Mathematical ingenuity. Research work "math savvy" Children's math savvy

Preface to the second edition 3

Chapter first
FUN CHALLENGES

Section I
1. Observant pioneers 9,385
2. "Stone flower" 10 385
3. Moving checkers 11 385
4. In three moves 11,386
5. Count! 12 386
6. Way of the gardener 12,386
7. You need to understand 13 386
8. Without hesitation 13,386
9. Down - up 13 387
10. Crossing the river (old problem) 14,387
11. Wolf, goat and cabbage 14,387
12. Roll out black balls 15 388
13. Chain repair 15 388
14. Fix bug 16,390
15. Out of three - four (joke) 16,390
16. Three and two - eight (another joke) 16,390
17 Three squares 16 390
18. How many parts? 17 390
19. Try it! 17 391
20. Flagging 17 391
21. Preserve parity 18,391
22. "Magic" number triangle 18 391
23. How 12 girls played ball 19 392
24. Four straight lines 20 392
25. Separate the goats from the cabbage 20,392
26. Two trains 21,392
27. At high tide (joke) 21,393
28. Dial 22 393
29. Broken dial 22 393
30. Amazing clock (Chinese puzzle) 23,393
31. Three in a row 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. Curly cancer 27 396
37. Cost of the book 27,396
38. Restless fly 27,396
39. Less than 50 years 28,396
40. Two jokes 28 396
41. How old am I? 29 396
42. Evaluate "at a glance" 29 397
43. Speed ​​addition - 29 397
44. In which hand? (math focus) 31,397
45. How many are there? 31 398
46. ​​Same digits 31 398
47. One hundred 31 398
48. Arithmetic duel 32 398
49. Twenty 33 398
50. How many routes? 33 399
51. Change the arrangement of numbers 35 400
52. Different actions, same result 35402
53. Ninety-nine and one hundred 36,402
54. Demountable chessboard 36 402
55. Search for mines 36 402
56. Collect in groups of 2 38 402
57. Collect in groups of 3 39 402
58. The clock has stopped 39 404
59. Four operations of arithmetic 39 404
60. Perplexed driver 40 404
61. For the Tsimlyansk hydroelectric complex 41,404
62. Bread delivery on time 41 405
63. In a suburban train 41 405
64. From 1 to 1,000,000,000 41,405
65. A football fan's nightmare 42,406

Section II
66. Hours 43 406
67. Staircase 43 407
68. Puzzle 43 407
69. Interesting fractions 43 407
70. What is the number? 44 407
71. Way of the schoolboy 44 407
72. At the stadium 44,407
73. Did you win? 44 407
74. Alarm clock 44 407
75. Instead of small shares, large 45,407
76. Bar of soap 45 408
77. Arithmetic nuts 45 408
78. Dominoes 46 409
79. Misha's 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. For dinner - 3 toasted slices 50 410

Chapter Two
CONFIDENTIAL PROVISIONS

87. Wits of blacksmith Hecho 51 410
88. Cat and mice 53 410
89. Matches around the coin 54 411
90. The lot fell on the siskin and the robin 54 411
91. Arrange coins 55 411
92. Pass passenger1 55 412
93. A problem arising from the caprice of three girls 56 412
94. Further development tasks 57 413
95. Jumping checkers 57 415
96. White and black 57 415
97. Complicating the problem 58 415
98. Cards are stacked in numerical order 58 415
99. Two location puzzles 59 417
100. Mysterious box 59 417
101. Brave "garrison" 60 417
102. Fluorescent lamps in TV room 61 419
103. Placement of guinea pigs 62,421
104. Preparation for the holiday 63 422
105. Seating oak trees differently 65 423
106. geometric games 65 423
107. Even and odd (puzzle) 68 424
108. Arrange the arrangement of checkers 69 424
109. Puzzle gift 69 425
110. Knight move 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 problems for the placement of letters 73 427
115. Layout of colorful squares 74 429
116. Last chip 74 430
117. Ring of disks 75 431
118. Skaters on the rink artificial ice 76 431
119. Joke problem 77 432
120. One hundred and forty-five doors (puzzle) 77 432
121. How was the prisoner released? 79 432

Chapter Three
GEOMETRY ON MATCHES

122. Five puzzles 85 433
123. Eight more puzzles 86 433
124. From nine matches 86 433
125. Spiral 87 433
126. Joke 87 433
127. Remove two matches 87 433
128 Facade of the "house" 87 433
129 Joke 88 433
130 Triangles 88 433
131 How many matches should be removed? 88 433
132 Joke 88 433
133 Fence 88 433
134. Joke 89 433
135. "Arrow" 89 433
136. Squares and diamonds 89 433
137. Different polygons in one figure 89 433
138 Garden planning 89 433
139 Equal parts 90 433
140. Parquet 91 433
141 Area ratio maintained 91 441
142. Find the outline of a figure 91 441
143 Find proof 92 441
144. Construct and prove 92 441

Chapter Four
TRY SEVEN TIMES, CUT ONCE

145. In equal parts 93 442
146. Seven roses on a cake 95 443
147. Figures that have lost their shape 95 445
148. Advise 96 445
149. Lossless! 96 445
150. When the Nazis encroached on our land 97 447
151. Memoirs of an electrician 98 447
152. Everything goes to work 99 447
153. Puzzle 99 447
154. Cut a horseshoe 99 447
155. In each part - a hole 99 448
156. From the "jug" - a square 100 448
157. Square from the letter "E" 100 448
158. Beautiful transformation 100 449
159. Carpet restoration 101449
160. Expensive reward 101 449
161. Help out the poor man! 102 449
162. Gift for grandmother 103 451
163. Carpenter's problem 104 451
164. And the furrier has geometry! 104 452
165. Each horse, one stable 105 453
166. More! 105 453
167. Transformation of a polygon into a square 106 453
168. Transformation of a regular hexagon into an equilateral triangle 107 453

Chapter Five
SKILL WILL FIND A USE EVERYWHERE

169. Where is the target? 109 454
170. Five minutes to think 110 455
171. Unforeseen meeting 110 455
172. Travel triangle Ш 456
173. Try to weigh 111 458
174. Transfer 112 458
175. Seven triangles 112458
176. Paintings of the artist 112 458
177. How much does a bottle weigh? 113 459
178. Cubes 113 460
179. Can of shot 114 461
180. Where did the sergeant come? 114 461
181. Determine the diameter of the log 115 461
182. Unexpected difficulty 115 461
183. The story of a student of a technical school 116 461
184. Is it possible to get 100% savings? 116 463
185. On spring scales 117 463
186. Design ingenuity 117 463
187. Misha's failure 117 465
188. Find the center of the circle 119 465
189. Which box is heavier? 119 466
190. The art of carpentry 120 466
191. Geometry on a ball 120 466
192. Great ingenuity is needed 121 467
193. Difficult conditions 121 468
194. Prefabricated polygons 122 468
195. An interesting method of composing similar figures 125 469
196. Hinge mechanism for constructing regular polygons 127 471

Chapter Six
DOMINO AND CUBE

A. Domino
197. How many points? 132 471
198. Two tricks 133 471
199. Winning the game is guaranteed 134 471
200. Frame 135 472
201. Frame within a frame 136 472
202. "Windows" 136 473
203. Magic squares of domino bones 137 473
204. Magic square with hole 141 473
205. Domino multiplication 141473
206. Guess the planned domino 142 473

B. Cube
207. Arithmetic trick with playing dice 144 473
208. Guessing the sum of points on hidden faces 145 477
209. What order are the cubes in? 145 478

Chapter Seven
PROPERTIES OF THE NINE

210. What number is crossed out? 149 478
211. Hidden property 152 479
212. A few more fun ways to find the missing number 152,480
213. Based on one digit of the result, determine the remaining three 154 480
214. Guessing the difference 154 481
215. Determination of age 154 481
216. What is the secret? 154 482

Chapter Eight
WITH AND WITHOUT ALGEBRA

217. Mutual assistance 159,482
218. Loafer and devil 160 483
219. Smart kid 161 483
220. Hunters 161,483
221. Oncoming trains 162,484
222. Faith is typing a manuscript 162,484
223. Mushroom story 163 484
224. Who will return first? 164 484
225. Swimmer and hat 164,486
226. Two ships 165 486
227. Test your ingenuity! 165 487
228. Embarrassment prevented 166,488
229. How many times more? 166 488
230. Motor ship and hydroplane 167,488
231. Cyclists in the arena 167,489
232. The speed of the turner Bykov 168 489
233. Jack London trip 168,489
234. Due to unsuccessful analogies, errors are possible169 490
235. Legal incident 170 491
236. In pairs and threes 171,491
237. Who rode a horse? 171 491
238. Two motorcyclists 171,492
239. In which plane is Volodin's father? 172 492
240. Shatter into pieces 173 493
241. Two candles 173 493
242. Amazing insight 173 493
243. Right Time 174 493
244. Hours 174 494
245. What time is it? 174 495
246. At what time did the meeting begin and end? 175 496
247. Sergeant trains scouts 175,497
248. According to two reports 176 498
249. How many new stations were built? 176 498
250. Choose four words 177 498
251. Is such weighing permissible? 177 499
252. Elephant and mosquito 178 500
253. Five digit number 179500
254. You will grow up to a hundred years without old age 179 500
255. Luke's problem 181 501
256. Peculiar walk, .181 502
257. One property of simple fractions 182 504

Chapter Nine
MATH WITH ALMOST NO CALCULATIONS

In a dark room
Apples
Weather forecast (joke)
forest day
Who has a name?
Competition in accuracy
Purchase
Passengers in one compartment
Final of the Soviet Army Chess Tournament
Sunday
What is the name of the driver?
criminal history
Herb Gatherers
Hidden division
Encrypted actions (numeric puzzles)
Arithmetic tiling
Motorcyclist and horseman
On foot and by car
"From the opposite"
Detect counterfeit coin
Logic draw
three wise men
Five questions for students
Reasoning instead of an equation
By common sense
Yes or no?

