Children's mathematical savings. Research work "Mathematical seam"

Chapter Six
Domino and Kubic
A. Domino
197. How many points?
198. Two focus
199. Winning party provided
200. Frame
201. Frame in the frame
202. "Winds"
203. Magic Squares from Domino Bones
204. Magic Square with Hole
205. Multiplication in Domino
206. Guess the intended domino bone
B. Kubik
207. Arithmetic focus with playing cubes
208. Gaying out the amount of glasses on hidden edges
209. In what order are the cubes are located?

Head seventh
Properties of nine
210. What figure is crossed out?
211. Hidden property
212. A few fun ways to find a missing number
213. According to one digit of the result, determine the remaining three
214. Gaying the difference
215. Definition of age
216. What is the secret?

Chapter Eighth
With algebra and without it
217. Mutual assistance
218. Loafer, and damn
219. Criminal baby
220. Hunters
221. Counter trains
222. Vera Prints Manuscript
223. History with mushrooms
224. Who will come back before?
225. Swimmer and hat
226. Two shipping
227. Check your seamless!
228. Confuez prevented
229. How many times more?
230. Ship and seaplane
231. Velofigurists in the arena
232. Tokary speed of Bykova
233. Jack London's trip
234. Due to unsuccessful analogies, errors are possible
235. Legal incident
236. Couples and Things
237. Who rode a horse?
238. Two motorcyclist
239. In what aircraft Volodin Dad?
240. Varnish on parts
241. Two candles
242. Amazing insight
243. "Ordinary time"
244. Hours
245. In which hours?
246. What time began and the meeting ended?
247. Sergeant trains scouts
248. By two messages
249. How many new stations have built?
250. Choose four words
251. Is such weighing?
252. Elephant and Komar
253. Five-digit number
254. years to a hundred growing you without old age
255. Luke task
256. A kind of walk
257. One property of simple frains

Ninth chapter
Mathematics almost without computing
258. In a dark room
259. Apples
260. Weather forecast (joke).
261. Forest Day
262. Who has any name?
263. Competition
264. Purchase
265. Passengers of one coupe
266. Final of the Tournament of Chessters Soviet army
267. Resurrection
268. How is the surname of the driver?
269. Coal story
270. Herbs collectors
271. Hidden division
272. Encrypted actions (numerical rebuses)
273. Arithmetic Mosaic
274. Motorcyclist and Horse
275. Walking and by car
276. "From the opposite"
277. Detect a fake coin
278. Logic draw
279. Three wise men
280. Five questions for schoolchildren
281. reasoning instead of equation
282. For common sense
283. Yes, or not?

Chapter Tenth
Mathematical games and tricks
A. Games
284. Eleven items
285. Take matches last
286. Wins Chet.
287. Jiangsitse
288. How to win?
289. Put the square
290. Who will first say "hundred"?
291. Squares game
292. OUA
293. "Mathematics" (Italian game)
294. Game in Magic Squares
295. Intersection of numbers
B. Focuses
296. Guessing the intended number (7 focus)
297. Guess the result of calculations, not asking anything
298. Who took some time, and found out
299. One, two, three attempts ... and I guess
300. Who took a gum, and who is a pencil?
301. Guessing the three conceived terms and sums
302. Guessing a few conceived numbers
303. How old are you?
304. Guess age
305. Geometric focus (mysterious disappearance)

Chapter eleventh
Dividitude of numbers
306. Number on the tomb
307. Gifts for the new year
308. Can there be such a number?
309. Basket of eggs (from the ancient French problem book)
310. Three-digit number
311. Four shipping
312. Cashira error
313. Numeric Rus
314. Sign of divisibility on 11
315. Joint Sign of Destinations at 7, 11 and 13
316. Simplification of the sign of divisibility on 8
317. The striking memory
318. Jointed Signal of Dividivity by 3, 7 and 19
319. Delicious of bicon
320. Old and new about divisibility on 7
321. Distribution of a sign on other numbers
322. Generalized sign of divisibility
323. Curious divisibility

Chapter twelve
Cross-sum and magic squares
A. Cross-Am
324. Interesting groupings
325. "Star"
326. "Crystal"
327. Decoration for showcase
328. Who will be able to?
329. "Planetarium"
330. "Ornament"
B. Magic squares
331. Aliens from China and India
332. How to make a magic square yourself?
333. On the approaches to the general methods
334. EXAM EMP
335. "Magic" game in "15"
336. Nearby Magic Square
337. What in the central cell?
338. "Magic" works
339. "Casket" arithmetic curiosities
B. Elements of the theory of magical squares
340. "In addition"
341. "Right" Magic Squares of Fourth Arrangements
342. Selection of numbers for the magical squares of any order

Chapter thirteenth.
Curious and serious in numbers
343. Ten digits (observations).
344. A few more advanced observations
345. Two interesting experience
346. Number carousel
347. Instant multiplication disk
348. Mental gymnastics
349. Patterns numbers
350. One for everyone and all for one
351. Numerical finds
352. Watching a number of natural numbers
353. Interesting difference
354. Symmetric amount (inconspicuous nuts)

Chapter Fourteenth
Ancient numbers, but forever young
A. The initial numbers
355. Numbers are simple and composite
356. "Eratosthenovo Decorating"
357. New "Detection" for prime numbers
358. Fifty first simple numbers
359. Another way to get simple numbers
360. How many simple numbers?
B. Fibonacci numbers
361. Public testing
362. A number of Fibonacci
363. Paradox
364. Properties of numbers of a number of Fibonacci
B. Figure numbers
365. Properties of curious numbers
366. Pythagoras numbers

Fifteenth chapter
Geometric Cooker In Labor
367. Geometry Seva
368. Rationalization in laying bricks for transportation
369. Working geometers

Municipal budgetary educational institution

Saranpaul Secondary School

Research work in mathematics

Prepared:

student 3- A class of Frolov Nikolai,

Leader:

Arteev Antonina Andreevna,

primary school teacher.

Saranspaul, 2017

Content

P.

Introduction

The value of tasks to the smell

Leonardo Fibonacci - mathematician who has contributed to the solution of tasks to the smell

Classification of tasks for "Rezkalka"

Logic tasks

Tasks for crossing

Tasks for transfusion

Task of fabulous character

Objective tasks, on a mixture

Numeric rows, rebuses

Conclusion

Bibliography

Introduction

Creative activity is the most powerful impetus in the development of the child. Potential genius lives in each person, but not always a person feels the presence of genius. It is necessary to start developing creative abilities as early as possible.

Any mathematical task for a smelter, for what age it is meant, carries a certain mental load, which is most often disguised as an entertaining plot, external data, the condition of the task, etc. In the tasks of different degree of complexity, the input attracts the attention of children, activates the thought , causes steady interest in the upcoming solution to the solution. The nature of the material is determined by its purpose: developing common mental and mathematical abilities in children, to engage in the subject of mathematics, to entertain that is not definitely the main one.Development, resourcefulness, initiative is carried out in active mental activity based on direct interest.

