CHAPTER SIX
DOMINO AND CUBIC
A. Domino
197. How many points?
198. Two focuses
199. The winning of the party is guaranteed
200. Frame
201. Frame within a frame
202. "Windows"
203. Domino Magic Squares
204. Magic square with a hole
205. Multiplication by dominoes
206. Guess the intended domino bone
B. Kubik
207. Arithmetic trick with dice
208. Guessing the amount of points on the hidden edges
209. In what order are the cubes located?
CHAPTER SEVEN
PROPERTIES OF 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's the secret?
CHAPTER EIGHT
WITH AND WITHOUT ALGEBRA
217. Mutual assistance
218. Loser, and Damn
219. Smart Kid
220. Hunters
221. Oncoming trains
222. Vera prints the manuscript
223. Mushroom Story
224. Who will return earlier?
225. Swimmer and the Hat
226. Two ships
227. Test your wits!
228. Confusion 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 Ride
234. Errors are possible due to unfortunate analogies
235. Legal incident
236. In pairs and threes
237. Who was riding the horse?
238. Two Motorcyclists
239. What plane is Volodin's dad on?
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 have been built?
250. Choose four words
251. Is such weighing permissible?
252. The Elephant and the Mosquito
253. Five-digit number
254. Up to a hundred years you grow up without old age
255. Luke's Problem
256. A Peculiar Walk
257. One Property of Simple Fractions
CHAPTER NINE
MATH WITH ALMOST CALCULATION
258. In a dark room
259. Apples
260. Weather forecast (just kidding).
261. Forest Day
262. Who has a name?
263. Contest in Accuracy
264. Purchase
265. Passengers of one compartment
266. Final of the tournament of chess players of the Soviet Army
267. Sunday
268. What is the name of the driver?
269. Coal History
270. Herbal Pickers
271. Hidden division
272. Encrypted Actions (Number Rebus)
273. Arithmetic Mosaic
274. Motorcyclist and Rider
275. On foot and by car
276. By Contrary
277. Detect Fake Coin
278. Logical Draw
279. Three Wise Men
280. Five Questions for Students
281. Reasoning instead of an equation
282. Common Sense
283. Yes or no?
CHAPTER TEN
MATH GAMES AND FOCUS
A. Games
284. Eleven Items
285. Take the last matches
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. The game of squares
292. Oah
293. "Mathematics" (Italian game)
294. The Magic Square Game
295. Number Crossing
B. Tricks
296. Guessing the intended number (7 tricks)
297. Guess the result of calculations without asking anything
298. Who took how much and found out
299. One, two, three tries ... and I got it right
300. Who took the eraser and who took the pencil?
301. Guessing the three conceived terms and sums
302. Guess a few of the conceived numbers
303. How old are you?
304. Guess Age
305. Geometric Focus (Mysterious Disappearance)
CHAPTER ELEVEN
DIVISIBILITY OF NUMBERS
306. The number on the tomb
307. Gifts for the New Year
308. Could there be such a number?
309. Egg Basket (From an Old French Problem Book)
310. Three-digit number
311. Four ships
312. Cashier's Error
313. Number Rebus
314. Divisibility by 11
315. The combined criterion of divisibility by 7, 11 and 13
316. Simplification of the divisibility criterion by 8
317. Amazing Memory
318. The combined criterion of divisibility by 3, 7 and 19
319. Divisibility of a binomial
320. Old and new about divisibility by 7
321. Extension of the sign to other numbers
322. Generalized divisibility criterion
323. Divisibility Curiosity
CHAPTER TWELVE
CROSS-SUM AND MAGIC SQUARES
A. Cross-sums
324. Interesting groupings
325. Asterisk
326. "The Crystal"
327. Showcase Decoration
328. Who will succeed earlier?
329. Planetarium
330. "Ornament"
B. Magic squares
331. Aliens from China and India
332. How to make a magic square yourself?
333. Approaches to general methods
334. Tricky Test
335. The Magic Game of 15
336. Unconventional Magic Square
337. What's in the central cell?
338. "Magic" works
339. "Box" of arithmetic curiosities
B. Elements of magic square theory
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 digits (observations).