Chapter Ten
MATH GAMES AND TOCKS

A. Games
284. Eleven items 201
285. Take matches last 202
286. Even wins 202
287. Jianshizi 202
288. How to win? 204
289. Lay out a square 205
290. Who will be the first to say "one hundred"? 206
291. Playing squares 206
292. Oya 209
293. "Matezatico" (Italian game) 212
294. Magic squares game 213
295. Intersection of numbers 215

B. Tricks
296. Guessing the planned number (7 tricks) 219
297. Guess the result of calculations without asking anything 224
298. Who took how much, I found out 226
299. One, two, three attempts and I guessed right 226 537
300. Who took the gum and who took the pencil? 227 537
301. Guessing three conceived terms and the sum 227 537
302. Guess several conceived numbers 228 538
303. How old are you? 229 538
304. Guess the age 229 538
305. Geometric focus (mysterious disappearance) 230 538

Chapter Eleven
DIVISIBILITY OF NUMBERS

306. Number on the 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 an old French problem book) 233 540
310. Three-digit number 234 540
311. Four ships 234 540
312. Cashier's mistake 234 540
313. Numerical puzzle 234 541
314. Sign of divisibility by 11 235 541
315. Combined sign of divisibility by 7, 11 and 13 237 541
316. Simplification of the test of divisibility by 8 239 541
317. Amazing memory 240 542
318. Combined sign of divisibility by 3, 7 and 19. 242 543
319. Divisibility of a binomial 242 543
320. Old and new about divisibility by 7,247,544
321. Extension of a sign to other numbers 251 -
322. Generalized sign of divisibility 252 -
323. Curiosity of divisibility 254 -

Chapter Twelve
CROSS-SUMS AND MAGIC SQUARES

A. Cross sums
324. Interesting groupings 256 545
325. "Asterisk" 257 545
326. "Crystal" 257 545
327. Showcase decoration 258 545
328. Who will succeed first? 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. At the entrance to common methods 266 549
334. Examination of ingenuity 271 549
335. "Magic" game of "15" 271 551
336. Non-traditional magic square 272 553
337. What is in the central cell? 273 553
338. "Magic" works 275 553
339. "Casket" of arithmetical curiosities 278 -
340. "In addition" 280 -
341. "Regular" magic squares of the fourth order 283 -
342. Selection of numbers for a magic square of any order 287 -

CHAPTER THIRTEEN CURIOUS AND SERIOUS IN NUMBERS
343. Ten figures (observations) 298 554
344. A few more interesting observations 300 555
345. Two interesting experiences 302 555
346. Number carousel 306 -
347. Instant multiplication disk 309 -
348 Mental gymnastics 310 -
349. Patterns of numbers 312 557
350 One for all and all for one 316 558
351. Numerical finds 319 559
352. Observing a series of natural numbers 326 560
353. Annoying Difference 339 -
354. Symmetric sum (unbroken nut) 340 -

Chapter Fourteen
NUMBERS ANCIENT BUT FOREVER YOUNG

A. Initial numbers
355. Prime and composite numbers 341 -
356. "Sieve of Eratosthenes" 342 -
357. New "sieve" for prime numbers 344 563
358. Fifty first primes 345 -
359. Another way to get prime numbers. 345-
360. How many prime numbers? 347

B. Fibonacci numbers
361. Public trial 347 -
362. Fibonacci Series 351 -
363. Paradox 352 564
364. Properties of numbers in the Fibonacci series 355 -

B. Curly numbers
365. Properties of curly numbers 360 -
366. Pythagorean numbers 369 -

CHAPTER FIFTEEN GEOMETRIC INTENT IN WORK
367. Sowing geometry 372 -
368. Rationalization in laying Brick for transportation 375 -
369. Working geometers 377

Recognized two chapters:

PREFACE TO THE SECOND EDITION
In work, in learning, in play, in any creative activity, a person needs ingenuity, resourcefulness, conjecture, the ability to reason - all that our people aptly defines in one word "savvy". Ingenuity can be brought up and developed by systematic and gradual exercises, in particular, by solving mathematical problems both in the school course and problems arising from practice related to observations of the world of things and events around us.
“Mathematics,” said M. I. Kalinin, addressing secondary school students, “disciplines the mind, accustoms to logical thinking. No wonder they say that mathematics is the gymnastics of the mind.
Every family in which parents are concerned about organizing mental development children and adolescents feel the need for selected material to fill their leisure time with useful, reasonable and not boring mathematical exercises.
It is for this kind of extracurricular activities, conversations and entertainment on a free evening, in the family circle and with friends, or at school at extracurricular meetings, that “Mathematical Ingenuity” is intended - a collection of mathematical miniatures: various tasks, math games, jokes and tricks that require the work of the mind, developing intelligence and the necessary logic in reasoning.
In pre-revolutionary times, the collections of E. I. Ignatiev “In the realm of ingenuity” were widely known. Now they are outdated for our reader and therefore are not republished. Nevertheless, in these collections there are problems that have not yet lost their pedagogical and educational value. Some of them entered the Mathematical Ingenuity unchanged, others with changed or completely new content.
For Mathematical Ingenuity, I also selected and, if necessary, processed problems from among those that are scattered across the pages of extensive domestic and foreign popular literature, trying, however, not to repeat the problems included in Ya. I. Perelman's popular books on entertaining mathematics.
This kind of "small form" mathematical problems sometimes arise as a by-product of a scientist's serious research; many tasks are invented by amateurs, as well as teachers, as special exercises for “mental gymnastics”. They, like riddles and proverbs, usually do not retain their authorship and become public property.
"Mathematical Wits" is intended for readers with a wide variety of degrees. mathematical training:
for a teenager of 10 - 11 years old, making the first attempts at independent thinking;
for a high school student who is passionate about mathematics,
and for an adult who wants to test and exercise his guess.
The systematization of tasks by chapters, of course, is very arbitrary; Each chapter has both easy and difficult tasks.
The book has fifteen chapters.
The first chapter consists of various types of initial exercises of an “intriguing” nature, based on a guess or direct physical actions (experiment), sometimes on simple calculations within the arithmetic of integers (first section of the chapter) and fractional numbers (second section). Somewhat violating the classificatory harmony of the book, I singled out in the first chapter some of the simple problems that thematically belong to subsequent chapters. This is done in the interests of those readers who still find it difficult to independently distinguish a feasible task from an impossible one. Solving different types of tasks in the first chapter in a row, they will be able to try their hand, and then transfer the interest in a particular topic to the corresponding tasks of subsequent chapters.
To solve the problems of the second chapter, one's own mathematical ingenuity and perseverance must overcome all sorts of obstacles and suggest a way out of difficult situations.
The third chapter - "Geometry on matches" - contains a number of geometric problems - puzzles.
The chapter "Try on seven times, cut once" consists of tasks for cutting shapes.
The content of the tasks of the chapter "Skill will find application everywhere" is connected with practical activities, with technology.
The chapter entitled "Mathematics with almost no calculations" contains problems that require a chain of skillful and subtle reasoning to solve.
Games and tricks are collected in a separate chapter, and also placed throughout the book. They contain a mathematical basis and are undoubtedly included in the "realm of ingenuity."
Three chapters: "Cross-sums and magic squares", "Curious and serious in numbers" and "Numbers ancient but eternally young", are devoted to some curious observations on numerical ratios that have accumulated in mathematics from ancient times to our time.
Final chapter- two short essays on the labor ingenuity of the people of our Motherland, workers in the fields and factories.
In various places in the book, the reader is offered small topics for independent research.
At the end of the book are solutions to problems, but you should not rush to look into them.
Any task for "ingenuity" is fraught with some "zest" and in most cases is a tough nut to crack, which is not so easy to crack, but all the more tempting.
If you fail to solve a problem right away, you can temporarily skip it and move on to the next one or to the tasks of another section, another chapter. Return to the missed task later.
"Mathematical Ingenuity" is not a book for easy reading "in one sitting", but for work over perhaps a number of years, a book for regular mental gymnastics in small portions, a reader's companion in his gradual mathematical development.
All the material of the book is subject to an educational and educational goal: to encourage the reader to independent creative thinking, to further improve their mathematical knowledge.
The second edition of Mathematical Wits is not a stereotypical repetition of the first. The required changes have been made to the text and to the solutions of some problems; separate tasks are replaced by new ones - more meaningful; the book has been redesigned.
Great efforts aimed at improving the book were made by the editor of the publishing house M. M. Hot.
Solving problems on their own, readers in some cases found additional or simpler solutions and kindly communicated their results to me. The authors of the most interesting solutions are mentioned in the appropriate places in the book.
I hope to receive feedback and suggestions from the readers of "Smekalka" on further improvement of the book, as well as my own original problems and mathematical materials of folk art.
Address: Moscow, B-64, st. Chernyshevsky, 31, apt. 53, Boris Anastasievich Kordemsky.
B. Kordemsky.

TASKS

"A book is a book, And move your brains"
V. Mayakovsky.

CHAPTER FIRST. FUN CHALLENGES

SECTION I
Test and exercise your ingenuity at first on such tasks, the solution of which requires only purposeful perseverance, patience, quick wits and the ability to add, subtract, multiply and divide whole numbers.

1. Observant pioneers
Schoolchildren - a boy and a girl - have just made meteorological measurements.
Now they are resting on a hillock and watching a freight train go by.
The locomotive on the rise frantically smokes and puffs. Along the canvas railway evenly, without gusts the wind blows.
- What wind speed did our measurements show? the boy asked.
- 7 meters per second.
- Today, this is enough for me to determine how fast the train is going.
- Well, yes, - the girl doubted.
- And you take a closer look at the movement of the train.
The girl thought a little and also realized what was the matter.
And they saw exactly what our artist painted (Fig. 1). What was the speed of the train?
Rice. 1. How fast is the train?

2. "Stone Flower"
Remember the talented "craftsman" master Danila from P. Bazhov's fairy tale "The Stone Flower"?
They say in the Urals that Danila, while still a student, carved two such flowers (Fig. 2), the leaves, stems and petals of which were separated, and from the resulting parts of the flowers it was possible to fold a plate in the shape of a circle.
Try it! Redraw danilina flowers on paper or cardboard, cut out the petals, stems and leaves and fold the circle.

3. Moving checkers
Place 6 checkers on the table in a row alternately - black, white, another black, another white, etc. (Fig. 3).
Rice. 3. White checkers should be on the left, followed by black ones.
Left or right leave free place, sufficient for four checkers.
It is required to move the checkers so that all white ones are on the left, and after them all black ones. At the same time, you need to move two nearby checkers to an empty place at once, without changing the order in which they lie. To solve the problem, it is enough to make three movements (three moves) *).
If you do not have checkers, use coins or cut pieces of paper or cardboard.
*) The theme of this problem is further developed in problems 96 and 97 (pp. 57 and 58).

4. In three moves
Put 3 piles of matches on the table. Put 11 matches in one pile, and 7 in the other, 6 in the third. When shifting matches from any pile to any other, you need to equalize all three piles so that each has 8 matches. This is possible, since the total number of matches - 24 - is divisible by 3 without a remainder; in this case, it is required to observe the following rule: it is allowed to add exactly as many matches to any pile as there are in it. For example, if there are 6 matches in a pile, then only 6 can be added to it, if there are 4 matches in a pile, then only 4 can be added to it.
The problem is solved in 3 moves.

5. Count!
Check your geometric observation: count how many triangles are in the figure shown in fig. four.

6. Way of the gardener
On fig. 5 is a plan of a small apple orchard (points - apple trees). The gardener processed all the apple trees in a row.
Rice. 5. Plan of the apple orchard.
He began from the cell marked with an asterisk, and walked one by one through all the cells, both occupied by apple trees and
free, never returning to the passed cell. He did not walk along the diagonals and was not on the shaded cells, since various buildings were placed there.
Having finished the tour, the gardener ended up on the same square from which he began his journey.
Draw the gardener's path in your notebook.

7. Need to be smart
There are 5 apples in the basket. How to divide these apples among five girls so that each girl gets one apple and one apple remains in the basket?

8. Without hesitation
Tell me, how many cats are in the room, if one cat sits in each of the four corners of the room, 3 cats sit opposite each cat, and a cat sits on the tail of each cat?