Interesting mathematical material attached game elementscontained in each task, logical exercise, entertainment, be it chess or the most elementary puzzle. For example, in the question: "How to fold the square on the table with two sticks?" - The unusualness of its production makes thinking in search of a response, get into the game of imagination.

The manifold of entertaining material - games, tasks, puzzles, gives the basis for their classification, although it is quite difficult to smash such a diverse material created by mathematics into groups.

It is possible to classify it in different signs: on the content and value, the nature of the mental operations, as well as the sign of generality, the focus on the development of certain skills. The basis for the allocation of such groups is the nature and purpose of the material of a particular type.

Purpose: Studying methods for solving problems in the smell.

Tasks:

1. To explore the topic "Solving tasks for a mixture", the types of tasks on the smelting and the methods of solving them.

2. Solve several types of tasks for the smelting, independently make an algorithm for solving such tasks.

The value of tasks to the smell

Creative activity of students in the process of studying mathematics is primarily in solving problems. The ability to solve problems is one of the level criteria mathematical Development Pupils, characterizes primarily the ability of students to apply their theoretical knowledge in a specific situation.

When solving traditional school tasks, certain knowledge, skills and skills in the narrow circle of software issues are used to solve them. At the same time, the well-known solutions limits the creative search for students.

The task of the smelter in contrast to the traditional cannot be directly solved according to any law. Tasks for smelting These are those for whom in the course of mathematics does not have the general rules and provisions that determine the exact program of their solution. Therefore, the need to find a solution arises, which requires creative work of thinking and contributes to its development.

The solution of tasks to the smelter generates the stroke of the search and the joy of discovery - the most important factors of development, creative achievement.

The value of tasks to the smelter is very large - the ability of students to solve non-standard tasks shows:

1. ​ The ability to think original, and is also of great importance in the formation and development of them creative abilities;

2. ​ The ability to summarize mathematical material, to identify the main thing, be distracted from insignificant, to see common in externally different;

3. ​ The ability to operate the numerical and sign symbolism;

4. ​ The ability to "serial, logical reasoning" associated with the need for evidence, justification, conclusions;

5. ​ The ability to reduce the process of reasoning, to think over the coolest structures;

6. ​ The ability to reversibility of the thought process (to the transition from direct for the opposite thoughts);

7. ​ The flexibility of thinking, the ability to switch from one mental operation to another, freedom from the processing influence of templates and stencils. This feature of thinking is important in the creative work of mathematicians;

8. ​ The ability to develop mathematical memory ... This is a memory for generalization, logic schemes;

9. Ability to spatial representations.

Another K.D. Shushinsky wrote that "... learning, devoid of all interest and taken only by force of coercion ... Kills in the student a hunt for teaching, without which he will not leave."

Interest is a powerful activity intensity, under his influence, all mental processes proceed especially intensively, and the activity becomes exciting and productive. His essence consists in the desire of a schoolboy to penetrate the learned area more deeply and thoroughly, in constant motivation to engage in the subject of his interest.

From the history of the appearance of tasks for a mixture

It is not surprising that the tasks for the smelter became entertainment "for all times and peoples."The first, which came to us the textbook of mathematics, more precisely, his ku​ juice of 5 meters long, known in the world as "London Papyrus", or "Papirus Akhmes", contains 84 accompanied by the solution of the problem. There were classes in the school of state scribes. Already ancient Egyptians understood how important role in the process of​ the element is played by the element of entertainment, and among the included in "​ rus Akhmes "tasks were a lot of such. So, for millennia from one collection​ nick entertaining tasks of mathematics to another wagging "The challenge​ c cats "from this papyrus. Despite the existence of thirteentomatic "started" Euclid (III century. BC), which became more than two millennia with a sample of scientific rigor, and in Ancient Greece An entertaining element in mathematics did not disappear and is most brightly represented in the "arithmetic" of Diophanta Alexandria (probably III century). In the Middle Ages, the Italians Leonardo (Fibonacci) from Pisa (XIII century) and Niccolo Tartalia (XVI century) were left in solving problems on the smelter.

Collections mathematical entertainmentsimilar to modern, began to appear from the XVII century. Among them were especially popular with "pleasant and entertaining tasks considered in the numbers" Mathematics and the poet Gaspara Claude Bashe Sier de Mesiriak and "Mathematical and physical entertainment" of another French mathematics and writer Jacques Ozanama.

In the XIX century French mathematician, a specialist in the theory of numbers Eduard Luke published four-volume work on entertaining mathematics, which became classic. At the turn of the XIX and XX centuries. A great contribution to the treasury of entertaining mathematics was made by outstanding inventors of games and puzzles - Talented self-taught us America Sam Loyd and Englishman Henry Ernest Diaudeni. Entertaining mathematics The second half of the XX century. You can not submit without a whole series of wonderful books belonging to Peru of the famous American mathematics Martin Gardner. It is his diverse mathematical essays that harmoniously combining scientific depths and the ability to entertain, millions of people around the world (including me) have come to accurate sciences and, of course, to entertaining mathematics.

In Russia, such collections of tasks as "arithmetic" L. F. Magitsky, "in the kingdom of the Smekalki" E. I. Ignatiev, "Living Mathematics", "Entertaining Arithmetic", "Entertaining Algebra" and "Entertaining Geometry" I. I. Perelman and "Mathematical Mathematical Mathematics" B. A. Kordemsky

Leonardo Fibonacci - Mathematician who has contributed to the solution of tasks to the smell.

Leonardo Fibonacci Born and lived in Italy in the city of Pisa in 12-13 centuries. His father was a merchant, and therefore the young Leonardo traveled a lot. In the east, he met the Arabic system of numbers; Subsequently, he analyzed, described and presented it with European society in his famous book "Liber Abaci. » (« Book invoice "). Recall that in Europe at that time, Roman numbers were used, which were terribly inconvenient to operate both with complex mathematical and physical computing and when working with and accounting.

Leonardo Fibonacci presented to Europe Arabic figures which enjoys almost all Western world to this day.The transition from the Roman system to Arabic produced a revolution in mathematics and other sciences , closely with it connected.

It is difficult to imagine what the world would be, if then, in the 13th century, Fibonacci would not publish their book and did not outlined the Europeans to Europeans. Interestingly, we use the Arabic figures without thinking, perceiving them as granted. But if it were not for Leonardo Fibonacci, who knows how to develop a course of history. After all, the representation of I.the treatise of Arab numbers significantly changed the medieval mathematics in best side; He advanced it forward, and with it and other sciences, such as physics, mechanics, electronics, etc. Notice, because these sciences are progress forward. That is why, in many respects, the course of history,the development of European civilization and science as a whole is obliged to Leonard Fibonacci .

A number of Fibonacci numbers

The second outstanding merit Leonardo Fibonacci isa number of Fibonacci numbers . It is believed that this series was known in the East, but it was Leonardo Fibonacci that published this number of numbers in the above-mentioned book "Liber Abaci" (he did it to demonstrate the reproduction of the rabbits population).