344. Some More Interesting Observations
345. Two interesting experiences
346. Numeric Carousel
347. Disk 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. Bothersome Difference
354. Symmetrical Amount (Unsized Nut)
CHAPTER FOURTEEN
ANCIENT NUMBERS, BUT ETERNALLY YOUNG
A. Original numbers
355. Numbers, prime and composite
356. "Eratosthenes Sieve"
357. New "sieve" for prime numbers
358. First Fifty Prime Numbers
359. Another way to get prime numbers
360. How many primes?
B. Fibonacci numbers
361. Public trial
362. Fibonacci Series
363. Paradox
364. Properties of numbers of the Fibonacci series
B. Curly numbers
365. Properties of curly numbers
366. Pythagorean Numbers
CHAPTER FIFTEEN
GEOMETRIC BRIGHTNESS IN WORK
367. Sowing geometry
368. Rationalization in the laying of bricks for transportation
369. Working-geometers
Preface to the second edition 3
Chapter first
CONSISTENT TASKS
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! 12386
6. The Gardener's Path 12 386
7. It is necessary to realize 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 the black balls 15 388
13. Repair of the chain 15 388
14. Fix error 16 390
15. Out of three - four (just kidding) 16 390
16. Three and two - eight (still a joke) 16 390
17 Three squares 16 390
18. How many parts? 17 390
19. Try it! 17391
20. Arrangement of flags 17 391
21. Keep parity 18 391
22. "Magic" number triangle 18 391
23. How 12 girls played with a ball 19 392
24. Four straight lines 20 392
25. Separate goats from 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. Fast, but careful 26 396
36. Curly cancer 27 396
37. The cost of the book is 27 396
38. Restless fly 27 396
39. In 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. High-speed folding - 29 397
44. In which hand? (math focus) 31 397
45. How many are there? 31398
46. \u200b\u200bThe same numbers 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, one result 35402
53. Ninety nine and one hundred 36 402
54. Dismountable chessboard 36 402
55. Search for Mine 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 actions of arithmetic 39 404
60. Puzzled chauffeur 40 404
61. For the Tsimlyansk hydroelectric complex 41 404
62. Bread delivery on time 41 405
63. On a suburban train 41 405
64. From 1 to 1,000,000,000 41 405
65. Terrible dream of a football fan 42 406
Section II
66. Clock 43 406
67. Ladder 43 407
68. Puzzle 43 407
69. Interesting Fractions 43 407
70. What number? 44407
71. The Way of the Student 44 407
72. At the stadium 44 407
73. Did you guess? 44407
74. Alarm 44 407
75. Instead of small shares, large 45 407
76. Soap Bar 45 408
77. Arithmetic nuts 45 408
78. Fraction-Dominoes 46 409
79. Misha's kittens 48 409
80. Average speed 48 409
81. Sleeping Passenger 48 409
82. How long is the train? 48409
83. Cyclist 48 409
84. Competition 49 409
85. Who is right? 49409
86. For dinner - 3 toasted slices 50 410
Chapter two
DIFFICULT PROVISIONS
87. Hecho Blacksmith Savvy 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. Spread out coins 55 411
92. Skip passenger 1 55 412
93. A problem arising from the whim of three girls 56 412
94. Further development of task 57 413
95. Jumping checkers 57 415
96. White and Black 57 415
97. Increasing the complexity of 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 the TV room 61 419
103. Placement of experimental rabbits 62 421
104. Preparing for the holiday 63 422
105. Plant the oaks 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's 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 letter placement problems 73 427
115. Layout of colored squares 74 429
116. The Last Piece 74 430
117. Ring of discs 75 431
118. Skaters on an artificial ice rink 76 431
119. Joke Problem 77 432
120. One hundred forty-five doors (puzzle) 77 432
121. How was the prisoner released? 79432
Chapter three
GEOMETRY ON MATCHES
122. Five Puzzles 85 433
123. Eight More Puzzles 86 433
124. Of 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 do you need to remove? 88 433
132 Joke 88 433
133 "Fence" 88 433
134. Joke 89 433
135. "Arrow" 89 433
136. Squares and rhombuses 89 433
137. Different polygons in one figure 89 433
138 Garden layout 89 433
139 Into equal parts 90 433
140. Parquet 91 433
141 The area ratio is preserved 91 441
142. Find the outline of the figure 91 441
143 Find Evidence 92 441