9. Down - up
The boy pressed the edge of the blue pencil firmly against the edge of the yellow pencil. One centimeter (in length) of the pressed edge of the blue pencil, counting from the bottom end, is stained with paint. The boy holds the yellow pencil motionless, and the blue one, continuing to press it against the yellow one, lowers it by 1 cm, then returns it to its previous position, lowers it again by 1 cm and again returns to its previous position; 10 times he lowers and raises the blue pencil 10 times (20 movements).
If we assume that during this time the paint does not dry out and does not deplete, then how many centimeters in length will the yellow pencil be soiled after the twentieth movement?
Note. This problem was invented by the mathematician Leonid Mikhailovich Rybakov on the way home after a successful duck hunt. What prompted him to write the problem, you will read on page 387 after you solve the problem.

10. Crossing the river (an old problem)
A small military detachment approached the river through which it was necessary to cross. The bridge is broken and the river is deep. How to be? Suddenly, the officer notices two boys near the shore, having fun in a boat. But the boat is so small that only one soldier or only two boys can cross it - no more! However, all the soldiers crossed the river on this boat. How?
Solve this problem "in your head" or practically - using checkers, matches or something like that and moving them around the table through an imaginary river.

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

13. Chain repair
Do you know what the young master thought about (Fig. 8)? In front of him are 5 links of the chain, which must be connected into one chain without using additional rings. If, for example, you unforge ring 3 (one operation) and hook it on ring 4 (one more operation), then unforge ring 6 and hook on ring 7, etc., then there will be eight operations in total, and the master strives to forge the chain at the help of only six operations. He succeeded. How did he act?

14. Fix the bug
Take 12 matches and lay out of them the "equality" shown in fig. 9.
Rice. 9. Correct the mistake by shifting only one match.
Equality, as you can see, is incorrect, since it turns out that 6 - 4 = 9.
Move one match so that you get the correct equality.

15. Out of three - four (joke)
There are 3 matches on the table.
Without adding a single match, make three to four. You can't break matches.

16. Three yes two - eight (another joke)
Here's another similar joke. You can offer it to your friend.
Put 3 matches on the table and invite a friend to add 2 more to them so that you get eight. Of course, you can't break matches.

17. Three squares
From 8 sticks (for example, matches), four of which are half as long as the other four, you need to make 3 equal squares.

18. In the turning shop of the plant, parts are turned from lead blanks. From one blank - a detail. The shavings resulting from the dressing of six parts can be: melted down and prepared for another blank. How many parts can be made in this way from 36 lead blanks?

19. Try it!
In a square dance hall, place 10 chairs along the walls so that there are equal number of chairs on each wall.

20. Arranging flags
A small inter-kolkhoz hydroelectric power station was built by Komsomol members. By the day of its launch, the pioneers decorate the outside of the power plant on all four sides with garlands, light bulbs and flags. There were few flags, only 12.
The pioneers first placed them 4 on each side, as shown in the diagram (Fig. 10), then they realized that they could place the same 12 flags 5 or even 6 on each side. They liked the second project more, and they decided put 5 checkboxes.
Show on the diagram how the pioneers arranged 12 flags, 5 on each of the four sides, and how they could arrange them 6 flags.

21. Preserve Parity
Take 16 of some objects (paper, coins, plums or checkers) and arrange them 4 in a row (Fig. 11). Now remove 6 pieces, but so that there is an even number of items left in each horizontal and in each vertical row. By removing different 6 pieces, you can get different solutions.

22. "Magic" number triangle
At the vertices of the triangle, I have placed the numbers 1, 2, and 3, and you will place the numbers 4, 5, 6, 7, 8, 9 on the sides of the triangle so that the sum of all the numbers along each side of the triangle is 17. This is not difficult, as I suggested What numbers should be placed at the vertices of the triangle. 2
Much longer will you have to tinker if I do not tell you in advance which numbers should be placed at the vertices of the triangle, and suggest placing the numbers again
1, 2, 3, 4, 5, 6, 7, 8, 9,
each one once, along the sides and at the vertices of the triangle so that the sum of the numbers on each side of the triangle is 20.
When you get the desired arrangement of numbers, look for more and more new arrangements. The conditions of the problem can be fulfilled for a wide variety of arrangements of numbers.

23. How 12 girls played ball
Twelve girls stood in a circle and began to play ball. Each girl threw the ball to her neighbor on the left. When the ball went around the whole circle, it was thrown in the opposite direction. After a while, one girl said:
- Let's better throw the ball through one person.
“But since there are twelve of us, half of the girls will not participate in the game,” Natasha objected vividly.
- Then we will throw the ball through two! (Every third catches the ball.)
- Even worse: only four will play ... If you want all the girls to play, you need to throw the ball through four (the fifth catches). There is no other combination.
- And if you throw the ball through six people?
- It will be the same combination, only the ball will go in the opposite direction.
- And if you play in ten (every eleventh catches the ball)? the girls asked.
We've already played this way...
The girls began to draw diagrams of all the proposed ways of playing and very soon became convinced that Natasha was right. Only one scheme of the game (except the initial one) covered all the participants without exception (Fig. 13, a).
Now, if there were thirteen girls playing, the ball could be thrown through one (Fig. 13, b), and through two (Fig. 13, c), and through three (Fig. 13, d), and through four (Fig. 13, e), and every time the game would cover all the participants. Find out if, with thirteen players, it is possible to throw the ball through five people?
Is it possible to throw the ball through six people with thirteen players? Think and draw the appropriate diagrams for clarity.

24. Four straight lines
Take a sheet of paper and draw ca Fig. 14. It has nine points so that they are arranged in the shape of a square, as shown in fig. 14. Now cross out all the dots with four straight lines, without lifting the pencil from the paper.

25. Separate the goats from the cabbage
Now solve a problem that is in some sense the opposite of the previous one. There we connected the points with straight lines, and here we need to draw 3 straight lines so as to separate the goats from the cabbage (Fig. 15). Straight lines should not be drawn in the drawing of the book.
Redraw the layout of the goats and cabbages in your notebook and then try to solve the problem. You can not draw lines at all, but use knitting needles or thin wires.

26. Two trains
The fast train left Moscow for Leningrad and went non-stop at a speed of 60 kilometers per hour. Another train came out to meet him from Leningrad to Moscow and also went non-stop at a speed of 40 kilometers per hour.
How far will these trains be 1 hour before they meet?

27. At high tide (joke)
Not far from the coast there is a ship with a rope ladder launched along the side. The stairs have 10 steps; the distance between the steps is 30 cm. The lowest step touches the surface of the water. The ocean is very calm today, but the tide is coming in and lifting
There were two numbers, and water for every hour by 15 cm. After how long will the third step of the rope ladder be covered with water?

28. Dial
a) Divide the clock face by two straight lines into three parts so that, by adding the numbers, in each part you get the same amount.
b) Can this dial be divided into 6 parts so that in each part the sums of these two numbers in each of the six parts would be equal to each other?

29. Broken dial
In the museum, I saw an old clock with Roman numerals on the dial, and instead of the familiar number four (IV), there were four sticks (IIII). The cracks formed on the dial divided it into 4 parts, as shown in Fig. 17. The sums of numbers in each part were not the same: in one - 21, in the other - 20, in the third - 20, in the fourth - 17.
I noticed that with a slightly different arrangement of cracks, the sum of the numbers in each of the four parts of the dial would be 20. With a new arrangement of cracks, they may not pass through the center of the dial. Redraw the clock face in your notebook and find this new location of cracks.
Rice. 17. Cracks divided the dial into 4 parts.

30. Amazing clock (Chinese puzzle)
Once, a watchmaker was urgently asked to come into one house.
- I'm sick, - answered the watchmaker, - and I can't go. But if the repair is simple, I will send you my apprentice.
It turned out that it was necessary to replace the broken arrows with others.
“My apprentice can handle this,” said the master. - He will check the mechanism of your watch and select new hands for it.
The apprentice did his work very diligently, and by the time he finished examining the clock, it was already dark. Considering the work completed, he hastily put on the picked up hands and put them on his watch: a large hand on the number 12, and a small one on the number 6 (it was exactly 6 pm).
But shortly after the apprentice returned to the tinkering room to inform the foreman that the job was done, the phone rang. The boy picked up the phone and heard the angry voice of the customer:
- You fixed the clock badly, it shows the time incorrectly.
The master's apprentice, surprised by this message, hurried to the customer. When he arrived, the clock he had repaired showed the beginning of the ninth. The student took out his pocket watch and handed it to the angry owner of the house:
- Check, please. Your clock is never behind.
The stunned customer was forced to agree that his watch was in this moment really shows the correct time.
But the next morning, the customer called again and said that the hands of the clock, obviously, had gone crazy and walked around the dial as they pleased. The master's apprentice ran to the customer. The clock showed the beginning of the eighth. Checking the time on his watch, he was seriously angry:
- You are laughing at me! Your clock shows the exact time!
The clock really showed the exact time. The indignant disciple of the master wanted to leave immediately, but the master restrained him. And after a few minutes, they found the cause of such incredible incidents.
Haven't you guessed what's going on here?

31. Three in a row
Arrange 9 buttons on the table in the shape of a square, 3 buttons on each side and one in the center (fig. 18). Note that if there are two or more buttons along any straight line, then we will always call such an arrangement a “row”. So, AB and CD are rows, each of which has 3 buttons, and EF is a row containing two buttons.
Rice. 18. How many rows are there?
Determine how many rows of 3 buttons each are in the picture and how many such rows, each of which has only 2 buttons.
Now remove any 3 buttons and arrange the remaining 6 in 3 rows so that there are 3 buttons in each row.

32. Ten rows
It is easy to guess how to arrange 16 checkers in 10 rows of 4 checkers in each row. It is much more difficult to arrange 9 checkers in 10 rows so that there are 3 checkers in each row.
Solve both problems.

33. Location of coins
On a sheet of blank paper, draw the figure shown in Fig. 19, while increasing its size by 2-3 times, and prepare 17 coins of the following denomination:
20 kopecks - 5 pieces,
15 kopecks - 3 pieces,
10 kopecks - 3 pieces,
5 kopecks - 6 pieces.
Rice. 19. Arrange the coins on the squares of this figure.
Arrange the prepared coins on the squares of the drawn figure so that the sum of the kopecks along each straight line shown in the figure is 55.

34. From 1 to 19
In nineteen circles of fig. 20 is required to arrange 19 so that the sum of the numbers in any three circles lying on the same straight line is equal to 30.

35. Fast but careful
Solve the following 4 problems “at speed” - who will give the correct answer faster:

Task 1. At noon, a bus with passengers leaves Moscow for Tula. An hour later, a cyclist leaves Tula for Moscow and rides along the same highway, but, of course, much slower than the bus.
When the bus passengers and the cyclist meet, which of them will be further from Moscow?
Problem 2. What is more expensive: a kilogram of hryvnias or half a kilogram of two hryvnias?
Problem 3. At 6 o'clock the wall clock struck 6 strokes. I noticed from my pocket watch that the time elapsed from the first strike to the sixth was exactly 30 seconds.
If it took the clock 30 seconds to strike 6 times, how long will the clock continue to strike at noon or at midnight, when the clock strikes 12 times?
Task 4. 3 swallows flew out from one point. When will they be on the same plane?

Now, with calm reasoning, check your decisions and look at the "Answers" section.
- Well, how? Have you fallen into those little traps that are contained in these simple tasks?
Such tasks are attractive because they sharpen attention and teach to be careful in the usual train of thought.
all integers from 1 to
Rice. 20. Fill in the circles with numbers from 1 to 19.