Subsequently, it turned out thatthis sequence of numbers is important. not only in mathematics, economy, and finance, but also in Botanic, zoology, physiology, medicine, art, as well as philosophy, aesthetics and many other things. Because Civilization This number of numbers became known from Leonardo Fibonacci, and called him, "Fibonacci row " or "Fibonacci numbers ».

Formula and example of a number of Fibonacci numbers

In Fibonacci sequenceeach element starting from the third is the sum of the two previous elements. , despite the fact that the row begins with numbers 0 and 1. It turns out: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025 …

Fibonacci is a legendary personality in mathematics, economics and finance ; He made public Arab numbers and presented a magical number of numbers.

The task is invented by the Italian scientist Fibonacci, who lived in the 13th century.
"Someone acquired a couple of rabbits and put them in the pen fenced on all sides. How many rabbits will be in a year, if we assume that each month the couple gives a new pair of rabbits as the selection, which from the second month of life also begin to bring the rating? "

Answer: 377 pairs. In the first month of rabbits, 2 pairs will be: 1 original pair, which has given the rating, and 1 born steam. In the second month of rabbits there will be 3 pairs: 1 initial, again granted the rats, 1 growing and 1 born. In the third month - 5 pairs: 2 pairs, giving outlits, 1 growing and 2 born. In the fourth month - 8 pairs: 3 pairs, giving outlits, 2 growing couples, 3 born pairs. Continuing consideration by months, it is possible to establish a link between the amounts of rabbits in the current month and two previous ones. If you designate the number of pairs through N, and through M - the sequence number of the month, then n m. \u003d N. m-1. + N. m-2 . With the help of this expression, the number of rabbits for months of the year is calculated: 2, 3, 5, 8, 13, 21, 34.55, 89, 144, 233, 377.

Classification of tasks for smelting

Tasks on weighing and transfusion

In such tasks from decisive, it is required for a limited number of weighing to localize the object differing from the rest of the weems by weight. Also in this heading, transfusion challenges are considered, in which it is necessary to obtain a certain amount of fluid using the container capacity.

Finding too much

Requires the ability to combine object groups by specific features.

Text tasks for calculations

Simple vital processes, ability to apply mathematical knowledge in life.

Tasks for finding logical errors, tasks with trick

Develop the valuable and very necessary quality of a successful person - critical thinking. Learning to analyze the condition. Sometimes the answer is contained in the task itself.

Task on the properties of numbers and operations with them

The property of even and odd numbers, the correct layout of the brackets, the alignment of numbers among the number corresponding to certain conditions. Dividitude of numbers. Operations on numbers.

Cryptarifami

Mathematical rebus, in which an example is encrypted for performing one of the arithmetic action. At the same time, the same numbers are encrypted by the same letter, and different letters correspond to different numbers.

Tasks for logic and reasoning

Tasks directly related to calculations, but actively developing thinking.

About the time

Calculate the date using the prompts, recall the pattern of work hours or determine someone's age only by hints.

On the sequence of numbers

In these tasks, it is necessary to solve the principle on which a certain sequence is specified and continued.

Tasks with matches

Making manipulations on matches, it is necessary to achieve the desired result. Most of these tasks refers to the number of "non-standard", requiring the skill to "assess the situation with unexpected for most of the point of view or see the possibility of using non-obvious data."

Rebuses

The game in which words, phrases or whole statements are encrypted with drawings in combination with letters and signs.

Chess

As a rule, each course of the course includes several classes (minimum 2) in chess. Main figures. Learning to build effective strategies, think, make weighted and rational solutions

Logic tasks

When solving logical tasks to a mutually unambiguous compliance, it is convenient to write data to the table, where at the intersection of the line and column put the "+" or sign "-".

1. Five classmates - Irena, Timur, Camilla, Eldar and the winners of the Olympiad of schoolchildren in physics, mathematics, computer science, literature and geography. It is known that

The winner of the Olympiad on computer science teaches Irene and Timur to work on a computer;

Camilla and Eldar also became interested in computer science;

Timur has always been afraid of physics;

Camilla, Timur and the winner of the Olympiad on the literature are swimming;

Timur and Camilla congratulated the winner of the Olympiad in mathematics;

Iren regrets that she remains little time for literature.

Winner, what kind of Olympics each of these guys has become?

1 solution solution using a table

2 solution method using graphs

And t to e z

F M and L G

Answer: Irena is the winner of the Olympiad in mathematics. Timur - in geography.

Camilla - in Physics Eldar - in literature. We will live in computer science

2. Three girls - Rosa, Margarita and Anuta presented at the competition baskets from roses grown, daisies and pansies. The girl who growed daisies turned the attention of roses to the fact that neither one of the girls had a name with the name of his favorite colors. What flowers grow each of the girls?

Solution: With the help of reasoning

a) Anya raised not pansies. b) Margarita raised not daisies c) Rosa raised not roses. Rose could grow either roses or pansies. Rose has not grown roses. Conclusion: Rosa raised pansies. Margarita raised roses. Anya raised daisies.

3. Four buddies - Zhenya, Kostya, Dima and Vadim - made decorations for the holiday. Someone made garlands of golden paper, someone - red balls, someone sterling paper garlands, and someone - gold paper crackers. Kostya and Dima worked with the paper of the same color, Zhenya and Kostya did the same toys. Who did what decorations?

Answer:

Logical tasks for bringing into mutually - unambiguous compliance of elements of three sets is convenient to solve with a three-dimensional table

4. Masha, Lida, Zhenya and Katya playing different tools - the accordion, piano, guitar, violin, but each on one. They own foreign languages \u200b\u200b- English, French, German, Spanish, but each one who plays on what instrument and what foreign language owns?

Tasks for crossing

In the tasks of the crossing, you need to specify a sequence of actions at which the required crossing is carried out and all the conditions of the task are made.

Wolf, goat and cabbage. On the banks of the river there is a peasant with a boat, and next to him are the wolf, goat and cabbage. The peasant should cross and transport the wolf, goat and cabbage to the other side. However, in the boat except the peasant is placed either only a wolf or only a goat or cape. Leave a wolf with a goat or a goat with a cabbage without supervision, it is impossible - the wolf can eat a goat, and the goat - cabbage. How should the peasant behave?

Answer: The peasant can follow one of two algorithms:

2. Two soldiers approached the river, on which two boys ride on the boat. How the soldiers cross to another shore, if the boat holds only one soldier, or two boys, and the soldier and the boy no longer holds out?

Answer: Let M1 and M2 be boys, C1 and C2 - soldiers. The crossing algorithm may be:

1. M1 and M2 -\u003e
2. M1.<–
3. C1 -\u003e
4. M2.<–
5. M1 and M2 -\u003e
6. M1.<–
7. C2 -\u003e
8. M2.<–

Tasks for transfusion

Thesetasks are practical. The solution to such tasks is developing logical thinking, makes thinking, approach to solve any problem from different sides, choose from a variety of ways to solve the easiest, easy way. To do this, with the help of vessels of well-known containers, it is necessary to measure a certain amount of fluid. The simplest technique of solving the tasks of this class consists in interactive options.And you need to specify the sequence of actions at which the required transfusion is carried out and all conditions are performed.