144. Build and Prove 92 441
Chapter four
TRY SEVEN TIMES, CUT ONCE
145. On equal parts 93 442
146. Seven roses on the cake 95 443
147. Figures that have lost their shape 95 445
148. Suggest 96 445
149. No loss! 96 445
150. When the Nazis invaded our land 97 447
151. Memoirs of an electrician 98 447
152. Everything goes into action 99 447
153. Puzzle 99 447
154. Cut the Horseshoe 99 447
155. Each part has a hole 99 448
156. From the "jug" - square 100 448
157. Square from the letter "E" 100 448
158. Beautiful transformation 100 449
159. Carpet Restoration 101449
160. Expensive Award 101 449
161. Help the poor fellow out! 102 449
162. Gift for grandmother 103 451
163. Carpenter's Problem 104 451
164. And the furrier has geometry! 104452
165. Each horse, a stable 105 453
166. Even 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
LESS WILL FIND APPLICATION EVERYWHERE
169. Where is the goal? 109,454
170. Five minutes for reflection 110 455
171. An unexpected meeting 110 455
172. Travel triangle Ш 456
173. Try to weigh 111 458
174. Pass 112 458
175. Seven Triangles 112 458
176. Canvases of the artist 112 458
177. How much does the bottle weigh? 113,459
178. Cubes 113 460
179. Can with fraction 114 461
180. Where did the sergeant come? 114 461
181. Determine the diameter of the log 115 461
182. An Unexpected Difficulty 115 461
183. The story of a student of a technical school 116 461
184. Can you 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 the Carpenter 120 466
191. Geometry on the ball 120 466
192. You need a lot of ingenuity 121 467
193. Difficult conditions 121 468
194. Prefabricated Polygons 122 468
195. A curious method of drawing up similar figures 125 469
196. Hinge mechanism for constructing regular polygons 127 471
Chapter six
DOMINO AND CUBIC
A. Domino
197. How many points? 132 471
198. Two focuses 133 471
199. The winning of the party is guaranteed 134 471
200. Frame 135 472
201. Framed Frame 136 472
202. "Windows" 136 473
203. Domino Magic Squares 137 473
204. Magic square with hole 141 473
205. Domino Multiplication 141 473
206. Guess the intended domino bone 142 473
B. Kubik
207. Arithmetic trick with dice 144 473
208. Guessing the amount of points on the hidden edges 145 477
209. In what order are the cubes located? 145 478
Chapter Seven
PROPERTIES OF NINE
210. What number is crossed out? 149,478
211. Hidden Property 152 479
212. A few more funny ways to find the missing number 152 480
213. By 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's the secret? 154,482
Chapter Eight
WITH AND WITHOUT ALGEBRA
217. Mutual assistance 159 482
218. Loafer and the Devil 160 483
219. Smart Kid 161 483
220. Hunters 161 483
221. Oncoming trains 162 484
222. Vera prints the manuscript 162 484
223. The Mushroom Story 163 484
224. Who will return earlier? 164,484
225. The Swimmer and the Hat 164 486
226. Two motor ships 165 486
227. Test your wits! 165,487
228. Confusion prevented 166 488
229. How many times more? 166488
230. Motor ship and seaplane 167 488
231. Cyclists in the arena 167 489
232. The speed of the turner Bykov 168 489
233. Jack London's Ride 168 489
234. Due to unfortunate analogies, errors are possible 169 490
235. Legal incident 170 491
236. In pairs and triples 171 491
237. Who was riding the horse? 171,491
238. Two motorcyclists 171 492
239. What plane is Volodin's dad on? 172,492
240. Break 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 have been built? 176,498
250. Choose four words 177 498
251. Is such weighing permissible? 177499
252. The Elephant and the Mosquito 178 500
253. Five-digit number 179 500
254. You grow up to one hundred years without old age 179 500
255. Luke's Problem 181 501
256. A Peculiar Walk, .181 502
257. One Property of Simple Fractions 182 504
Chapter nine
MATH WITH ALMOST CALCULATION
In a dark room
Apples
Weather forecast (kidding)
Forest day
Who has a name?
Contest in marksmanship
Purchase
Passengers of one compartment
Final of the tournament of chess players of the Soviet Army
Sunday
What is the name of the driver?
Criminal history
Herbal pickers
Latent division
Encrypted actions (numerical puzzles)
Arithmetic mosaic
Motorcyclist and Rider
On foot and by car
"By contradiction"
Detect fake coin
Logical draw
Three wise men
Five questions for students
Reasoning instead of an equation
By common sense
Yes or no?