36. Curly Cancer
Figured cancer, shown in Fig. 21, composed of 17 pieces.
Fold two figures at once from the pieces of this cancer: a circle and a square next to it.

37. The cost of the book
For the book they paid 1 ruble and another half of the cost of the book. How much does a book cost?

38. Restless fly
On the highway Moscow - Simferopol, two athletes simultaneously started a training bike ride towards each other.
At that moment, when only 300 km remained between the cyclists, the fly became very interested in the mileage. Having flown off the shoulder of one cyclist and ahead of him, she rushed towards another. Having met the second cyclist and making sure that everything was safe, she immediately turned back. The fly flew to the first athlete and again turned to the second.
So she flew between approaching cyclists until the cyclists met. Then the fly calmed down and sat down on one of them on the nose.
The fly flew between the cyclists at a speed of 100 km per hour, and the cyclists all this time were traveling at a speed of 50 km per hour.
How many kilometers did the fly fly?

39. Less than 50 years later
Will there be such a year in this century that if it is written in numbers, and the paper is turned upside down, then the number formed on the turned paper will express the same year?

40. Two jokes
First joke. Dad called his daughter, asked her to buy some of the things he needed for his departure, and said that the money was in an envelope on his desk. The girl, glancing briefly at the envelope, saw the number 98 written on it, took out the money and, without counting it, put it in
bag, and crumpled the envelope and threw it away.
In the store she bought things for 90 rubles, and when she wanted to pay off, it turned out that not only did she not have eight rubles left, as she expected, but she even lacked four rubles.
At home, she told her father about this and asked if he had made a mistake when he counted the money. The father replied that he counted the money correctly, but she herself made a mistake and, laughing, pointed out to her the mistake. What was the girl's mistake?

Second joke. Prepare 8 pieces of paper with the numbers 1, 2, 3, 4, 5, 7, 8 and 9 and arrange them in two columns as in fig. 22.
By moving only two pieces of paper, ensure that the sums of the numbers in both columns are the same.
Rice. 22. Equalize unequal amounts.

41. How old am I?
When my father was 31, I was 8 years old, and now my father is twice my age. How old am I now?

42. Rate "at a glance"
You have two columns of numbers:
123456789 1
12345678 21
1234567 321
123456 4321
12345 54321
1234 654321
123 7654321
12 87654321
1 987654321
Take a closer look: the numbers of the second column are formed from the same numbers as the numbers of the first column, but with the opposite order of their arrangement. (For clarity, the zeros in the left column have been omitted.)
Which column, when added together, will give the greater result?
First compare these sums "at a glance", that is, without adding yet, try to determine whether they should be the same or whether one should be greater than the other, and then check by addition.

43. Speed ​​addition
Eight six-digit terms (...) are selected in such a way that, by reasonably grouping them, you can “in your mind” find the sum in 8 seconds. Can you handle this speed?
There are instructions in the "Answers" section, but ... you will look for them longer.
And show your friends two tricks, which you can also jokingly call "speed addition."

First focus. Say: “Without showing me, write as many multi-digit numbers as you like in a column. Then I’ll come], I’ll write the same number of numbers very quickly and add them all up instantly. ”
Let's say friends wrote:
7621
3057
2794
4518
And you assign such numbers, each of which complements up to 9999 one by one all the written numbers. These numbers will be:
5481
7205
6942
2378
Really: (...)
Now it is not difficult to figure out how to quickly calculate the entire amount: (...)
It is necessary to take 9999 4 times, that is, 9999X4, and such a multiplication is quickly done in the mind. Multiply 10,000 by 4 and subtract the extra 4 units. It turns out:
10,000 X 4 - 4 = 40,000 - 4 = 39,996.
That's the whole trick secret!

Second focus. Write one under the other any 2 numbers of any size. I will add the third and instantly, from left to right, I will write the sum of all three numbers.
Let's say you wrote:
72 603 294
51 273 081
I will assign, for example, the following number: 48 726 918 and immediately tell you the amount.
What number should be attributed and how to quickly find the sum in this case, figure it out for yourself!

44. In which hand? (math trick)
Give your friend two coins: one with an even number of kopecks and the other with an odd number (for example, two kopecks and three kopecks). Let him, without showing you, take one of these coins (any) in his right hand, and the second in his left. You can easily guess which hand he has which coin.
Invite him to triple the number of kopecks contained in the coin held in his right hand and double the number of kopecks contained in the coin held in his left hand. Let him add up the results and tell you only the resulting amount.
If the amount named is even, then there are 2 kopecks in the right hand, if it is odd, then 2 kopecks in the left hand.
Explain why it always works out this way, and think of ways to diversify this trick.

45. How many are there?
A boy has as many sisters as brothers, and his sister has half as many sisters as brothers.
How many brothers and sisters are in this family?

46. ​​Same numbers
Using only addition, write the number 28 with five twos, and the number 1000 with eight eights.

47. Hundred
Using any arithmetic operations, make the number 100 either from five ones or from five fives, and from five fives, 100 can be made in two ways.

48. Arithmetic duel
At one time there was such a custom in the mathematical circle of our school. To each new member of the circle, the chairman of the circle offered a simple task - a sort of mathematical nut. If you solve the problem, you immediately become a member of the circle, and if you don’t cope with the nut, then you can visit the circle as an auditor.
I remember once our chairman suggested to one newcomer Vitya the following task: (...)

49. Twenty
From four odd numbers it is easy to make a sum equal to 10, namely:
1 + 1+3 + 5=10,
or like this:
1 + 1 + 1+7 = 10.
A third solution is also possible:
1 + 3 + 3 + 3= 10.
There are no other solutions (changes in the order of the terms, of course, do not form new solutions).
The following problem has much more different solutions:
Compose the number 20 by adding exactly eight odd numbers, among which it is also allowed to have the same terms.
Find all different solutions to this problem and determine how many of them will be such sums that contain the largest number of unequal terms?
Little advice. If you pick numbers at random, you will still come up with several solutions, but random trials will not give you confidence that you have exhausted all the solutions. If, however, you introduce some order, a system into the “method of trials”, then not one of the possible solutions will escape you.

50. How many routes?
From a letter from schoolchildren: “While studying in a mathematical circle, we drew a plan of sixteen quarters of our city. On the attached scheme of the plan (Fig. 23), all quarters are conventionally depicted as identical squares.
We are interested in the following question:
How many different routes can be planned from point A to point C if we move along the streets of our
Rice. 23. How many routes lead from L to S?
cities only forward and to the right, to the right and forward? The routes may coincide in their separate parts (see dotted lines on the plan diagram).
We have the impression that this is not an easy task. Did we solve it correctly if we counted 70 different routes?”
What should be the answer to this letter?

52. Different actions, one result
If between two twos the sign of addition is replaced by the sign of multiplication, then the result will not change. Indeed: 2+ 2 = 2X2. It is easy to pick and 3 numbers with the same property, namely: 1+2 + 3 = = 1X2X3. There are also 4 single-digit numbers that, when added or multiplied by each other, give the same result.
Who will pick up these numbers faster? Ready? Keep up the competition! Find 5, and then 6, then 7, and so on, single-digit numbers that have the same property. Keep in mind that, starting with a group of 5 numbers, the answers may be different.

53. Ninety-nine and one hundred
How many plus signs (+) do you need to put between the digits of 987654321 to add up to 99?
Two solutions are possible. Finding at least one of them is not easy, but you will gain experience that will help you quickly place the plus signs between the seven numbers 1 2 3 4 5 6 7 so that the total is 100. (The location of the numbers is not allowed to be changed). A schoolgirl from Kemerovo claims that two solutions are possible here too.

54. Demountable chessboard
The cheerful chess player cut his cardboard chessboard into 14 pieces, as shown in fig. 25. It turned out a collapsible chessboard. To the comrades who came to him to play chess, he first offered a puzzle: to make a chessboard from these 14 parts. Cut out the same figures from checkered paper and see for yourself whether it is difficult or easy to make a chessboard out of them.

60. Perplexed driver
What did the driver think when he looked at the speedometer of his car (Fig. 29)? The counter showed the number 15951. The driver noticed that the number of kilometers traveled by the car was expressed as a symmetrical number, that is, one that was read the same way both from left to right and from right to left:
15951.
- Interesting! .. - the driver muttered. - Now, probably not soon, another number will appear on the counter, which has the same feature.
However, exactly 2 hours later the counter showed a new number, which was also read the same in both directions.
Determine how fast the driver was driving during these 2 hours?

61. For the Tsimlyansk hydroelectric complex
In fulfilling an urgent order for the manufacture of measuring instruments for the Tsimlyansk hydroelectric complex, a team of excellent quality took part, consisting of a foreman - an old, experienced worker - and 9 young workers who had just graduated from a vocational school.
During the day, each of the young workers mounted 15 devices, and the foreman - 9 devices more than the average of each of the 10 members of the team.
How many measuring instruments were installed by the team in one working day?

62. Delivery of bread on time
Starting the delivery of grain to the state, the board of the collective farm decided to deliver a train with grain to the city exactly by 11 o'clock in the morning. If the cars drive at a speed of 30 km / h, then the convoy will arrive in the city at 10 am, and if at a speed of 20 km / h, then at 12 noon.
How far from the collective farm to the city and at what speed should you drive to arrive just in time?

63. In the suburban train
In an electric train car, two schoolgirl friends were traveling from the city to the dacha.
- I notice, - said one of her friends, - that we meet the return suburban trains every 5 minutes. How many suburban trains do you think arrive in the city in one hour if the speeds of trains in both directions are the same?
- Of course, 12, since 60:5 = 12, - said the second friend.
But the schoolgirl who asked the question did not agree with her friend's decision and gave her her thoughts.
What do you think about this?

65. Football fan's nightmare
The “fan”, upset by the defeat of “his” team, slept restlessly. He dreamed of a large square room without furniture. The goalkeeper was training in the room. He kicked the soccer ball against the wall and then caught it.
Suddenly, the goalkeeper began to shrink, shrink, and finally turned into a small celluloid ball from "table tennis", and the soccer ball turned out to be a cast-iron ball. The ball swirled wildly across the smooth floor of the room, trying to crush the small celluloid ball. The poor ball in desperation rushed from side to side, exhausted and unable to bounce.
Could he, without leaving the floor, still hide somewhere from the persecution of the cast-iron ball?
Rice. 30. The ball tried to crush the ball.
To solve the problems of the second section, familiarity with operations on simple and decimal fractions is required.
The reader who has not yet studied fractions can temporarily skip the problems in this section and move on to the following chapters.

66. Clock
Traveling through our great and wonderful Motherland, I found myself in such places where the difference in air temperatures day and night is so great that when I spent days and nights in the open air, this began to affect the course of the clock. I noticed that the temperature changes during the day made the clock go ahead by 1 minute, and during the night they were behind by 1 minute.
On the morning of May 1, the clock still showed the correct time. By what date will they be 5 minutes ahead?

67. Staircase
The house has 6 floors. Tell me, how many times is the path up the stairs to the sixth floor longer than the path along the same stairs to the third floor, if the spans between floors have the same number of steps?

