section contains problems in which the cutting of figures (mainly squares and rectangles) goes along the sides of the cells. The paragraph contains 4 lessons, we recommend them for studying by students of the 5th grade.

6 6 Introduction 2. Pentomino. This paragraph contains tasks related to pentomino figures, so for these lessons it is advisable to distribute sets of these figures to the children. There are two lessons here, we recommend them for studying by students of 5-6th grades. 3. Difficult cutting tasks. Here are collected tasks for cutting shapes of a more complex shape, for example, with borders that are arcs, and more complex tasks for cutting. There are two lessons in this paragraph, we recommend that they be taught in the 7th grade. 4. Splitting the plane. Here are collected problems in which you need to find solid partitions of rectangles into rectangular tiles, problems for compiling parquets, problems for the most dense packing of shapes in a rectangle or square. We recommend that you study this paragraph in grades 6-7. 5. Tangram. Here are collected tasks related to the ancient Chinese puzzle "Tangram". For this lesson, it is desirable to have this puzzle, at least made of cardboard. This section is recommended for studying in the 5th grade. 6. Problems for cutting in space. Here, students are introduced to the development of a cube, a triangular pyramid, parallels are drawn and differences are shown between figures on a plane and three-dimensional bodies, which means differences in solving problems. The paragraph contains one lesson, which we recommend for studying by students of the 6th grade. 7. Tasks for coloring. It shows how coloring a shape helps solve a problem. It is not difficult to prove that the solution of the problem of cutting some figure into parts is possible, it is enough to provide some way of cutting. But to prove that cutting is impossible is more difficult. Coloring the figure helps us to do this. There are three lessons in this paragraph. They are recommended for study by students of the 7th grade. 8. Tasks with coloring in the condition. Here are collected tasks in which you need to color a figure in a certain way, answer the question: how many colors are needed for such a coloring (the smallest or largest number), etc. There are seven lessons in the paragraph. We recommend them for studying by students of the 7th grade. The second section includes tasks that can be solved in additional classes. It contains three paragraphs.

7 Introduction 7 9. Transformation of figures. It contains tasks in which one figure is cut into parts from which another figure is composed. There are three lessons in this paragraph, the first one deals with the "transformation" of various figures (quite easy tasks are collected here), and the second lesson deals with the geometry of the transformation of a square. 10. Different tasks for cutting. This includes various cutting tasks that are solved by various methods. There are three lessons in this section. 11. The area of ​​figures. There are two lessons in this section. In the first lesson, problems are considered, in the solution of which it is necessary to cut the figures into parts, and then prove that the figures are equally composed, in the second lesson, problems in the solution of which it is necessary to use the properties of the areas of the figures.

8 Section 1 1. Tasks on checkered paper Lesson 1.1 Topic: Tasks for cutting on checkered paper. Purpose: To develop combinatorial skills (to consider various ways of constructing a cut line of figures, the rules that allow not to lose solutions when constructing this line), to develop ideas about symmetry. We solve problems in the lesson, problem 1.5 for the house The square contains 16 cells. Divide the square into two equal parts so that the cut line runs along the sides of the cells. (The ways of cutting a square into two parts will be considered different if the parts of the square obtained with one method of cutting are not equal to the parts obtained with another method.) How many solutions does the problem have? Instruction. Finding several solutions to this problem is not so difficult. On fig. 1, some of them are shown, and solutions b) and c) are the same, since the figures obtained in them can be combined by superposition (if you rotate the square c) by 90 degrees). Rice. 1 But finding all the solutions and not losing any solution is already more difficult. Note that the broken line dividing the square into two equal parts is symmetrical with respect to the center of the square. This observation allows us to step