Chapter ten
MATH GAMES AND FOCUS
A. Games
284. Eleven Items 201
285. Take the last matches 202
286. Even 202 wins
287. Jianshizi 202
288. How to win? 204
289. Lay out square 205
290. Who will be the first to say "one hundred"? 206
291. The Square Game 206
292. Oah 209
293. Matezatico (Italian game) 212
294. The Magic Square Game 213
295. Number Intersection 215
B. Tricks
296. Guessing the intended 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 tries and I guess 226 537
300. Who took the eraser and who took the pencil? 227,537
301. Guessing the three planned terms and the sum 227 537
302. Guess a few of the 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. Could there be such a number? 233540
309. A basket of eggs (from an old French problem book) 233 540
310. Three-digit number 234 540
311. Four motor ships 234 540
312. Cashier error 234 540
313. Number Rebus 234 541
314. Divisibility criterion by 11 235 541
315. The combined criterion of divisibility by 7, 11 and 13 237 541
316. Simplification of the divisibility criterion by 8 239 541
317. Amazing Memory 240 542
318. The combined criterion 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 the sign to other numbers 251 -
322. Generalized divisibility criterion 252 -
323. Divisibility Curiosity 254 -
Chapter twelve
CROSS-SUM AND MAGIC SQUARES
A. Cross-sums
324. Interesting groupings 256 545
325. Asterisk 257 545
326. "Crystal" 257 545
327. Decoration for a showcase 258 545
328. Who will succeed earlier? 258545
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? 264548
333. At the entrance to general methods 266 549
334. Tricky Test 271 549
335. "Magic" game of "15" 271 551
336. Unconventional Magic Square 272 553
337. What's in the central cell? 273,553
338. "Magic" works 275 553
339. "Box" of arithmetic curiosities 278 -
340. "By 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 Digits (Observations) 298 554
344. Some More Interesting Observations 300 555
345. Two interesting experiences 302 555
346. Numeric Carousel 306 -
347. Instant Multiplication Disc 309 -
348 Mental gymnastics 310 -
349. Number Patterns 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. An annoying difference 339 -
354. Symmetrical sum (unshelled nut) 340 -
Chapter fourteen
ANCIENT NUMBERS, BUT ETERNALLY YOUNG
A. Original numbers
355. Prime and composite numbers 341 -
356. "Eratosthenes sieve" 342 -
357. New "sieve" for prime numbers 344 563
358. First fifty prime numbers 345 -
359. Another way to get prime numbers. 345 -
360. How many primes? 347
B. Fibonacci numbers
361. Public Trial 347 -
362. Fibonacci Series 351 -
363. Paradox 352 564
364. Properties of numbers of the Fibonacci series 355 -
B. Curly numbers
365. Properties of curly numbers 360 -
366. Pythagorean Numbers 369 -
CHAPTER FIFTEEN A GEOMETRIC BRIGHTNESS IN WORK
367. Sowing geometry 372 -
368. Rationalization in the laying of Bricks for transportation 375 -
369. Work Geometers 377
Recognized two chapters:
FOREWORD TO THE SECOND EDITION
In work, in learning, in play, in any creative activity, a person needs intelligence, resourcefulness, guesswork, the ability to reason - all that our people aptly define in one word "ingenuity." The 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 observing the world of things and events around us.
“Mathematics,” said MI Kalinin, addressing the students of the secondary school, “disciplines the mind, teaches them to think logically. No wonder they say that mathematics is the gymnastics of the mind. "
Every family in which parents are concerned with organizing the mental development of children and adolescents feels 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, and the "Matematic ingenuity" is intended - a collection of mathematical miniatures: various problems, mathematical games, jokes and tricks that require work of the mind, developing intelligence and the necessary consistency in reasoning.
In pre-revolutionary times the collections of EI Ignatiev "In the kingdom of ingenuity" were widely known. Now they are outdated for our reader and therefore not reissued. Nevertheless, these collections also contain tasks that have not yet lost their pedagogical and educational value. Some of them entered the "Mathematical Ingenuity" without change, others with changed or completely new content.
For Mathematical Savvy, 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 the popular books of Ya. I. Perelman on entertaining mathematics.
Mathematical problems of this kind of "small form" sometimes arise as a by-product of the scientist's serious research; many problems are invented by amateurs and teachers as special exercises for "mental gymnastics". They, like riddles and proverbs, usually do not retain authorship and become the property of society.
"Math Savvy" is intended for readers with a wide range of mathematical backgrounds:
for a teenager 10 - 11 years old making the first attempts at independent reflection;
for a high school student keen on mathematics,
and for the adult who wants to test and practice their guess.
The systematization of tasks by chapters is, of course, rather arbitrary; each chapter contains both easy and difficult tasks.
The book has fifteen chapters.