68. Puzzle
What sign should be placed between the numbers 2 and 3 written next to each other to get a number greater than two, but less than three?
69. Interesting fractions
If the denominator of 1/3 is added to the numerator and denominator, the fraction will double.
Find a fraction that, by adding the denominator to its numerator and denominator, would increase: a) three times, b) four times.
(Algebraic people can generalize the problem and solve it with an equation.)

70; What number?
Half past two. What is this number?

71. Way of a schoolboy
Borya does a pretty good job every morning. a long way to school.
At a distance from the house to the school there is an MTS building with an electric clock on the facade, and at a distance from the entire path there is a railway station. When he passed the MTS, it was usually 7:30 on the clock, and when he reached the station, the clock showed 25 minutes to 8:00.
When did Borya leave the house and at what time did he come to school?

72. At the stadium
12 flags are placed along the treadmill at equal distances from each other. Start at the first flag. The athlete was at the eighth flag 8 seconds after the start of the run. After how many seconds at a constant speed will he be at the twelfth flag? Don't get in trouble!

73. Did you win?
Ostap was returning home from Kyiv. He traveled the first half of the journey by train 15 times faster than if he walked. However, he had to drive the second half of the way on oxen - 2 times slower than if he walked.
Did Ostap gain any time compared to walking?

74. Alarm clock
The alarm clock is 4 minutes behind. in hour; 3.5 hours ago it was delivered exactly. Now the clock showing the exact time is exactly 12.
In how many minutes will the alarm clock also show 12?

75. Instead of small shares, large
There is a very exciting profession in machine-building factories; It's called the scriber. The scriber marks on the workpiece those lines along which this workpiece should be processed in order to give it the necessary shape.
The scriber has to solve interesting and sometimes difficult geometric problems, perform arithmetic calculations, etc.
It was necessary to somehow distribute 7 identical rectangular plates in equal shares between 12 parts. They brought these 7 records to the scriber and asked him, if possible, to mark the records so that none of them had to be crushed into very small pieces. This means that the simplest solution - cutting each record into 12 equal parts - was not good, since this resulted in many small parts. How to be?
Is it possible to divide these records into larger parts? The scaler thought, made some arithmetic calculations with fractions and nevertheless found the most economical way to divide these plates.
Subsequently, he easily crushed 5 plates to distribute them in equal shares among six parts, 13 plates for 12 parts, 13 plates for 36 parts, 26 for 21, etc.
How did the spreader do it?

76. Bar of soap
A bar of soap is placed on one scale pan, and another kg of the same bar on the other. Scales in balance.
How much does the bar weigh?

79. Misha's kittens
If Misha sees an abandoned kitten somewhere, he will certainly pick it up and bring it home. He always brought up several kittens, and he did not like to say exactly how many, so that they would not laugh at him.
Sometimes they ask him:
- How many kittens do you have now?
“A little,” he replies. - Three quarters of their number, and even three quarters of one kitten.
The comrades thought he was just joking. Meanwhile, Misha asked them a problem that was not difficult to solve at all. Try!

80. Medium speed
Half the way the horse walked empty at a speed of 12 km / h. She walked the rest of the way with a cart, making 4 km / h.
What is the average speed, that is, at what constant speed would the horse have to move in order to use the same amount of time for the whole journey?

81. Sleeping passenger
When the passenger traveled half of the entire journey, he went to bed and slept until there was no more left - to travel half the distance that he had traveled sleeping. How much of the entire journey did he travel sleeping?

82. What is the length of the train?
Two trains go towards each other on parallel tracks; one at a speed of 36 km/h, the other at a speed of 45 km/h. A passenger sitting on the second train noticed that the first train had passed him for 6 seconds. What is the length of the first train?

83. Cyclist
When the cyclist drove 2/3 of the way, the tire burst.
On the rest of the journey, he spent twice as much time on foot as on a bicycle ride.
How many times did the cyclist ride faster than he walked?

84. Competition
Turners Volodya A. and Kostya B. - students of the vocational school of metal workers, having received from the master the same outfit for the manufacture of a batch of parts, wanted to complete their tasks at the same time and ahead of schedule.
After some time, however, it turned out that Kostya had done only half of what Volodya had left to do, and Volodya had only half of what he had already done left to do.
By how many times would Kostya now have to increase his daily output compared to Volodya in order to complete his task at the same time?

Chapter Two
CONFIDENTIAL PROVISIONS

87. Wits of Blacksmith Hecho
Traveling in Georgia last summer, we sometimes amused ourselves by inventing all sorts of extraordinary stories inspired by some ancient monument.
Once we came to a lonely ancient tower. Examined her, sat down to rest. And there was a mathematics student among us; he immediately came up with an interesting problem:
“300 years ago, an evil and arrogant prince lived here. The prince had a daughter-bride, Darijan by name. The prince promised his Darijan as a wife to a rich neighbor, and she fell in love with a simple guy, the blacksmith Khecho. Darijan and Khecho tried to escape into the mountains from captivity, but their servants Knyazevs caught them.
The prince became furious and decided to execute both the next day, but for the night he ordered them to be locked up in this tall, gloomy, abandoned, unfinished tower, and with them also the maid Darijan, a teenage girl who helped them escape.
He was not at a loss in the Hecho tower, looked around, climbed the stairs to the upper part of the tower, looked out the window - it is impossible to jump, you will break. Then Hecho noticed near the window a rope forgotten by the builders, thrown over a rusty block, reinforced higher.
window. Empty baskets were tied to the ends of the rope, and a basket to each end. Hecho recalled that with the help of these baskets, masons lifted bricks up and lowered rubble down, and if the weight of the load in one basket exceeded the weight of the load in the other by about 5-6 kg (translated into modern measures), then the basket fell rather smoothly to the ground; another basket at that time was going up to the window.
Hecho determined by eye that Darijan weighs about 50 kg, the maid no more than 40 kg. Hecho knew his weight - about 90 kg. In addition, he found a chain weighing 30 kg in the tower. Since a person and a chain or even 2 people could fit in each basket, all three of them managed to descend to the ground, and they descended in such a way that the weight of the lowering basket with a person never exceeded the weight of the rising basket by more than 10 kg.
How did they get out of the tower?

88. Cat and mice
Purr's cat has just "helped" his young owner solve problems. Now he sleeps sweetly, and in a dream he sees himself surrounded by thirteen mice. Twelve mice are gray and one is white. And the cat hears, someone says in a familiar voice: “Purr, you must eat every thirteenth mouse, counting them in a circle all the time in the same direction, so that the last white mouse is eaten.”
But which mouse to start with in order to solve the problem correctly?
Help Purr.

89. Matches around a coin
Let's replace the cat with a coin, and the mice with matches. It is required to remove all matches, except for the one that faces the coin (Fig. 35), observing the following condition: first remove one match, and then, moving to the right in a circle, remove every thirteenth match.
Think about which match you need to remove first.

90. The lot fell on the siskin and the robin
At the end of the summer camp period, the pioneers decided to release the feathered inhabitants of fields and groves caught by young birders. There were 20 birds in total, each in a separate cage. The leader suggested the following:
- Put all the cages with birds in one row and, starting from left to right, open every fifth cage. Having reached the end of the row, transfer the score to the beginning of the row, but open cells do not count any more, and so continue until all the cells are open, except for some of the last two. Birds in these cages can be taken with you to the city.
The offer was accepted.
Most of the children did not care which two birds to take away with them (if it was already impossible to take all of them), but Tanya and Alik wanted the lot to fall without fail on the siskin and the robin. When they helped arrange the cells in a row, they remembered the cat and mice problem (problem 88). They quickly figured out where to place the cages with the siskin and the robin so that these particular cages would remain unopened, and put them on ...
However, you can easily determine for yourself where Tanya and Alik put the cages with the siskin and the robin.

91. Spread coins
Prepare 7 matches and 6 coins. Arrange matches on the table with an asterisk, as shown in fig. 36. Starting from any match, count the third by the movement of the clock hand and put a coin near its head. Then again count the third match in the same direction, starting from any match against which there is not yet a coin, and also put a coin near the head.
Proceeding in this way, try to arrange all 6 coins near the heads of six matches. When counting matches, one should not skip those near which a coin has already been placed;
it is necessary to start the countdown with a match that does not have a coin near it; Do not put two coins in one place.
What rule should be followed in order to surely solve the problem?

92. Skip the passenger!
At the half-station of a single-track railway, a train consisting of a steam locomotive and five wagons stopped, delivering a team of workers for the construction of a new branch. So far, at this stop there was only a small dead end, in which, if necessary, a steam locomotive with two cars could hardly fit.
Rice. 37. How to skip the passenger?
Soon after the train with the construction team, a passenger train approached the same half-station.
How to skip the passenger?

93. A problem that arose from the caprice of three girls
The topic of this problem has a respectable prescription. Three girls, each with their dad, were walking. All six approached a small river and wished to cross from one side to the other. At their disposal was only one boat without a rower, raising only two people. The crossing would, of course, not be difficult to carry out if the girls had not declared, either out of a whim, or out of prank, that not one of them would agree to ride in a boat, or be on the shore with one or two other people's dads without their dad. The girls were small, but not very small, so that each of them could drive the boat on their own.
Thus, unexpectedly additional terms crossings, but for the sake of fun, the travelers decided to try to complete them. How did they act?

94. Further development of the problem
Funny company safely crossed to the opposite bank of the river and sat down to rest. The question arose: would it be possible under the same conditions to organize the crossing of four couples? It soon became clear that if the conditions put forward by the girls were preserved (see the previous problem), the crossing of four couples could be carried out only if there was a boat that could lift three people, and in just 5 steps.
How?
Developing the theme of the problem even further, our travelers found that even on a boat that can accommodate only two people, it is possible to cross four girls with their dads from one bank to another, if there is an island in the middle of the river where you can make an intermediate stop and disembark. In this case, for the final crossing, at least 12 crossings are required, subject to the same condition, that is, that not a single girl will be in a boat, or on an island, or on the shores with someone else's dad without her dad.
Find this solution too.

95. Jumping Checkers
Place 3 white checkers on squares 1, 2, 3 (fig. 38), and 3 black checkers on squares 5, 6, 7. Using the free square 4, move the white checkers to the place of the black ones, and the black ones to the place of the white ones; at the same time, adhere to the following rule: checkers can be moved to an adjacent free square; it is also allowed to jump over an adjacent checker if there is a free square behind it. White and black checkers can move towards each other. Moves in the opposite direction are not allowed. The problem is solved in 15 moves.

96. White and black
Take four white and four black checkers (or 4 copper and 4 silver coins) and put them on the table in a row, alternating colors: white, black, white, black, and so on. On the left or right, leave such free space that could fit no more than 2 checkers (coins). Using free space, you can mix each time only two adjacent checkers (coins), without changing their relative position.
It is enough to make 4 such movements of pairs of checkers so that all black checkers are in a row, followed by all white checkers.
Check it out!

97. Complicating the task
With an increase in the number of initially taken checkers (coins), the task becomes more complicated.
So, if you place 5 white and 5 black checkers in a row, alternating their color, it will take 5 moves to arrange black checkers with black, and white checkers with white.
In the case of six pairs of checkers, 6 moves will be required; in the case of seven pairs - 7 moves, etc. Find solutions to the problem for five, six and seven pairs of checkers.
Remember that during the initial layout of the checkers, you should leave free space on the left (or right) for no more than two checkers and move 2 checkers each time without changing their relative position.