1. How, having two buckets with a capacity of 3 and 5 liters, dial from a water tap 7 liters of water?

Answer:

Total in two buckets of 7 liters of water.

2. The evil stepmother sent a fever to a spherian for water and said: "In our buckets include 5 and 9 liters of water. Take them and bring exactly 3 liters of water. " How should Padryman act to fulfill this order?

Answer:

In the problems considered on problem problems, two vessels were given and the water was poured from a water tap.There are more complex tasks not two vessels, but three or more. Water takes not from the water tap. In such tasks, water is already in some vessel, for example, in the largest. And we will overflow with small capacles. It is impossible to pour water. If it is necessary to free the vessel, then excess water is poured into another vessel. Usually, a larger vessel is a repository where water comes from and is merged into it too much.

Task of fabulous character

Solution of such tasks revives mathematics. The desire to help the hero in the trouble stimulates mental activity, in the future it is a desire to read the work. Sympathy in such tasks on the side of the positive hero. Good triumph, evil is punished, negative qualities are ridicule.

on one of them you will meet your death,

on the other, nothing happens to you

the third road will lead you to Vasilis beautiful.

To keep in mind that all three inscriptions are made by the blazing immortal. Threw Ivan the tangle to the ground. He rolled, Ivan behind him. How long whether Ivan went briefly, but he came to a huge stone. On the stone it is written:

"Will you go left - you will meet your death,"

"Will you go right - you will rescue Vasilisa beautiful out of Nilo," you will go straight - something happens to you. "

Solution: The third entry is incorrect - on the road directly with Ivan nothing will happen. The second entry is also incorrect, i.e. On the way, Ivan will not call Vasilis beautiful. So, on the remaining road (the road to the left) Ivan will cause Vasilis beautiful.

2. Six robbers robbed Tsar Dadon. Production turned out to be rich - less than a hundred of the same ingots. Rogue began to share the prey to equally, but one ingot turned out to be superfluous. Rogue worried and one of them in a fight was killed. The remaining again began to share gold and again one piece turned out to be superfluous. And again in a fight died one of the robbers. And so on: each time one ingot was superfluous and one of the robbers died in a fight. In the end, one robber stayed, who died from the Russian Academy of Sciences. How many ingots were?

Decision: If it would initially, it would be less for one ingot, then the divide would take place. A number that is less than 100 and sharing on 2, 3, 4, 5, 6 - 60. So, the entire ingots are 60 + 1 \u003d 61.

Tasks for sponderfulness

1. Two mothers, two daughters and grandmother with granddaughter. How much is all?

2. There were 3 rooms in the apartment. From one did two. How many rooms have become in the apartment?

3. How to place 8 chairs from four walls of the room so that each wall has 3 stools?

Tasks on the mixture

How many hours together last day and night?

On the table lay an apple. It was divided into 4 parts. How many apples lies on the table?

Tasks for changing the built figure

The skill in modeling plane geometric shapes develops. 1. Make out of the sticks of the same figure as in the figure. Frames 2 sticks so that it turned out 2 squares.

2. Make out of the sticks of the same figure as in the figure. Remove 2 sticks to get 6 squares.

Numeric rows

1,2,3,4,5,6…

1,4,16…

45,39,33,27…

0,3,8,15,24…

112,56,28,14…

Rebuses

Replace the asterisks with numbers so that equalities are performed in all rows and each number of the last line was equal to the sum of the number of the column number under which it is located. Decision:

* 1 x ** \u003d ** 0

11x10 \u003d 110.

6* : *7 = *

68:17 = 4

** +** =20

10+10= 20

* 2 -* = *

12- 4 = 8

*** +**=1**

101 +41+142

Tasks with geometrically content (unicuristic figures)

Known Parable: Someone gave a million rubles to everyone who draws the following figure. But when drawing, one condition was set. It was necessary that the figure would be drawn by one continuous stroke, that is, not a totable pen or pencil from paper and without doubling a single line, in other words, during the line it was impossible to go through the second time.

Conclusion

In mathematics, there are various types of tasks for the smelter:

On weighing and transfusion

Logical tasks

Transfusion tasks

Tasks for crossing

Tasks with geometric content,

Rebuses, numeric rows.

Methods for solving such tasks is to logically analyze the conditions, the choice of the relevant laws of mathematics and the optimal solution of the solution.

There is no universal way to solve all types of tasks for the smelting, each task is solved by its way.

Tasks for smelting help learn to think independently, develop logic, interest in mathematics. With their help, you can feel the connection of mathematics with the problems of real life.

Solved the tasks facing the author of the work, namely:

Examine the topic "Solving tasks for smelting", types of tasks for smelting and methods of solving them;

Solve several types of tasks for the smelting, independently make an algorithm for solving such tasks.

Bibliography

1. etc. Gavrilova: "Entertaining Mathematics". Publishing House "Teacher" 2008

2. E.G. Kozlova: "Fairy Tales and Tips." Publisher "MIRO" 1995

3. B. A. Kordemsky: "Mathematical Matcheckan". Is the "State Publishing House of Technical and Theoretical Literature" 1958

4. Ya. I. Perelman: "Entertaining algebra". Publishing "Century" 1994

5.r.m.smallian "How is this book called?". Publishing House "House Mescheryakova"

2007

7. .http: // Matematika.Gyn

8.www.smekalka.pp.

See also:

Preface to the second edition 3

Chapter first
Cleaning tasks

Section I.
1. Observation pioneers 9 385
2. "Stone flower" 10 385
3. Moving checkers 11 385
4. In three stroke 11 386
5. Consider! 12 386.
6. Gardener's path 12 386
7. It is necessary to cut 13 386
8. Without thinking not long 386
9. Down - Up 13 387
10. Crossing through the river (an old task) 14 387
11. Wolf, goat and cabbage 14 387
12. Relote black balls 15 388
13. Repair of the chain 15 388
14. Correct the error 16 390
15. Of the three - four (joke) 16 390
16. Three yes two - eight (still joke) 16 390
17 Three Square 16 390
18. How many details? 17 390.
19. Try! 17 391.
20. Alignment of flags 17 391
21. Save parity 18 391
22. "Magic" numerical triangle 18 391
23. How to play ball 12 girls 19 392
24. Four straight 20 392
25. Separate goats from cabbage 20 392
26. Two trains 21 392
27. During the tide (joke) 21 393
28. Dial 22 393
29. Broken Dial 22 393
30. Amazing clock (Chinese puzzle) 23 393
31. Three in series 24 395
32. Ten rows 24 395
33. Location of coins 25 395
34. From 1 to 19 26 395
35. Quickly, but carefully 26 396
36. Figure Cancer 27 396
37. Cost of the book 27 396
38. Restless Fly 27 396
39. Less than in 50 years 28 396
40. Two jokes 28 396
41. How old are I? 29 396.
42. Rate "At the glance" 29 397
43. High-speed addition - 29 397
44. In what hand? (mathematical focus) 31 397
45. How many of them? 31 398.
46. \u200b\u200bSame numbers 31 398
47. Hundred 31 398
48. Arithmetic duel 32 398
49. Twenty 33 398
50. How many routes? 33 399.
51. Change the location of the numbers 35 400
52. Different actions, one result 35402
53. Ninety nine and hundred 36 402
54. Collapsible Chess Board 36 402
55. Looking for mines 36 402
56. Collect in groups of 2 38 402
57. Collect in groups of 3 39 402
58. Clock stopped 39 404
59. Four actions of arithmetic 39 404
60. PUAGED SHOPER 40 404
61. For Tsimlyan Hydrogen 41 404
62. Khleboshdachu in-time 41 405
63. In the dacha train 41 405
64. From 1 to 1 000 000 41 405
65. Scary Song of the Football Fan 42 406