9 Lesson by step to draw a polyline from two ends. For example, if the beginning of the polyline is at point A, then its end will be at point B (Fig. 2). Make sure that for this problem, the beginning and end of the polyline can be drawn in two ways, shown in Fig. 2. When constructing a broken line, in order not to lose any solution, you can follow this rule. If the next link of the polyline can be drawn in two ways, then first you need to prepare a second similar drawing and perform this step on one drawing in the first way, and on the other in the second way (Fig. 3 shows two continuations of Fig. 2 (a)). Similarly, you need to act when there are not two, but three methods (Fig. 4 shows three continuations of Fig. 2 (b)). The specified procedure helps to find all solutions. Rice. 2 Fig. 3 Rice Rectangle 3 4 contains 12 cells. Find five ways to cut a rectangle into two equal parts so that the cut line goes along the sides of the cells (cutting methods are considered different if the parts obtained with one method of cutting are not equal to the parts obtained with another method) Rectangle 3 5 contains 15 cells and a central cell removed. Find five ways to cut the remaining figure

10 10 1. Tasks on checkered paper are divided into two equal parts so that the cut line goes along the sides of the cells. Square 6 6 is divided into 36 identical squares. Find five ways to cut a square into two equal parts so that the cut line goes along the sides of the squares Problem 1.4 has over 200 solutions. Find at least 15 of them. Lesson 1.2 Topic: Problems for cutting on checkered paper. Purpose: To continue to develop ideas about symmetry, preparation for the topic "Pentamino" (consideration of various figures that can be built from five cells). Problems Can a square of 5 5 cells be cut into two equal parts so that the cut line goes along the sides of the cells? Justify your answer Divide square 4 4 ​​into four equal parts so that the cut line goes along the sides of the cells. How many different ways of cutting can you find? 1.8. Divide the figure (Fig. 5) into three equal parts so that the cut line runs along the sides of the squares. Rice. 5 Fig. Fig. 6 Divide the figure (Fig. 6) into four equal parts so that the cut line goes along the sides of the squares Divide the figure (Fig. 7) into four equal parts so that the cut lines go along the sides of the squares. Find as many solutions as possible.

11 Lesson Divide a square of 5 5 cells with a cut out central cell into four equal parts. Lesson 1.3 Topic: Problems for cutting on checkered paper. Purpose: To continue to develop ideas about symmetry (axial, central). Tasks Cut the shapes shown in fig. 8, into two equal parts along the grid lines, and in each of the parts there should be a circle. Rice. 8 Figure The figures shown in fig. 9, it is necessary to cut along the grid lines into four equal parts so that there is a circle in each part. How to do it? Cut the figure shown in Fig. 10, along the grid lines into four equal parts and fold them into a square so that the circles and stars are arranged symmetrically about all the axes of symmetry of the square. Rice. 10

12 12 1. Tasks on checkered paper Cut this square (Fig. 11) along the sides of the cells so that all parts are of the same size and shape and each contains one circle and an asterisk. 12 into four identical parts so that each of them contains three filled cells. Lesson 1.4 11 Fig. 12 Topic: Problems for cutting on checkered paper. Goal: Learn to cut a rectangle into two equal parts, from which you can add a square, another rectangle. Learn to determine from which rectangles, cutting them, you can make a square. Tasks Additional tasks 1.23, 1.24 (these tasks can be considered at the beginning of the lesson for warm-up) Cut the rectangle 4 9 cells along the sides of the cells into two equal parts so that they can then be folded into a square Can a rectangle 4 8 cells be cut into two parts along sides of the cells so that they can form a square? From a rectangle of 10 7 cells, a rectangle of 1 6 cells was cut out, as shown in Fig. 13. Cut the resulting figure into two parts so that they can be folded into a square. Filled figures were cut out from a rectangle of 8 9 cells, as shown in fig. 14. Cut the resulting figure into two equal parts so that you can add a rectangle of 6 10 from them.