98. Cards are stacked in numerical order
Cut out 10 cards of 4X0 si from cardboard and number them with numbers from 1 to 10. Having stacked the cards, take them in your hand. Starting with the top card, place the first card on the table, the second under the bottom of the pile, the third card on the table, the fourth under the bottom of the pile. Do this all the time until you put all the cards on the table.
We can say with confidence that the cards will not be in numerical order.
Think about the sequence in which you need to initially put the cards in a pile so that, with the specified layout, they are arranged in the order of numbers from 1 to 10.

99. Two location puzzles
First puzzle. Twelve checkers (coins, pieces of paper, etc.) are easy to arrange on the table in the form of a square frame with 4 checkers along each side. But try to place these checkers so that there are 5 of them along each side of the square.
Second puzzle. Arrange 12 checkers on the table so that 3 rows are formed horizontally and 3 rows vertically, and so that each of these rows contains 4 checkers.

100. Mysterious box
Misha spent the summer in Artek and brought back a beautiful box decorated with 36 shells as a gift to his younger sister Irochka. Lines are burnt on the lid of the box so that they divide the lid into 8 sections.
Irochka doesn't go to school yet, but she can count up to 10. What she liked most about Misha's gift was that there were exactly 10 shells along each side of the lid of the box (Fig. 40). Counting the shells along the side, Irochka takes into account all the shells located in the section adjacent to this side. Shells located in the corner sections, Irochka counts on both sides.
Once my mother, wiping the box with a cloth, accidentally crushed 4 shells. Now there are no more 10 shells along each side of the lid. What a nuisance! Ira will come from kindergarten and very upset.
Rice. 40. Along each side of the lid of the box - 10 shells.
Rice. 39. How to put these checkers 5 on each side?
- The trouble is not great, - Misha reassured his mother.
He carefully peeled off part of the shells from the remaining 32 and so skillfully pasted them back onto the lid of the box that there were again 10 shells along each side of it.
Several days have passed. Trouble again. The box fell, 6 more shells broke; only 26 of them remained. But this time, too, Misha figured out how to arrange the remaining 26 shells on the lid so that Irochka would still have 10 shells along each side. True, the remaining shells in the latter case could not be distributed on the lid of the box as symmetrically as they had been arranged before, but Irochka paid no attention to this.
Find both Mishina's solutions.

101. Brave "garrison"
The snow fortress is protected by a brave "garrison". The guys repelled 5 assaults, but did not give up. At the beginning of the game, the "garrison" consisted of 40 people. The "commandant" of the snow fortress initially placed forces according to the scheme shown in the square box on the right (in the central square - the total number of the "garrison").
The "enemy" saw that each of the 4 sides of the fortress was defended by 11 people. According to the terms of the game, during the first, second, third and fourth assaults, the “garrison” “lost” 4 people each time. In the last, fifth, assault, the “enemy” disabled two more people with their snowballs. And yet, despite the losses, after each assault, either side of the snow fortress continued to be defended by 11 people.
How did the "commandant" of the snow fortress place the forces of his garrison after each assault?

104. Preparation for the holiday
The geometric meaning of the previous five tasks was to arrange objects along four straight lines (sides of a rectangle or square) in such a way that the number of objects along each straight line remained the same when their total number changed.
This arrangement was achieved due to the fact that all objects located at the corners were considered as if belonging to each of the sides of the corner, just as the point of intersection of two lines belongs to each of them.
If we assume that each of the objects placed on the sides of the figure occupies a certain point on the corresponding side, then all objects located at the corners must be imagined concentrated at one point (at the top of the corner).
Let us now refuse the possibility of even an imaginary accumulation of objects in one geometric point.
We will assume that each individual object (pebble, light bulb, tree, etc.) from among those located on a certain plane occupies a separate point of this plane, and we will not limit ourselves to the requirement to place these objects only along four straight lines.
lines. If these conditions are supplemented with the requirement that the solution be symmetrical in some sense, then the problems of placing objects along straight lines will acquire additional geometric interest. The solution of such problems usually leads to the construction of some geometric figure.
For example, how could you beautifully arrange 10 light bulbs in 5 rows of 4 light bulbs in each row when making a festive illumination?
The answer to this question is given by the five-pointed star shown in Fig. 44.
Practice solving similar problems; try to achieve symmetry in the desired location.
Problem 1. How to arrange 12 light bulbs in 6 rows of 4 light bulbs in each row? (This problem has two solutions.)
Task 2. Plant 13 decorative bushes in 12 rows of 3 bushes in each row.
Problem 3. On a triangular site (Fig. 45), the gardener has grown 16 roses arranged in 12 straight rows of 4 roses in each row. Then he prepared a flower bed and transplanted all 16 roses there in 15 rows of 4 roses each? How did he do it?
Task 4. Arrange 25 trees in 12 rows of 5 trees in each row.
Rice. 44. 5 rows of 4.
Rice. 45. How to do 15 rows of 4.

105. Seating oak trees differently
Beautifully planted 27 oak trees according to the scheme shown
in fig. 46, in 9 rows with 6 oak trees in each row, but the arborist would undoubtedly reject such a layout. The oak needs the sun only from above, and on the sides so that there is greenery.
He loves, as they say, to grow up in a fur coat, but without a hat, and then 3 oak trees jumped off somewhere to the side and stick out alone!
Try to plant these 27 oaks in a different way, also in 9 rows and also 6 oaks in each row, but so that all the trees are arranged in three groups and not from their own group; save and
none of them rebounded symmetry in arrangement.

109. Puzzle gift
There is such a toy: a box; you open it, and inside there is still a box; you open it, there is a box inside again.
Make such a toy out of four boxes. Put 4 candies in the smallest inner box, 4 candies in each of the next two boxes, and 9 candies in the largest one.
Thus, 21 candies will be placed in four boxes (Fig. 53).
Give this box of candies to your friend on his birthday with the condition that he does not eat candies until the “anniversary” redistributes 21 candies so that each box contains an even number of pairs of candies and one more.
Of course, before making this gift, you yourself must “bite through” this puzzle. Keep in mind that no rules of arithmetic will help here, you just need to be smart and have a little wit.

110. Knight's move
You don't need to know how to play chess to solve this fun chess puzzle. It is enough to know how the knight's piece moves on the board. Black pawns are placed on the chessboard (see diagram in Fig. 54). Place the white knight on any free square you want chessboard in such a way that this knight could remove all black pawns from the board, while making the least possible number of knight moves.

113. Eight stars
In one of the white cells in Fig. 57 I put an asterisk.
Place 7 more stars in white cells so that no 2 stars (out of eight) are on the same horizontal or vertical, or any diagonal.
To solve the problem, of course, it is necessary by trials, so the additional interest of the problem is also to introduce a known system into the process of necessary tests.

114. Two problems for the placement of letters
First task. In a square divided into 16 equal squares, arrange 4 letters so that in each horizontal row, in each vertical row and in each of the two diagonals of the large square, there is only one letter. How large is the number of solutions to this problem in the case when the placed letters are the same, and in the case when they are different?
Second task. In a square divided into 16 equal squares, arrange 4 times each of the four letters a, b, c and d so that there are no identical letters in each horizontal row, in each vertical row and in each of the two diagonals of the large square. How large is the number of solutions to this problem?

115. Layout of colorful squares
Prepare 16 squares of the same size, but four different colors, let's say white, black, red and green - 4 squares of each color. You have four sets of multi-colored squares. On each square of the first set write the number 1, on each square of the second set - 2, on the squares of the third set - 3 and on the squares of the fourth - 4.
It is required to arrange these 16 multi-colored squares also in the form of a square, and in such a way that in each horizontal row, in each vertical row and in each of the two diagonals there are squares with the numbers 1, 2, 3 and 4 in any arbitrary order and, moreover, without fail different colours.
The problem admits a lot of solutions. Think about a system for obtaining required locations.

119. Joke problem
Kolya Sinichkin, a 4th grade student of a secondary school, is diligently trying to move the chess knight from the lower left corner of the chessboard (from field a \) to the upper right corner (on field h8) so that the knight visits each square of the board once. Until he succeeds. But is he trying to solve an unsolvable problem?
Understand this theoretically and explain to Kolya Sinichkin what is the matter here.

120. One hundred and forty-five doors (puzzle)
Medieval feudal lords sometimes turned the cellars of their castles into prisons - labyrinths with all sorts of tricks and secrets: with sliding cell walls, secret passages, various traps.
You look at such an old castle and involuntarily there is a desire to dream up.
Imagine that in one of these cellars, the plan of which is shown in Figure 62, a man is thrown from those who fought against the feudal lord. Imagine such a secret in the construction of this basement. Of the 145 doors, only 9 are locked (they are indicated in Fig. 62 by bold stripes), and all the rest are wide open. It seems so easy to walk up to the door that leads outside and try to open it. It wasn't there. It is impossible to open a locked door, but it will open itself if it is exactly the ninth in a row, that is, if 8 open doors. In this case, all the locked doors of the dungeon must be opened and passed; each of them also opens itself if exactly eight open doors have been passed before. Correcting the mistake and going through 2 - 3 extra doors in the neighborhood to bring the number of doors passed to eight will also fail: as soon as any chamber is passed, all the doors previously open in it are tightly closed and locked - you will not pass through the chamber a second time. The feudal lords arranged it that way on purpose.
The prisoner knew about this secret of the dungeon, and on the wall of his cell (marked with an asterisk on the plan) he found the exact plan of the dungeon scratched with a nail. For a long time he puzzled over how to chart the right route so that each locked door would indeed be the ninth. Finally, he solved this problem and went free.
What solution did the prisoner find?

121. How was the prisoner released?
Those who wish can think about this version of the previous problem.
Imagine that the casemate in which the prisoner languishes consists of 49 cells.
In the seven chambers, marked on the dungeon plan (Fig. 63) with the letters A, B, C, D, E, F and G, there is one door each that can be opened only with a key, and the key to the door of chamber A is in chamber a, the key to the door of cell B is located in cell b, the keys to the doors of cells C, D, E, F and G are located in cells c, d, e, f and g, respectively.
The rest of the doors open with a simple push on the handle, but there is only a handle on one side of each door, and the door, after it has been passed, automatically slams shut. There is no handle on the other side of the door.
The dungeon map shows which way you can go through each door that opens without a key, but in what order the locked doors should be opened is unknown. It is allowed to pass through the same door any number of times, of course, observing the conditions under which it opens.
The prisoner is in cell O. Show him the path leading to the exit to freedom.

CHAPTER SIX
DOMINO AND CUBE
A. Domino
197. How many points?
198. Two tricks
199. Winning the game is guaranteed
200. Frame
201. Frame within a frame
202. "Windows"
203. Magic squares of domino bones
204. Magic square with a hole
205. Domino multiplication
206. Guess the intended domino bone
B. Cube
207. Arithmetic trick with dice
208. Guessing the sum of points on hidden faces
209. What order are the cubes in?

CHAPTER SEVEN
PROPERTIES OF THE NINE
210. What number is crossed out?
211. Hidden property
212. Some More Fun Ways to Find the Missing Number
213. By one digit of the result, determine the remaining three
214. Guessing the difference
215. Determination of age
216. What is the secret?