The first chapter consists of various types of initial exercises of an "intricate" nature, based on guesswork 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 classification harmony of the book, I singled out in the first chapter some of the simple problems that thematically belong to the subsequent chapters. This is done in the interests of those readers who still find it difficult to independently distinguish a feasible task from an unbearable one. Solving in a row the different types of problems of the first chapter, they will be able to try their hand, and then transfer their interest in a particular topic to the corresponding tasks of the subsequent chapters.
To solve the problems of the second chapter, your own mathematical ingenuity and perseverance must overcome all sorts of obstacles and prompt 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 associated with practical activities, with technology.
The chapter titled "Mathematics with Almost No Computation" contains problems that require a chain of skillful and subtle reasoning to be solved.
Games and tricks are collected in a separate chapter, and also placed throughout the book. They contain a mathematical basis and are undoubtedly part of the "area of \u200b\u200bingenuity."
Three chapters: "Cross-sums and magic squares", "Curious and serious in numbers" and "Ancient numbers, but forever young", are devoted to some interesting observations of the numerical ratios that have accumulated in mathematics from antiquity to our time.
The last chapter is two small essays about the labor ingenuity of the people of our Motherland, the workers of fields and factories.
In various places of the book, the reader is offered small topics for independent research.
At the end of the book there are solutions to problems, but you should not rush to look at them.
Any task of "quick wits" is fraught with some "zest" and in most cases is a tough nut to crack, which is not so easy to bite, but all the more tempting.
If you cannot solve the problem right away, you can temporarily skip it and go 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 companion of the reader in his gradual mathematical development.
All the material of the book is subordinated 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 Savvy is not a stereotypical repetition of the first. The required changes have been made to the text and to the solution of some problems; individual tasks were replaced by new ones - more meaningful; the design of the book has been done again.
Great efforts to improve the book were made by the editor of the publishing house M. M. Goryachaya.
Solving problems on their own, the readers in some cases found additional or simpler solutions and kindly informed me of their results. The authors of the most interesting solutions are mentioned in the corresponding places in the book.
I hope to receive feedback and suggestions from the readers of "Smile" for 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 Anastasevich Kordemsky.
B. Kordemsky.
TASKS
"Book - book, Move your brains"
V. Mayakovsky.
CHAPTER ONE. CONSISTENT TASKS
SECTION I
Test and practice your ingenuity first on problems that require only determined perseverance, patience, ingenuity and the ability to add, subtract, multiply and divide whole numbers.
1. Observant pioneers
The schoolchildren, a boy and a girl, have just taken meteorological measurements.
Now they are resting on a hillock and watching a freight train passing by.
The locomotive on the rise is desperately smoking and puffing. The wind blows smoothly along the railroad bed without gusts.
- What wind speed did our measurements show? the boy asked.
- 7 meters per second.
- Today this is enough for me to determine the speed of the train.
- 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). How fast was the train going?
Figure: 1.What is the speed of the train?
2. "Stone flower"
Do you 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 formed parts of the flowers it was possible to fold a plate in the shape of a circle.
Try it! Redraw the danilina flowers onto paper or cardboard, cut out the petals, stems and leaves, and fold the circle.
3. Moving checkers
Place 6 checkers in a row on the table alternately - black, white, black, white, etc. (Fig. 3).
Figure: 3. White checkers should be on the left, behind them - black.
Leave enough space on the right or left for four checkers.
It is required to move the checkers so that all the white ones are on the left, and after them all the black ones. At the same time, two adjacent checkers must be moved to an empty space 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 don't have checkers, use coins or cut pieces of paper, cardboard.
*) The topic of this problem is further developed in problems 96 and 97 (pp. 57 and 58).
4. In three moves
Place 3 piles of matches on the table. Put 11 matches in one pile, 7 matches in another, and 6. Putting matches from any pile to any other, you need to equalize all three piles so that there are 8 matches in each. 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!
Test your geometric observation: count how many triangles are in the figure shown in fig. 4.
6. The gardener's path
In fig. 5 is a plan of a small apple orchard (points - apple trees). The gardener has worked all the apple trees in a row.
Figure: 5. Plan of the apple orchard.
He began with 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.
After completing the walk, the gardener found himself in the same cage from which he began his journey.
Draw a gardener's path in your notebook.
7. It is necessary to realize
The basket contains 5 apples. How do you divide these apples between 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 there in the room, if there is one cat in each of the four corners of the room, 3 cats are sitting opposite each cat, and a cat is sitting 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 (long) 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, continuing to press against the yellow, lowers it by 1 cm, then returns to its previous position, again lowers it by 1 cm and again returns to its previous position; He lowers it 10 times and lifts the blue pencil 10 times (20 movements).