13 Lesson Fig. 13 Rice A 5 5 cell square is drawn on checkered paper. Show how to cut it along the sides of the cells into 7 different rectangles Cut the square into 5 rectangles along the sides of the cells so that all ten numbers expressing the lengths of the sides of the rectangles are different integers Divide the figures shown in fig. 15, into two equal parts. (You can cut not only along the cell lines, but also along their diagonals.) Fig. 15

14 14 2. Pentomino Cut out the figures shown in fig. 16, into four equal parts. 2. Pentomino Fig. 16 Lesson 2.1 Topic: Pentomino. Purpose: Development of combinatorial skills of students. Tasks Figures of dominoes, trominoes, tetraminoes (a game with such figures is called Tetris), pentominoes are made up of two, three, four, five squares so that any square has a common side with at least one square. From two identical squares, only one domino figure can be made (see Fig. 17). Trimino figures can be obtained from a single domino figure by attaching another square to it in various ways. You will get two tromino figures (Fig. 18). Rice. 17 Rice Make all kinds of tetramino figures (from the Greek word "tetra" four). How many did they get? (Shapes obtained by rotation or symmetrical display from any others are not considered new).

15 Lesson Make all possible figures of pentomino (from the Greek "penta" five). How many did they get? 2.3. Compose the figures shown in Fig. 19, from pentomino figurines. How many solutions does the problem have for each figure? Figure Fold a 3 5 rectangle of pentomino pieces. How many different solutions will you get? 2.5. Compose the figures shown in Fig. 20, from pentomino figurines. Rice. 20

16 16 2. Pentomino Lesson 2.2 Topic: Pentomino. Purpose: Development of ideas about symmetry. Problems In problem 2.2 we made up all possible pentomino pieces. Look at them in fig. 21. Fig. 21 Figure 1 has the following property. If it is cut out of paper and bent along a straight line a (Fig. 22), then one part of the figure will coincide with the other. The figure is said to be symmetrical about the straight a axis of symmetry. Figure 12 also has an axis of symmetry, even two of them are straight lines b and c, while figure 2 has no axes of symmetry. Figure How many axes of symmetry does each pentomino figure have? 2.7. Of all 12 pentomino figures, fold a rectangle. Unsymmetrical pieces are allowed to be turned over. Fold a 6 10 rectangle of twelve pentomino figures, and so that each element touches one side of this rectangle.

Lesson 17 Cut the rectangle shown in fig. 23 (a), along the internal lines into two such parts, from which it is possible to fold a figure with three square holes the size of one cell (Fig. 23 (b)). Fig. From the pentomino figures, fold a square 8 8 with a square 2 2 cut out in the middle. Find several solutions Twelve pentominoes are laid in a rectangle Restore the boundaries of the figures (Fig. 24) if each star falls into exactly one pentomino. Rice. 24 Fig. Twelve pentomino pieces are stacked in a box 12 10, as shown in fig. 25. Try to place another set of pentominoes on the remaining free field.

18 18 3. Difficult cutting problems 3. Difficult cutting problems Lesson 3.1 Topic: Problems for cutting shapes of a more complex shape with boundaries that are arcs. Purpose: To learn how to cut shapes of a more complex shape with borders that are arcs, and make a square out of the resulting parts. Tasks In fig. 26 shows 4 figures. With one cut, divide each of them into two parts and make a square out of them. Checkered paper will make it easier for you to solve the problem. Rice Cutting the square 6 6 into parts, add the figures shown in fig. 27. Fig. 27

19 Lesson 28 shows part of the fortress wall. One of the stones has such a bizarre shape that if you pull it out of the wall and put it differently, the wall will become even. Draw this stone. What will more paint be used for: painting a square or this unusual ring (Fig. 29)? Rice. 28 Rice Cut the vase shown in fig. 30, into three parts, from which a rhombus can be folded. Rice. 30 Fig. 31 Fig. 32 Lesson 3.2 Topic: More complex cutting problems. Objective: To practice solving more complex cutting problems. We solve problems in the lesson, problem 3.12 for home Cut the figure (Fig. 31) with two straight cuts into such parts from which you can add a square 32 figure into four equal parts, from which it would be possible to add a square Cut the letter E, shown in fig. 33, into five parts and fold them into a square. Do not turn parts upside down