CHAPTER EIGHT
WITH AND WITHOUT ALGEBRA
217. Mutual Aid
218. A slacker and a devil
219. Smart baby
220. Hunters
221. Oncoming trains
222. Faith is typing a manuscript
223. Mushroom Story
224. Who will return first?
225. Swimmer and hat
226. Two ships
227. Test your ingenuity!
228. Embarrassment averted
229. How many times more?
230. Motor ship and seaplane
231. Cyclists in the arena
232. The speed of the turner Bykov
233. Jack London's trip
234. Mistakes are possible due to unsuccessful analogies
235. Legal incident
236. In pairs and threes
237. Who rode a horse?
238. Two motorcyclists
239. In which plane is Volodin's father?
240. Break into pieces
241. Two candles
242. Amazing insight
243. "Right Time"
244. Clock
245. What time is it?
246. At what time did the meeting begin and end?
247. Sergeant trains scouts
248. According to two reports
249. How many new stations were built?
250. Choose four words
251. Is such weighing permissible?
252. Elephant and mosquito
253. Five digit number
254. You grow up to a hundred years without old age
255. Luke's Problem
256. Peculiar walk
257. One property of simple fractions

CHAPTER NINE
MATH WITH ALMOST NO CALCULATIONS
258. In a dark room
259. Apples
260. Weather forecast (joke).
261. Forest Day
262. Who has what name?
263. Competition in marksmanship
264. Purchase
265. Passengers of one compartment
266. Final of the Soviet Army Chess Tournament
267. Sunday
268. What is the name of the driver?
269. Coal history
270. Herb gatherers
271. Hidden division
272. Encrypted actions (numeric puzzles)
273. Arithmetic mosaic
274. Motorcyclist and rider
275. On foot and by car
276. "From the contrary"
277. Detect counterfeit coin
278. Logic draw
279. Three Wise Men
280. Five questions for schoolchildren
281. Reasoning instead of an equation
282. Common sense
283. Yes or no?

CHAPTER TEN
MATH GAMES AND TOCKS
A. Games
284. Eleven items
285. Take matches last
286. Even wins
287. Jianshizi
288. How to win?
289. Lay out a square
290. Who will be the first to say "one hundred"?
291. Playing squares
292. Owa
293. "Mathematics" (Italian game)
294. Magic squares game
295. Intersection of numbers
B. Tricks
296. Guessing the planned number (7 tricks)
297. Guess the result of calculations without asking anything
298. Who took how much and found out
299. One, two, three attempts... and I guessed right
300. Who took the gum and who took the pencil?
301. Guessing three conceived terms and sum
302. Guess a few conceived numbers
303. How old are you?
304. Guess the age
305. Geometric trick (mysterious disappearance)

CHAPTER ELEVEN
DIVISIBILITY 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 an old French problem book)
310. Three-digit number
311. Four ships
312. Cashier's mistake
313. Number puzzle
314. Sign of divisibility by 11
315. Combined sign of divisibility by 7, 11 and 13
316. Simplification of the criterion of divisibility by 8
317. Amazing Memory
318. Combined sign of divisibility by 3, 7 and 19
319. Divisibility of a binomial
320. Old and new about divisibility by 7
321. Extension of a sign to other numbers
322. Generalized test of divisibility
323. Curiosity of divisibility

CHAPTER TWELVE
CROSS-SUMS AND MAGIC SQUARES
A. Cross sums
324. Interesting groupings
325. "Asterisk"
326. "Crystal"
327. Showcase decoration
328. Who will succeed first?
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 common methods
334. Examination of ingenuity
335. "Magic" game of "15"
336. Non-traditional magic square
337. What is in the central cell?
338. "Magic" works
339. "Casket" of arithmetic curiosities
B. Elements of the theory of magic squares
340. "By addition"
341. "Regular" magic squares of the fourth order
342. Selection of numbers for magic squares of any order

CHAPTER THIRTEEN.
CURIOUS AND SERIOUS IN NUMBERS
343. Ten figures (observation).
344. Some More Interesting Observations
345. Two interesting experiences
346. Number carousel
347. Disc of Instant Multiplication
348. Mental gymnastics
349. Patterns of numbers
350. One for all and all for one
351. Numerical finds
352. Observing a series of natural numbers
353. An annoying difference
354. Symmetric sum (unbroken nut)

CHAPTER FOURTEEN
NUMBERS ANCIENT BUT FOREVER YOUNG
A. Initial numbers
355. Prime and Composite Numbers
356. "Sieve of Eratosthenes"
357. New "sieve" for prime numbers
358. Fifty First Prime Numbers
359. Another way to get prime numbers
360. How many prime numbers?
B. Fibonacci numbers
361. Public trial
362. Fibonacci Series
363. Paradox
364. Properties of numbers in the Fibonacci series
B. Curly numbers
365. Properties of curly numbers
366. Pythagorean numbers

CHAPTER FIFTEEN
GEOMETRIC INTELLIGENCE IN WORK
367. Sowing geometry
368. Rationalization in laying bricks for transportation
369. Geometer Workers

Municipal budgetary educational institution

Saranpaul secondary school

Research work mathematics

Prepared by:

3rd grade student Frolov Nikolay,

Supervisor:

Arteeva Antonina Andreevna,

primary school teacher.

Saranpaul, 2017

Page

Introduction

The value of smart tasks

Leonardo Fibonacci- a mathematician who contributed to the solution of problems with ingenuity

Classification of tasks into "ingenuity"

Logic tasks

Crossing tasks

Tasks for transfusions

Fairy Tale Tasks

Tasks for ingenuity, for ingenuity

Number series, puzzles

Conclusion

Bibliography

Introduction

Creative activity is the most powerful impulse in the development of a child. Potential genius lives in every 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 ingenuity, no matter what age it is intended for, carries a certain mental load, which is most often disguised by an entertaining plot, external data, the condition of the problem, etc. In tasks of varying degrees of complexity, entertaining attracts the attention of children, activates thought , causes a steady interest in the upcoming search for a solution. The nature of the material determines its purpose: to develop general mental and mathematical abilities in children, to interest them in the subject of mathematics, to entertain, which is, of course, not the main one.The development of ingenuity, resourcefulness, initiative is carried out in active mental activity based on direct interest.

Entertaining mathematical material is given by the game elements contained in each task, logical exercise, entertainment, whether it is chess or the most elementary puzzle. For example, in the question: “How to fold a square on the table with two sticks?” - the unusualness of his production makes you think in search of an answer, get involved in a game of imagination.

The variety of entertaining material - games, problems, puzzles - provides a basis for their classification, although it is rather difficult to divide into groups such a diverse material created by mathematicians.

It can be classified according to various criteria: according to the content and meaning, the nature of mental operations, as well as the sign of generality, 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: To study methods for solving problems with ingenuity.

Tasks:

1. To study the topic "Solving problems with ingenuity", types of tasks for ingenuity and methods for solving them.

2. Solve several types of tasks for ingenuity, independently draw up an algorithm for solving such problems.

The value of smart tasks

The creative activity of students in the process of studying mathematics consists, first of all, in solving problems. The ability to solve problems is one of the criteria for the level mathematical development students, characterizes, first of all, the ability of students to apply their theoretical knowledge in a particular situation.

When solving traditional school problems, certain knowledge, skills and abilities are used to solve them in a narrow range of issues of program material. Wherein known ways solutions limits the creative search of students.

The task of ingenuity, unlike the traditional one, cannot be directly solved according to any law. Tasks for ingenuity are those for which in the course of mathematics there is no general rules and provisions defining the exact program for their solution. Consequently, there is a need to find a solution, which requires creative thinking and contributes to its development.

Solving problems with ingenuity gives rise to the tension of search and the joy of discovery - the most important factors of development, creative achievement.

The value of tasks for ingenuity is very high - the ability of students to solve non-standard tasks shows:

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

2. ​ The ability to generalize mathematical material, to isolate the main thing, to be distracted from the insignificant, to see the general in the outwardly different;

3. ​ Ability to operate numerical and symbolic symbols;

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

5. ​ The ability to reduce the process of reasoning, to think in folded structures;

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

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

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

9. Ability for spatial representations.

Even K.D.Ushinsky wrote that "... learning, devoid of any interest and taken only by force of coercion ... kills the student's desire for learning, without which he will not go far."

Interest is a powerful motivator of activity, under its influence all mental processes proceed especially intensively, and activity becomes exciting and productive. Its essence lies in the student's desire to penetrate into the cognizable area more deeply and thoroughly, in a constant urge to engage in the subject of his interest.

From the history of the appearance of tasks for ingenuity

It is not surprising that tasks for ingenuity have become entertainment "for all times and peoples."The first mathematics textbook that has come down to us, or rather, its​ juice 5 meters long, known in the world as "London papyrus", or "Ahmes papyrus", contains 84 accompanied by the solution of the problem. According to him, classes were conducted at the school of state scribes. Already the ancient Egyptians understood how important the role in the process of learning​ value plays an element of entertainment, and among those included in the "papi​ Rus Ahmes "there were many such tasks. So, for millennia, from one collection​ nick of entertaining problems of mathematics in another roams "the problem of se​ my cats" from this papyrus. Despite the existence of the thirteen-volume “Beginnings” of Euclid (3rd century BC), which became a model of scientific rigor for more than two millennia, the entertaining element in mathematics did not disappear in Ancient Greece and is most clearly represented in the “Arithmetic” by Diophantus of Alexandria ( probably 3rd century). In the Middle Ages, the Italians Leonardo (Fibonacci) from Pisa (XIII century) and Niccolò Tartaglia (XVI century) left the deepest mark in solving problems with ingenuity.

Collections of mathematical entertainment, similar to modern ones, began to appear from the 17th century. Among them, “Pleasant and entertaining tasks considered in numbers” by the mathematician and poet Gaspard Claude Bache sieur de Meziriac and “Mathematical and physical entertainments” by another French mathematician and writer Jacques Ozanam.

In the 19th century Edouard Lucas, a French mathematician and number theorist, published a four-volume work on entertaining mathematics, which has become a 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 - the talented self-taught American Sam Loyd and the Englishman Henry Ernest Dudeney. Entertaining mathematics second half of the 20th century can not be imagined without a whole series of wonderful books written by the famous American mathematician Martin Gardner. It was his diverse mathematical essays, harmoniously combining scientific depth and the ability to entertain, that introduced millions of people around the world (including me) to the exact sciences and, of course, to entertaining mathematics.

In Russia, such collections of problems as “Arithmetic” by L. F. Magnitsky, “In the Realm of Ingenuity” by E. I. Ignatiev, “Live Mathematics”, “Entertaining Arithmetic”, “Entertaining Algebra” and “Entertaining Geometry” by Ya. I. Perelman and “Mathematical ingenuity” by B. A. Kordemsky

Leonardo Fibonacci - a mathematician who contributed to the solution of problems with ingenuity.

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

Leonardo Fibonacci introduced Arabic numerals to Europe , which is used by almost the entire Western world to this day.The transition from the Roman system to the Arabic system revolutionized mathematics and other sciences closely related to it.