Section II.
66. Watch 43 406
67. Staircase 43 407
68. Puzzle 43 407
69. Interesting fractions 43 407
70. What number? 44 407.
71. Schoolboy's path 44,407
72. At the stadium 44,407
73. Did you guess? 44 407.
74. Alarm clock 44 407
75. Instead of small fractions large 45 407
76. Soap Bar 45 408
77. Arithmetic nuts 45 408
78. Domino Domino 46 409
79. Mishina Kittens 48 409
80. Average speed 48 409
81. Sleeping passenger 48 409
82. What is the length of the train? 48 409.
83. Cyclist 48 409
84. Competition 49 409
85. Who is right? 49 409.
86. To dinner - 3 roasted slice 50 410

Chapter Second
Difficult provisions

87. Moekler Blacksmith Chcho 51 410
88. Cat and mice 53 410
89. Matches around the coin 54 411
90. Lot fell on Chizhi and Malinovka 54 411
91. Decay coins 55 411
92. Skip Passenger1 55 412
93. The task that arose from the whim of three girls 56 412
94. Further development of the problem 57 413
95. Jumping checkers 57 415
96. White and black 57 415
97. Completeness of the task 58 415
98. Cards are stacked in order of numbers 58 415
99. Two puzzles location 59 417
100. Mysterious box 59 417
101. Brave "garrison" 60 417
102. Daylight lamps in the room for television gear 61 419
103. Placement of experimental rabbits 62 421
104. Preparation for the holiday 63 422
105. Sear the oaks differently 65 423
106. Geometric Games 65 423
107. Chet and Unit (puzzle) 68 424
108. Sort the location of checkers 69 424
109. Puzzle gift 69 425
110. Stroke Horse 70 425
111. Moving checkers (2 puzzles) 71 425
112. Original grouping of integers from 1 to 15 72 426
113. Eight stars 73 426
114. Two tasks for placement of letters 73 427
115. Laying of multicolored squares 74 429
116. Last chip 74 430
117. Ring from disks 75 431
118. Figurers on the rink of artificial ice 76 431
119. Joke Task 77 432
120. One hundred forty-five doors (puzzle) 77 432
121. How did the prisoner come to freedom? 79 432.

CHAPTER THREE
Geometry on matches

122. Five puzzles 85 433
123. More eight puzzles 86 433
124. Of the nine matches 86 433
125. Spiral 87 433
126. Joke 87 433
127. Remove two matches 87 433
128 Facade "Houses" 87 433
129 Joke 88 433
130 triangles 88 433
131 How many matches should be removed? 88 433.
132 Joke 88 433
133 "Hedge" 88 433
134. Joke 89 433
135. "Strela" 89 433
136. Squares and diamonds 89 433
137. In one figure, different polygons 89 433
138 Garden layout 89 433
139 on isometric parts 90 433
140. Parquet 91 433
141 The ratio of the square is preserved 91,441
142. Find out the shape of the figure 91 441
143 Find Proof 92 441
144. Build and prove 92 441

Chapter Fourth
Seven times for example, once again

145. On equal parts 93 442
146. Seven roses on the cake 95 443
147. Figures that have lost their outline 95 445
148. Advise 96 445
149. Without loss! 96 445.
150. When the fascists have encroached 97 447 on our land
151. Memories of the electrician 98 447
152. Everything goes into business 99 447
153. Puzzle 99 447
154. Cut the horseshoe 99 447
155. In each part - the hole 99 448
156. From "Jug" - Square 100 448
157. Square from the letter "E" 100 448
158. Beautiful transformation 100 449
159. Carpet Restoration 101449
160. Dear Award 101 449
161. Check the poor man! 102 449.
162. Gift Grandma 103 451
163. Joiner's task 104 451
164. And the speed of the geometry! 104 452.
165. Each horse, on the stable 105 453
166. Yeshe more! 105 453.
167. Transformation of the polygon per square 106 453
168. Turning the right hexagon in equilateral triangle 107 453

Chapter Fifth
Decrease everywhere will find application

169. Where is the goal? 109 454.
170. Five minutes to think 110 455
171. Unforeseen meeting 110 455
172. Travel triangle sh 456
173. Try itching 111 458
174. Transfer 112 458
175. Seven triangles 112 458
176. Artist's cloth 112 458
177. How much does a bottle weigh? 113 459.
178. Cubes 113 460
179. Bank with fraction 114 461
180. Where did the sergeant come? 114 461.
181. Determine the log diameter 115 461
182. Unexpected difficulty 115 461
183. Student Student of Technical School 116 461
184. Is it possible to get 100 ° / about savings? 116 463.
185. Spring scales 117 463
186. Design cutter 117 463
187. Mishina failure 117 465
188. Find the center of the circumference 119 465
189. Which box is heavier? 119 466.
190. The art of the joiner 120 466
191. Geometry on a ball 120 466
192. Need a large seaming 121 467
193. Hard conditions 121 468
194. Prefabricated polygons 122 468
195. Curious reception of the compilation of such figures 125 469
196. Hinged mechanism for building the right polygons 127 471

Chapter Six
Domino and Kubic

A. Domino
197. How many points? 132 471.
198. Two focus 133 471
199. Winning party provided 134 471
200. Frame 135 472
201. Frame in the frame 136 472
202. "Winds" 136 473
203. Magic squares from Domino Bones 137 473
204. Magic Square with Hole 141 473
205. Multiplication in Domino 141 473
206. Guess the intended domino bone 142 473

B. Kubik
207. Arithmetic focus with playing cubes 144 473
208. Gaying the amount of points on hidden edges 145 477
209. In what order are the cubes are located? 145 478.

Head seventh
Properties of nine

210. What figure is crossed out? 149 478.
211. Hidden property 152 479
212. Another few funny ways to find a missing number 152 480
213. According to one digit of the result, determine the remaining three 154 480
214. Gaying the difference 154 481
215. Definition of age 154 481
216. What is the secret? 154 482.