20 20 4. Dividing the plane is allowed. Is it possible to get by with four parts, if you allow the parts to be turned upside down? 3.9. The cross, made up of five squares, needs to be cut into such parts, from which it would be possible to make one equal-sized cross (that is, equal in area) square. Two chessboards are given: an ordinary one, in 64 cells, and another in 36 cells. It is required to cut each of them into two parts so that from all the four parts obtained to make a new chessboard of cells. The cabinetmaker has a piece of a chessboard of 7 7 cells made of precious mahogany. He wants without wasting material and swiping Fig. 33 cuts only along the edges of the cells, cut the board into 6 parts so that they make three new squares, all of different sizes. How to do it? Is it possible to solve Problem 3.11 if the number of parts must be 5 and the total length of the cuts is 17? 4. Dividing a plane Lesson 4.1 Topic: Solid partitions of rectangles. Purpose: To learn how to build solid partitions of rectangles with rectangular tiles. Answer the question under what conditions the rectangle admits such a division of the plane. Tasks (a) are solved in the lesson. Tasks 4.5 (b), 4.6, 4.7 can be left at home. Suppose we have an unlimited supply of 2 1 rectangular tiles, and we want to use them to lay out a rectangular floor, and no two tiles should overlap Lay 2 1 tiles on the floor in a 5 6 room. It is clear that if the floor in a rectangular room p q is tiled 2 1, then p q is even (since the area is divisible by 2). And vice versa: if p q is even, then the floor can be laid out with tiles 2 1.

21 Lesson Indeed, in this case one of the numbers p or q must be even. If, for example, p = 2r, then the floor can be laid out as shown in Fig. 34. But in such parquets there are break lines that cross the entire “room” from wall to wall, but do not cross the tiles. But in practice, parquets without such lines are used - solid parquets. Fig Lay out tiles 2 1 solid parquet of the room Try to find a continuous tiling 2 1 a) rectangle 4 6; b) square Lay tiles 2 1 solid parquet a) rooms 5 8; b) rooms 6 8. Naturally, the question arises for which p and q the rectangle p q admits a continuous partition into tiles 2 1? We already know the necessary conditions: 1) p q is divisible by 2, 2) (p, q) (6, 6) and (p, q) (4, 6). One more condition can also be verified: 3) p 5, q 5. It turns out that these three conditions turn out to be sufficient as well. Tiles of other sizes Lay tiles 3 2 without gaps a) rectangle 11 18; b) rectangle Lay out without gaps, if possible, a square with tiles. Is it possible, taking a square of checkered paper measuring 5 5 cells, to cut out 1 cell from it so that the rest can be cut into plates of 1 3 cells? Lesson 4.2 Topic: Parquets.

22 22 4. Dividing the plane Purpose: To learn how to cover the plane with various figures (moreover, parquets can be with break lines or solid), or prove that this is impossible. Problems One of the most important questions in the theory of partitioning a plane is: “What shape should a tile be so that its copies can cover the plane without gaps and double coverings?” Quite a few obvious forms immediately come to mind. It can be proved that there are only three regular polygons that can cover the plane. This is an equilateral triangle, square and hexagon (see Fig. 35). There is an infinite number of irregular polygons that can cover the plane. Fig Divide an arbitrary obtuse triangle into four equal and similar triangles. In problem 4.8 we split the triangle into four equal and similar triangles. Each of the four resulting triangles can in turn be divided into four equal and similar triangles, etc. If we move in the opposite direction, that is, add four equal obtuse triangles so that we get one triangle similar to them, but four times larger , etc., then such triangles can tile the plane. The plane can be covered with other figures, for example, trapezoids, parallelograms. Cover the plane with the same figures shown in fig. 36.