It is hard to imagine what the world would be like if then, in the 13th century, Fibonacci had not published his book and presented Arabic numerals to Europeans. It is interesting that we use Arabic numerals without hesitation, taking them for granted. But if not for Leonardo Fibonacci, who knows how the course of history would have developed. After all, the presentationtreatise Arabic numbers significantly changed medieval mathematics for the better; he advanced it, and with it other sciences such as physics, mechanics, electronics, and so on. Note that it is these sciences that lead progress forward. That is why, in many ways, the course of history,the development of European civilization and science in general is due to Leonard Fibonacci .

Series of Fibonacci numbers

The second outstanding merit of Leonardo Fibonacci isseries of fibonacci numbers . It is believed that this series was known in the East, but it was Leonardo Fibonacci who published this series of numbers in the aforementioned book "Liber Abaci" (he did this to demonstrate the reproduction of a population of rabbits).

Later it turned out thatthis sequence of numbers is important not only in mathematics, economics, and finance, but also in botany, zoology, physiology, medicine, art, as well as philosophy, aesthetics and much more. because civilization, this series of numbers became known from Leonardo Fibonacci, he was nicknamed, “Fibonacci series» or "Fibonacci numbers ».

Formula and example of a series of Fibonacci numbers

In the Fibonacci sequence,each element, starting from the third, is the sum of the previous two elements , despite the fact that the series begins with the numbers 0 and 1. The total is obtained: 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 figure in mathematics, economics and finance ; he promulgated the Arabic numerals and presented the magical series of numbers.

The problem was invented by the Italian scientist Fibonacci, who lived in the 13th century.
“Someone purchased a pair of rabbits and placed them in a paddock fenced on all sides. How many rabbits will there be in a year, if we assume that every month a pair gives birth to a new pair of rabbits as offspring, which also begin to bear offspring from the second month of life?

Answer: 377 pairs In the first month, there will already be 2 pairs of rabbits: 1 initial pair that gave birth, and 1 born pair. In the second month of rabbits there will be 3 pairs: 1 initial, again giving birth, 1 growing and 1 born. In the third month - 5 pairs: 2 pairs that gave birth, 1 growing and 2 born. In the fourth month - 8 pairs: 3 pairs that gave birth, 2 growing pairs, 3 born pairs. Continuing the consideration by months, it is possible to establish a relationship between the numbers of rabbits in the current month and in the previous two. If we denote the number of pairs through N, and through m - the ordinal number of the month, then N m = N m-1 + N m-2 . Using this expression, the number of rabbits is calculated by the months of the year: 2, 3, 5, 8, 13, 21, 34.55, 89, 144, 233, 377.

Classification of tasks for ingenuity

In such problems, the solver is required for a limited number of weighings to localize an object that differs from other objects in weight. Also in this section, transfusion tasks are considered, in which it is necessary to obtain a certain amount of liquid using containers of a given volume.

Finding the excess

The ability to combine groups of objects according to certain characteristics is required.

Text problems for calculations

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

Tasks for finding logical errors, tasks with a catch

They develop a valuable and very necessary quality of a successful person - critical thinking. Learning to analyze the condition. Sometimes the answer is in the problem itself.

Assignment to the properties of numbers and operations with them

The property of even and odd numbers, the correct placement of brackets, the placement of digits in a number that meets certain conditions. Divisibility of numbers. Operations on numbers.

Cryptocurrencies

A mathematical rebus in which an example is encrypted for performing one of the arithmetic operations. In this case, the same numbers are encrypted with the same letter, and different numbers correspond to different letters.

Tasks for logic and reasoning

Tasks that are not directly related to calculations, but actively develop thinking.

About the time

Calculate a date using hints, remember how a clock works, or determine someone's age just by hints.

On a sequence of numbers

In these tasks, it is necessary to unravel the principle by which a certain sequence is set, and continue it.

Problems with matches

When manipulating matches, it is necessary to achieve the desired result. Most of these tasks are among the "non-standard" ones, requiring the skill "to assess the situation from a point of view that is unexpected for the majority or to see in the condition the possibility of using non-obvious data."

puzzles

A game in which words, phrases or entire statements are encrypted using drawings combined with letters and signs.

Chess

As a rule, each stage of the course includes several lessons (minimum 2) in chess. Basic figures. We learn to build effective strategies, think, make informed and rational decisions

Logic tasks

When solving logical problems for one-to-one correspondence, it is convenient to write data to a table, where we put a “+” sign or a “-” sign at the intersection of a row and a column.

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

The winner of the Olympiad in Informatics teaches Irena and Timur how to work on the computer;

Camilla and Eldar also became interested in computer science;

Timur was always afraid of physics;

Camilla, Timur and the winner of the Literature Olympiad go swimming;

Timur and Kamilla congratulated the winner of the Mathematics Olympiad;

Irena regrets that she has little time left for literature.

Which Olympiad did each of these guys win?

1 way to solve, using a table

I T C E Z

F M I L G

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

Camille - in physics Eldar - in literature. Zalim - in computer science

2. Three girls - Rosa, Margarita and Anyuta presented at the competition baskets of roses, daisies and pansies grown by them. The girl who raised the daisies drew Rosa's attention to the fact that none of the girls' names match the names of their favorite flowers. What flowers did each of the girls grow?

Solution: by reasoning

a) Anya did not grow pansies. b) Margarita did not grow daisies c) Rose did not grow roses. Rose could grow either roses or pansies. The rose didn't grow roses. Conclusion: Rose has grown pansies. Margarita grew roses. Anya grew daisies.

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

Answer:

4. Masha, Lida, Zhenya and Katya play different instruments - button accordion, piano, guitar, violin, but each on one. They also speak foreign languages ​​- English, French, German, Spanish, but each one is the same. Who plays which instrument and which foreign language does he speak?

Crossing tasks

In tasks for crossings, it is required to indicate the sequence of actions in which the required crossing is carried out and all the conditions of the task are met.

Wolf, goat and cabbage. On the bank of the river stands a peasant with a boat, and next to him are a wolf, a goat and a cabbage. The peasant must cross himself and transport the wolf, goat and cabbage to the other side. However, in addition to the peasant, either only the wolf, or only the goat, or only the cabbage is placed in the boat. You cannot leave a wolf with a goat or a goat with cabbage unattended - a wolf can eat a goat, and a goat can eat cabbage. How should a peasant behave?

Answer: A peasant can follow one of two algorithms:

Answer: Let M1 and M2 be boys, C1 and C2 be soldiers. The crossing algorithm can be as follows:

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

Tasks for transfusions

Thesetasks are practical. Solving such problems develops logical thinking, makes you think, approach the solution of a problem from different angles, choose the simplest, easiest way from a variety of solutions. To do this, using vessels of known containers, it is required to measure a certain amount of liquid. The simplest method for solving problems of this class is to enumerate possible options.And it is required to indicate the sequence of actions in which the required transfusion is carried out and all conditions are met.

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

Answer:

2. The evil stepmother sent her stepdaughter to the spring for water and said: “Our buckets contain 5 and 9 liters of water. Take them and bring exactly 3 liters of water.” How should the stepdaughter act in order to fulfill this assignment?

Answer:

In the transfusion problems discussed above, two vessels were given and water was poured from a water tap.There are more difficult tasks, not two vessels, but three or more. Water is NOT taken from the tap. In such problems, water is already in some vessel, for example, in the largest one. And we will pour water in small containers. Water cannot be poured out. If it is necessary to empty the vessel, then the excess water is poured into another vessel. Usually a larger vessel is a storage from which water is taken and excess water is poured into it.

Fairy Tale Tasks

The solution of such problems enlivens mathematics. The desire to help the hero in trouble stimulates mental activity, in the future causes a desire to read the work. Sympathy in such tasks is on the side of the positive hero. Good triumphs, evil is punished, negative qualities are ridiculed.

on one of them you will meet your death,

nothing will happen to you,

the third road will lead you to Vasilisa the Beautiful.

Keep in mind that all three inscriptions were made by Koshchei the Immortal. Ivan threw the ball on the ground. He rolled, Ivan followed him. How long, how short Ivan walked, but he came to a huge stone. On the stone is written:

"If you go to the left, you will meet your death",

“If you go to the right, you will rescue Vasilisa the Beautiful from captivity”, “If you go straight, something will happen to you.”

Solution: The third entry is incorrect - nothing will happen to Ivan on the way. The second entry is also incorrect, i.e. on the way to the right, Ivan will not rescue Vasilisa the Beautiful. So, on the remaining road (the road to the left), Ivan will rescue Vasilisa the Beautiful.

2. Six robbers robbed King Dadon. The loot turned out to be rich - less than a hundred identical ingots. The robbers began to divide the booty equally, but one ingot turned out to be superfluous. The robbers fought and one of them was killed in a fight. The rest began to divide the gold again, and again one piece turned out to be superfluous. And again, one of the robbers died in a fight. And so on: each time one ingot was superfluous and one of the robbers died in a fight. In the end, one robber remained, who died from his wounds. How many ingots were there?

Solution:if initially there would have been one bar less, then the division would have taken place. A number that is less than 100 and divisible by 2, 3, 4, 5, 6 - 60. So the total number of ingots is 60+1=61.

Tasks for ingenuity

1. Two mothers, two daughters and a grandmother with a granddaughter. How many?

2. The apartment had 3 rooms. Made two from one. How many rooms are in the apartment?

3. How to arrange 8 chairs against the four walls of the room so that each wall has 3 chairs?

Tasks for ingenuity

How many hours are day and night together?

There was an apple on the table. It was divided into 4 parts. How many apples are on the table?

Tasks for changing the constructed figure

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

Number series

1,2,3,4,5,6…

1,4,16…

45,39,33,27…

0,3,8,15,24…

112,56,28,14…

puzzles

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

11x10=110

6* : *7 = *

68:17 = 4

** +** =20

10+10= 20

* 2 -* = *

12- 4 = 8

*** +**=1**

101 +41+142

Problems with geometric content (unicursal figures)

There is a well-known parable: someone gave a million rubles to everyone who draws the next figure. But when drawing, one condition was set. It was required that this figure be drawn in one continuous stroke, that is, without taking the pen or pencil off the paper and without doubling a single line, in other words, it was impossible to pass a line once drawn a second time.

Conclusion

In mathematics, there are different types of tasks for ingenuity:

For weighing and transfusion,

Logic tasks,

transfusion tasks,

crossing tasks,

Problems with geometric content,

Rebuses, number series.

Methods for solving such problems lies in the logical analysis of the conditions, the choice of the appropriate laws of mathematics and the optimal solution.

There is no universal way to solve all kinds of problems with ingenuity, each problem is solved in its own way.

Tasks for ingenuity help to 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.

The tasks facing the author of the work are solved, namely:

To study the topic "Solving problems with ingenuity", types of tasks for ingenuity and methods for solving them;

Solve several types of tasks for ingenuity, independently draw up an algorithm for solving such problems.

Bibliography

1. T.D. Gavrilova: "Entertaining mathematics." Publishing house "Uchitel" 2008

2. E.G. Kozlova: "Tales and hints". Miros Publishing House 1995

3. B. A. Kordemsky: “Mathematical ingenuity” Publishing house “State publishing house of technical and theoretical literature” 1958

4. Ya. I. Perelman: "Entertaining Algebra". Publishing house "Century" 1994

5.R.M.Smullyan "What is the name of this book?". Publishing house "Dom Meshcheryakova"

2007

7. http://matematika.gyn

8.www.smekalka.pp