Chapter Eighth
With algebra and without it

217. Mutual assistance 159 482
218. Slock and Chort 160 483
219. Clear baby 161 483
220. Hunters 161 483
221. Counter trains 162 484
222. Vera Prints Manuscript 162 484
223. History with mushrooms 163 484
224. Who will come back before? 164 484.
225. Swimmer and hat 164 486
226. Two ships 165 486
227. Check your seamless! 165 487.
228. Confuez prevented 166 488
229. How many times more? 166 488.
230. motor ship and seaplane 167 488
231. Velofigurists in the arena 167 489
232. The speed of Tokar Bykova 168 489
233. Jack London's trip 168 489
234. Due to unsuccessful analogies, errors are possible169 490
235. Legal Casus 170 491
236. Couples and Things 171 491
237. Who rode a horse? 171 491.
238. Two motorcyclists 171 492
239. In what aircraft Volodin Dad? 172 492.
240. Varnish on parts 173 493
241. Two Candles 173 493
242. Amazing insight 173 493
243. "Ordinary time" 174 493
244. Watch 174 494
245. In which hours? 174 495.
246. What time began and the meeting ended? 175 496.
247. Sergeant training intelligence officers 175 497
248. By two posts 176 498
249. How many new stations have built? 176 498.
250. Select four words 177 498
251. Is such weighing? 177 499.
252. Elephant and Komar 178 500
253. Five-digit number 179 500
254. years to one hundred to grow you without old age 179,500
255. Luke Task 181 501
256. A kind of walk, .181 502
257. One property of simple fractions 182 504

Ninth chapter
Mathematics almost without computing

In a dark room
Apples
Weather forecast (joke)
Forest day
Who has any name?
Competition in the accuracy
Purchase
Passengers of one coupe
Final tournament of chess players of the Soviet Army
Resurrection
Like the surname of the driver?
Criminal history
Collectors of herbal
Hidden division
Encrypted actions (numerical rebuses)
Arithmetic Mosaic
Motorcyclist and Horse
On foot and car
"From the opposite"
Detect a false coin
Logic draw
Three wisers
Five questions for schoolchildren
Reasoning instead of equation
In common sense
Yes or no?

Chapter Tenth
Mathematical games and tricks

A. Games
284. Eleven items 201
285. Take matches last 202
286. Wins Chet 202
287. Jiangsitse 202.
288. How to win? 204.
289. Lay Square 205
290. Who will first say "hundred"? 206.
291. Game in squares 206
292. OUA 209.
293. "Matecathico" (Italian Game) 212
294. Game in Magic Squares 213
295. Intersection of numbers 215

B. Focuses
296. Guessing the intended number (7 focus) 219
297. Guess the result of calculations, not asking anything 224
298. Who took, I learned 226
299. One, two, three attempts and I guess 226 537
300. Who took a gum, and who is a pencil? 227 537.
301. Guessing three conceived terms and amounts 227 537
302. Guessing somewhat intended numbers 228 538
303. How old are you? 229 538.
304. Guess Age 229 538
305. Geometric focus (mysterious disappearance) 230 538

Chapter eleventh
Dividitude of numbers

306. Number on Tomb 232 539
307. Gifts for the new year 233 540
308. Can there be such a number? 233 540.
309. Basket of eggs (from the ancient French problem book) 233 540
310. Three-digit number 234 540
311. Four ships 234 540
312. Cashier Error 234 540
313. Numeric Rus 234 541
314. Sign of divisibility by 11 235 541
315. Jointed Signal of Dividivity at 7, 11 and 13 237 541
316. Simplification of the sign of divisibility at 8 239 541
317. Striking memory 240 542
318. Jointed Signal of Dividivity by 3, 7 and 19. 242 543
319. Delibery 242 543
320. Old and new about divisibility at 7 247 544
321. Distribution of a sign on other numbers 251 -
322. Generalized Signal of Validity 252 -
323. Curious divisibility 254 -

Chapter twelve
Cross-sum and magic squares

A. Cross-Am
324. Interesting groupings 256 545
325. "Star" 257 545
326. "Crystal" 257 545
327. Decoration for showcase 258 545
328. Who will be able to? 258 545.
329. "Planetarium" 259 545
330. "Ornament" 259 545

B. Magic squares
331. Aliens from China and India 260 548
332. How to make a magic square yourself? 264 548.
333. On Vstaps to General Methods 266 549
334. Executive exam 271 549
335. "Magic" game in "15" 271 551
336. Nearby magic Square 272 553
337. What in the central cell? 273 553.
338. "Magic" works 275 553
339. "Casket" arithmetic curiosities 278 -
340. "In addition" 280 -
341. "Right" magic squares of the fourth order 283 -
342. Selection of numbers for the magical square of any order 287 -

Chapter thirteenth curious and serious in numbers
343. Ten digits (observations) 298 554
344. A few more advanced observations 300 555
345. Two interesting experiences 302 555
346. Number Carousel 306 -
347. Instant multiplication disk 309 -
348 Mental Gymnastics 310 -
349. Patterns numbers 312 557
350 one for everyone and all for one 316 558
351. Number finds 319 559
352. Watching a number of natural numbers 326 560
353. Interesting difference 339 -
354. Symmetric amount (inconspicuous nuts) 340 -

Chapter Fourteenth
Ancient numbers, but forever young

A. The initial numbers
355. Numbers are simple and composite 341 -
356. "Eratosphenovo Deuto" 342 -
357. New "Swelto" for simple numbers 344 563
358. Fifty first simple numbers 345 -
359. Another way to get simple numbers. 345 -
360. How many simple numbers? 347.

B. Fibonacci numbers
361. Public Test 347 -
362. Fibonacci row 351 -
363. Paradox 352 564
364. Properties of numbers of the Fibonacci row 355 -

B. Figure numbers
365. Properties of figured numbers 360 -
366. Pythagoras of Numbers 369 -

Chapter Fifteenth Geometric Cooker In Labor
367. Geometry Seva 372 -
368. Rationalization in laying bricks for transportation 375 -
369. 377 working and geometers

Recognized two chapters:

Preface to the second edition
In labor, in the teaching, in the game, in every creative activity, a person is needed by intelligence, resourcefulness, guess, decrease to reason - all that our people are aptive defines in a word "sedent". The mixture can be brought up and developing by systematic and gradual exercises, in particular by the solution of mathematical tasks of both the school course and the tasks arising from the practice related to the observations of the world around us and events.
"Mathematics," said M. I. Kalinin, referring to high school students, - disciplines the mind, teaching to logical thinking. No wonder they say that mathematics is the gymnastics of the mind. "
Each family in which parents are concerned about the organization of mental development of children and adolescents feeling the need for a selected material for filling leisure with useful, reasonable and mischievous mathematical exercises.
Here for this kind of out-of-promotional classes, conversations and entertainment in a free evening, in a family circle and with friends, or at school on extracurricular meetings and intended "M Amem" - a collection of mathematical miniatures: a variety of tasks, mathematical games, jokes and focus requirements requiring The work of the mind, developing intention and the necessary logicality in reasoning.
In the pre-revolutionary time, the collections of E. I. Ignatiev "in the kingdom of smelting" were widely known. Now they are outdated for our reader and therefore are not reprinted. Nevertheless, in these collections there are tasks that have not yet lost pedagogical and educational value. Some of them entered the "mathematical smelting" unchanged, others with a changed or completely new content.
For the "mathematical mixture", I also selected and, if necessary, handled the tasks from among those that were scattered on the pages of extensive domestic and foreign popular literature, striving, however, do not repeat the tasks included in the common books of Ya. I. Perelman on entertaining mathematics.
This kind of mathematical tasks of the "small form" sometimes arise as a by-product of serious research scholar; Many tasks are invented by lovers, as well as teachers as special exercises for "mental gymnastics". They, like mysteries and proverbs, usually do not retain authorship and become the property of society.
"Mathematical sedizer" is intended for readers with the most diverse degree of mathematical training:
For a teenager 10 - 11 years old, making the first attempts by independent reflection;
For a schoolboy of senior classes enthusiastic with mathematics,
And for an adult wishing to experience and practice their guess.
The systematization of tasks on chapters, of course, is very conditional; Each chapter has lungs and difficult tasks.
In the book fifteen chapters.
The first chapter consists of a variety of initial exercises of the "clautting" nature based on guesses or direct physical actions (experiment), sometimes on simple calculations within the range of integers (the first section of the chapter) and fractional numbers (the second section). Several violating classification of books, I allocated part of the simple tasks to the first chapter, thematically belonging to the subsequent chapters. This is done in the interests of those readers who are still difficult to independently distinguish the full task from the unbearable. By deciding in a row, the variety of tasks of the first chapter, they will be able to try their strength, and then have any interest in a certain topic to transfer to the relevant tasks of the following chapters.
To solve the tasks of the second chapter, its own mathematical seaming and perseverance must overcome all sorts of obstacles and suggest a way out of difficult provisions.
The third chapter is "geometry on matches" - contains a number of geometric tasks - puzzles.
Chapter "Seven times for example, a rejection once" consists of tasks for cutting figures.
The content of the tasks of chapter "decrease everywhere will find application" is associated with practical activities, with appliances.
In the chapter, called "mathematics almost without computing", contains tasks to solve which it is required to build a chain of skillful and subtle reasoning.
Games and focuses are collected in a separate chapter, and also posted throughout the book. They contain a mathematical basis and are undoubtedly included in the "region of the smelting".
Three chapters: "Cross-Amounts and Magic Squares", "curious and serious in numbers" and "Numbers ancient, but forever young," are devoted to some curious observations on the numerical ratios accumulated in mathematics from deep antiquity to our time.
The last chapter is two small essays about the labor mixture of people of our homeland, workers of fields and plants.
In different places of the book, small topics are offered to the reader for independent surveys.
At the end of the book, tasks are placed, but do not hurry in them to look.
Any task for "Clear" is in itself a "highlight" and represents in most cases a strong nuts, it is not so easy to cut, but the more tempting.
If the solution to the task is not possible at once, you can temporarily skip it and go to the next or to the tasks of another ROOM, another chapter. Later return to the missed task.
"Mathematical savings" - a book is not for easy reading "in one sitting", and for work throughout, maybe a number of years old, a book for regular mental gymnastics in small portions, a reader's companion in its gradual mathematical development.
The entire material of the book is subordinated to the educational and educational goal: to encourage the reader to independent creative thinking, to further improve its mathematical knowledge.
The second edition of the "mathematical smelting" is not a stereotypical repetition of the first. Required changes in text and solve some problems; Separate tasks are replaced by new - more substantive; Re-executed the design of the book.
Big efforts aimed at improving the book, put the editor of the Publishing House M. M. Hotly.
Independently solving the tasks, readers in some cases found additional or simpler solutions and kindly informed me their results. The authors of the most interesting decisions are mentioned in the relevant places of the book.
I hope to receive reviews and wishes from readers "Smekalki" for further improvement of the book, as well as our own original tasks and mathematical materials of folk art.
Address: Moscow, B-64, ul. Chernyshevsky, d. 31, square. 53, Boris Anastasyevich Kordemsky.
B. Kordemsky.

TASKS

"The book is a book, and the brains move"
V. Mayakovsky.

CHAPTER FIRST. Cleaning tasks

Section I.
Check and inspect your seamless at first such tasks, to solve only purposeful perseverance, patience, intelligence and decrease to add, deduct, multiply and divide integers.

1. Observation pioneers
Schoolchildren - boy and girl - just made meteorological measurements.
Now they are resting on the hillock and look at the commodity train passing by them.
The locomotive on the rise desperately smoke and puffs. Along the railway canvas, the wind blows the wind without gusts.
- What speed of wind showed our measurements? - asked the boy.
- 7 meters per second.
- Today it is enough for me to determine how fast the train is going.
"Well, yes," the girl doubted.
- And you look closer to the train movement.
The girl thought a little and also realized, what's the matter.
And they saw exactly what our artist drew (Fig. 1). What speed was the train?
Fig. 1. What speed is the train?

2. "Stone flower"
Remember the talented "craft" Master Danil from the fairy tale P. Bazhova "Stone Flower"?
They are told in the Urals, which Danila, being another student, drew two such flower (Fig. 2), the leaves, stalks and petals of which were released, and from the resulting flowers of flowers could be folded in the form of a circle.
Try! Redraw Danilina Flowers on paper or cardboard, cut the petals, stems and leaves and fold the circle.

3. Moving checkers
Put 6 checkers on a table in a row alternately - black, white, eating black, but white, etc. (Fig. 3).
Fig. 3. White checkers should be on the left, for them - black.
Right or left, leave free space sufficient for four checkers.
It is required to move checkers so that the left is all white, and after them all black. At the same time, it is necessary to move on a free space at once two next to the lying checkers, without changing the order in which they lie. To solve the problem, it suffices to make three movements (three strokes) *).
If you do not have checkers, use the coins or cut the pieces of paper, cardboard.
*) The topic of this task is further developed in tasks 96 and 97 (p. 57 and 58).

4. In three strokes
Put 3 handhes of matches on the table. In one bunch, put 11 matches, and to another - 7, in the third - 6. Shooting matches from any bunch of any other, you need to compose all three bugs so that each has 8 matches. This is possible, since the total number of matches - 24 - divides 3 without a balance; It is necessary to observe such a rule: it is allowed to add exactly so many matches to any pile as it is in it. For example, if 6 matches in a bunch, you can only add 6 to it, if 4 matches in a pile, then only 4 can be added to it.
The task is solved in 3 strokes.

5. Consider!
Check your geometric observation: count how many triangles in the figure shown in Fig. four.

6. The path of the gardener
In fig. 5 Dan a small apple orchard plan (dots - apple tree). Gardener handled all the apple tree in a row.
Fig. 5. Plan of apple orchard.
He began with a cell marked with an asterisk, and went around one another all cells, both engaged in apple tops and
Free, never returning to the cell passed. He did not go to diagonals and was not on the shaded cells, as various buildings were placed there.
Having finished around, the gardener was on the same cell from which he began his way.
Distribute the path of the gardener in your notebook.