23 Lesson Tile the plane with the same "brackets" shown in fig. 37. Fig. 36 Rice There are four squares with a side of 1, eight with a side of 2, twelve with a side of 3. Can you make one large square out of them? Is it possible to fold a square of any size from the wooden tiles indicated in fig. 38 kinds, using tiles of both kinds? Lesson 4.3 Topic: Problems of the most dense packing. Rice. 38 Purpose: To form the concept of the optimal solution. Tasks What is the largest number of strips of size 1 5 cells can be cut out of a square of checkered paper 8 8 cells? The master has a sheet of tin square. dm. The master wants to cut out as many rectangular blanks of 3 5 square meters from it as possible. dm. Help him. Is it possible to cut the rectangle of the cell into rectangles of size 5 7 without residue? If possible, how? If not, why not? On a sheet of checkered paper, mark the cuts with the size of the cells, with the help of which you can get as many whole figures as shown in Fig. 39. The figures depicted in fig. 39 (b, d), can be turned over.

24 24 5. Tangram Rice Tangram Lesson 5.1 Topic: Tangram. Purpose: To introduce students to the Chinese puzzle "Tangram". Practice geometric research, design. Develop combinatorial skills. Problems Speaking of cutting problems, one cannot fail to mention the ancient Chinese puzzle "Tangram", which arose in China 4 thousand years ago. In China, it is called "chi tao tu", that is, a seven-piece mental puzzle. Guidelines. To conduct this lesson, it is desirable to have handouts: a puzzle (which the students themselves can make), drawings of figures that need to be folded. Figure Make a puzzle yourself: transfer a square divided into seven parts (Fig. 40) onto thick paper and cut it Using all seven parts of the puzzle, make the figures shown in fig. 41.

25 Lesson Fig. 41 Fig. 42 Guidelines. Children can be given drawings of figures a), b) in full size. And so the student can solve the problem by putting parts of the puzzles on the drawing of the figure and thereby selecting the right parts, which simplifies the task. And figure drawings

26 26 6. Problems for cutting in space c), d) can be given on a smaller scale; consequently, these tasks will be more difficult to solve. On fig. 42 more figures for self-compilation are given. Try to come up with your own figure using all seven parts of the tangram In the tangram, among its seven parts, there are already triangles of different sizes. But from its parts you can still add various triangles. Fold a triangle using four parts of a tangram: a) one large triangle, two small triangles and a square; b) one large triangle, two small triangles and a parallelogram; c) one large triangle, one middle triangle and two small triangles Can you make a triangle using only two parts of a tangram? Three parts? Five parts? Six parts? All seven parts of the tangram? 5.6. It is obvious that a square is made from all seven parts of the tangram. Is it possible or impossible to make a square of two parts? Out of three? Out of four? 5.7. What different parts of a tangram can be used to make a rectangle? What other convex polygons can be made? 6. Problems for cutting in space Lesson 6.1 Topic: Problems for cutting in space. Purpose: To develop spatial imagination. Learn to build a sweep of a triangular pyramid, a cube, determine which sweeps are incorrect. Practice solving problems for cutting bodies in space (the solution of such problems differs from solving problems for cutting shapes on a plane). Tasks Pinocchio had paper, on one side pasted over with polyethylene. He made the piece shown in Fig. 43 to glue milk bags (triangular pyramids) out of it. And the fox Alice can make another blank. What?

27 Lesson Rice Cat Basilio also got this paper, but he wants to glue cubes (kefir bags). He made the blanks shown in Fig. 44. And the fox Alice says that some can be thrown away right away, because they are not good. Is she right? The Pyramid of Cheops has a square at the base, and its side faces are equal isosceles triangles. Pinocchio climbed up and measured the angle of the edge at the top (AMD, in Fig. 45). It turned out 100. And the fox Alice says that he overheated in the sun, because this cannot be. Is she right? 6.4. What is the minimum number of flat cuts needed to divide a cube into 64 small cubes? After each cut, it is allowed to shift the parts of the cube as you like. The wooden cube was painted on the outside with white paint, then each of its edges Fig. 45 was divided into 5 equal parts, after which it was sawn so that small cubes were obtained, in which the edge is 5 times smaller than that of the original cube. How many small cubes are there? How many cubes have three sides painted? Two edges? One edge? How many unpainted cubes are left? 6.6. The watermelon was cut into 4 pieces and eaten. It turned out 5 crusts. Could this be?

28 28 7. Tasks for coloring 6.7. What is the maximum number of pieces that a pancake can be cut into with three straight cuts? How many pieces can be obtained with three cuts of a loaf of bread? 7. Tasks for coloring Lesson 7.1 Topic: Coloring helps to solve problems. Purpose: To learn how to prove that some cutting problems do not have solutions using a well-chosen coloring (for example, coloring in a checkerboard pattern), thereby improving the logical culture of students. Problems It is not difficult to prove that the solution of the problem of cutting some figure into parts is possible: it is enough to provide some method of cutting. Finding all solutions, that is, all ways of cutting, is already more difficult. And to prove that cutting is impossible is also quite difficult. In some cases, the coloring of the figure helps us to do this. We took a square of checkered paper measuring 8 8, cut off two cells from it (lower left and upper right). Is it possible to completely cover the resulting figure with "dominoes" rectangles 1 2? 7.2. On the chessboard there is a “camel” figure, which with each move moves three cells vertically and one horizontally, or three horizontally and one vertically. Can a "camel" after making several moves get into a cell adjacent to its original side? 7.3. There is a beetle in each cell of the 5 5 square. On command, each beetle crawled onto one of the adjacent cells on the side. Can it then turn out that exactly one beetle will again sit in each cell? What if the original square had dimensions 6 6? 7.4. Is it possible to cut a square of 4x4 checkered paper into one pedestal, one square, one column and one zigzag (Fig. 46)?

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1. A figure was drawn on checkered paper. Divide it into 4 equal
parts along the lines of checkered paper. Find all possible figures for which
you can cut this figure according to the condition of the problem.
Solution.
2. From the square 5 5 cut out the central cell. Cut the resulting
figure into two equal parts in two ways.
Solution.

3. Divide the 3×4 rectangle into two equal parts. Find how you can
more ways. You can cut only along the side of a 1 × 1 square, and methods
are considered different if the resulting figures are not equal for each
way.
Solution.
4. Cut the figure shown in the figure into 2 equal parts.
Solution.
5. Cut the figure shown in the figure into 2 equal parts.

Solution.
6. Cut the figure shown in the figure into two equal parts along
grid lines, and in each of the parts there should be a circle.
Solution.
7. Cut the figure shown in the figure into four equal parts

Solution.

8. Cut the figure shown in the figure into four equal parts
along the grid lines, and in each of the parts there should be a circle.
Solution.
9. Cut this square along the sides of the cells so that all parts
be of the same size and shape, and that each contain one
mug and cross.
Solution.

10. Cut the figure shown in the figure along the grid lines into
four equal parts and fold them into a square so that the circles and crosses
located symmetrically about all axes of symmetry of the square.
Solution.
11. Cut the square 6  6 cells shown in the figure into four
identical parts so that each of them contains three filled cells.

Solution.
12. Is it possible to cut a square into four parts so that each part
was in contact with the other three (parts are in contact if they have a common
border area)?
Solution.
13. Is it possible to cut a rectangle of 9 4 cells into two equal parts along

then how to do it?
Solution. The area of ​​such a square is 36 cells, that is, its side is 6
cells. The cutting method is shown in the figure.

14. Is it possible to cut a rectangle of 5 10 cells into two equal parts along
the sides of the cells so that they could form a square? If yes,
then how to do it?
Solution. The area of ​​such a square is 50 cells, that is, its side is
more than 7, but less than 8 whole cells. So, to cut such a rectangle
in the required way on the sides of the cells is impossible.
15. There were 9 sheets of paper. Some of them were cut into three parts. Total
became 15 sheets. How many sheets of paper were cut?
Solution. Cut 3 sheets: 3 ∙ 3 + 6 = 15.