7. It is necessary to cut
In the basket lies 5 apples. How to divide these apples between five girls so that every girl got one apple and so that one apple is left in the basket?

8. Without thinking
Tell me how many in the room of cats, if in each of the four corners of the room sits on the same cat, against each cat sits on 3 cats and on the tail of every cat sits on a cat?

9. Down - Up
The boy tightly pressed the face of a blue pencil to the verge of a yellow pencil. One centimeter (in length) of the pressed face of a blue pencil, counting from the bottom end, the paint blur. Yellow pencil The boy holds motionless, and blue, continuing to press to yellow, lowers 1 cm, then returns to the previous position, again lowers 1 cm and returns again to the previous position; 10 times he lowers so and 10 times raises the blue pencil (20 movements).
If we assume that during this time the paint does not dry and is not depleted, then how many centimeters will be blurred yellow pencil after the twentieth movement?
Note. This task came up with mathematician Leonid Mikhailovich Fishermen on the way to home after a successful hunting for ducks. What served as a reason for the work of the task, you will read on page 387, after you decide the task.

10. Crossing through the river (an old task)
A small military squad approached the river through which it was necessary to cross. Bridge is broken, and the deep river. How to be? Suddenly, the officer notices the shore of two boys that chew in the boat. But the boat is so small that only one soldier can cross it or only two boys - no more! However, all the soldiers crossed the river on this boat. How?
Decide this task "in the mind" or practically - using checkers, matches or something like that and moving them on the table through the imaginary river.

11. Wolf, goat and cabbage
This is also an old task; It is found in the writings of the VIII century. It has fabulous content.
Fig. 6. It was impossible to leave a wolf and goat without a person ...
A certain person was supposed to carry in a boat across the Wolf River, Goat, and Cabbage. Only one person could fit in the boat, and with him or a wolf, or a goat, or cabbage. But if you leave a wolf with a goat without a person, then the wolf will eat the goat, if you leave the goat with a cabbage, then the goat will eat cabbage, and in the presence of a person "no one eating anyone." Man still transported his cargo across the river.
How did he do it?
In a narrow and very long chute are 8 balls: four black ones and four white slightly larger diameter on the right (Fig. 7). In the middle part of the gutter in the wall there is a small niche in which only one ball (any) can fit. Two balls can be located near the head of the gutter only in the place where Niche is located. The left end of the gutter is closed, and in the right end there is a hole through which any black ball can pass, but not white. How to roll out all the black balls from the gutter? It is not allowed to remove balls from the gutter.

13. Repair of chains
You know, what a young master thought about (Fig. 8)? In front of him, 5 links chains that need to be connected in one chain without drinking extra rings. If, for example, to freeze the ring 3 (one operation) and cling to them for the ring 4 (another operation), then freeze the ring 6 and cling to the ring 7, etc., then there will be eight operations, and the master seeks to sow the chain when help only six operations. He succeeded. How did he acted?

14. Correct the error
Take 12 matches and lay out "equality" from them, shown in Fig. nine.
Fig. 9. Correct the error by shifting only one match.
Equality, as you see, incorrect, as it turns out that 6 - 4 \u003d 9.
Place one match so that the correct equality it turns out.

15. Of the three - four (joke)
There are 3 matches on the table.
Without adding any matches, make out of three - four. It is impossible to break matches.

16. Three yes two - eight (another joke)
Here is another like a joke. You can offer it to your comrade.
Put 3 matches on the table and offer a comrade to add 2 more to them so that it turned out eight. Of course, it is impossible to break matches.

17. Three squares
Of 8 sticks (for example, matches), four of which twice the remaining four are required to be 3 equal squares.

18. In the turning shop of the plant, the details from lead blanks are pulled. From one billet - detail. Chips, which come with the sediment of six parts, can be used and prepare another workpiece. How many details can be done in this way out of 36 lead blanks?

19. Try!
In the square hall for dancing to put the chairs along the walls so that each wall stood the seats equally.

20. Alignment of flags
A small intercoleous hydroelectric station was built by Komsomol members. To the day of her start, pioneers decorate the power plant outside from all four sides by garlands, light bulbs and flags. The flags were a bit, only 12.
Pioneers first put them in 4 on each side, as shown in the diagram (Fig. 10], then realized that the same 12 flags they could place 5 and even 6 on each side. They liked the second project more, and they decided Plan 5 flags.
Show in the diagram as pioneers lay 12 flashes on 5 from each of the four sides and how they could arrange them for 6 flags.

21. Save parity
Take 16 of any items (paper, coins, plums or checkers) and place them 4 in a row (Fig. 11). Now remove 6 pieces, but so that it remains in each horizontal and in each vertical row along an even number of items. Restringing different 6 pieces, you can get different solutions.

22. "Magic" numerical triangle
In the vertices of the triangle, I placed the number 1, 2 and 3, and you place numbers 4, 5, 6, 7, 8, 9 along the sides of the triangle so that the sum of all numbers along each side of the triangle is 17. It is not difficult, since I suggested What numbers should be placed in the vertices of the triangle. 2.
Much longer you have to tinker, if I do not say in advance, what numbers should be placed in the vertices of the triangle, and offer again to place numbers
1, 2, 3, 4, 5, 6, 7, 8, 9,
Each one is one time, along the parties and in the vertices of the triangle so that the amount of numbers on each side of the triangle is 20.
When you get the desired location of the numbers, look for more and new locations. The conditions of the task can be performed with a wide variety of numbers.

23. How to play 12 girls
Twelve girls became a circle and started playing the ball. Each girl threw the ball of his neighbor on the left. When the ball went around the whole circle, he was thrown in the opposite direction. After a while, one girl said:
- We will better throw the ball through one person.
"But since we are twelve, then half of the girls will not participate in the game," Natasha objected.
- Then we will throw the ball through two! (Each third catches the ball.)
"Everything is worse: only four will play ... If you want all the girls to play, you need to throw the ball after four (the fifth catches). There is no other combination.
- And if you throw the ball in six people?
- It will be the same combination, only the ball will go in the opposite direction.
- And if you play ten (each eleventh catches the ball)? - Girls were pretty.
- In this way, we have already played ...
Girls began to draw diagrams of all the proposed ways of game and very soon made sure that Natasha was right. Only one game scheme (except initial) covered all participants without exception (Fig. 13, a).
Now, if the girls had been playing girls, the ball could be thrown through one (Fig. 13, b), and Cheree-two (Fig. 13, c), and through three (Fig. 13, d), and after four (Fig. 13, e), and every time the game would cover all participants. Find out if you can throw the ball with five people at thirteen?
Is it possible to throw the ball after six people with thirteen playing? Think for clarity to draw the appropriate schemes.

24. Four straight
Take a sheet of paper and apply a Central Asian. 14. It is nine points so that they are located in the form of a square, as shown in Fig. 14. List now all the points four straight lines, without taking a pencil from paper.

25. Separate goats from cabbage
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15951.
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Chapter Second
Difficult provisions

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