Assuming 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. You will read what prompted him to compose the problem on page 387 after solving the problem.
10. Crossing the river (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 at the shore, having fun in the 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 in this particular boat. How?
Solve this problem "in your head" or practically - using checkers, matches or something like that and moving them across the table across an imaginary river.
11. Wolf, goat and cabbage
This is also an old problem; found in the writings of the 8th century. It has fabulous content.
Figure: 6. It was impossible to leave a wolf and a goat without a man ...
A certain man had to transport a wolf, a goat, and a 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 the goat, if you leave the goat with cabbage, then the goat will eat the cabbage, and in the presence of a man "no one ate anyone." The man nevertheless transported his cargo across the river.
How did he do it?
In a narrow and very long groove there are 8 balls: four black ones on the left and four white ones of slightly larger diameter on the right (Fig. 7). In the middle part of the groove in the wall there is a small niche in which only one ball (any) can fit. Two balls can be located side by side across the groove only in the place where the niche is. The left end of the chute is closed, and the right end has a hole through which any black ball can pass, but not a white one. How to roll all the black balls out of the chute? It is not allowed to remove balls from the chute.
13. Chain repair
Do you know what the young master was thinking about (Fig. 8)? In front of him are 5 chain links, which must be connected into one chain without using additional rings. If, for example, you can open ring 3 (one operation) and hook it onto ring 4 (one more operation), then open ring 6 and hook onto ring 7, etc., then there will be eight operations in total, and the master seeks to forge the chain at assistance of only six operations. He succeeded. How did he act?
14. Correct the error
Take 12 matches and lay out the "equality" shown in fig. nine.
Figure: 9. Correct the mistake by shifting only one match.
Equality, as you can see, is incorrect, since it turns out that 6 - 4 \u003d 9.
Move one match so that you get the correct equality.
15. Out of three - four (just kidding)
There are 3 matches on the table.
Without adding a single match, make three to four. You cannot 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 your friend to add 2 more to them so that you get eight. Of course, you can't break matches.
17. Three squares
Of 8 sticks (for example, matches), four of which are half the length of the other four, you need to make 3 equal squares.
18. In the lathe shop of the plant, parts are turned from lead blanks. From one workpiece - a part. The shavings resulting from the manufacture of six parts can be: reflowed and prepared another workpiece. 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 chairs on each wall.
20. Arrangement of flags
A small inter-collective farm hydroelectric power station was built by the Komsomol members. On the day of its launch, the pioneers decorate the power plant from the outside on all four sides with garlands, bulbs and flags. There were few checkboxes, 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. The second project they liked more, and they decided place 5 flags each.
Show on the diagram how the pioneers placed 12 flags, 5 on each of the four sides, and how they could arrange them 6 flags.
21. Preserve parity
Take 16 any objects (pieces of 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 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 put the numbers 1, 2 and 3, and you 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 equals 17. It is not difficult, since I suggested what numbers should be placed at the vertices of the triangle. 2
You will have to tinker much longer if I do not say in advance which numbers should be placed at the vertices of the triangle, and I 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 locations. The conditions of the problem can be fulfilled with a wide variety of arrangement 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 over one person.
“But since there are twelve of us, half of the girls will not participate in the game,” Natasha objected briskly.
- Then we'll throw the ball in two! (Every third person catches the ball.)
- Even worse: only four will play ... If you want all the girls to play, you have to throw the ball after four (the fifth catches). There is no other combination.
- And if you throw the ball over 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)? - asked the girls.
- We have already played this way ...
The girls began to draw diagrams of all the suggested ways of playing, and very soon they became convinced that Natasha was right. Only one scheme of the game (except for 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 each time the game would cover all participants. Find out if it is possible with thirteen players to throw the ball over five people?
Is it possible to throw the ball over six people with thirteen players? Think and draw the appropriate diagrams for clarity.
24. Four straight lines
Take a sheet of paper and apply the picture. 14. Draw nine points so that they are arranged in the shape of a square, as shown in fig. 14. Now cross out all the points with four straight lines, without lifting the pencil from the paper.
25. Separate goats from cabbage
Now solve the problem, in a sense opposite to the previous one. There we connected the points with straight lines, but here it is required 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 cabbage 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 went 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 their meeting?
27. At high tide (kidding)
Not far from the coast there is a ship with a rope ladder lowered into the water along the side. The staircase has 10 steps; distance between steps 30 cm. The lowest step touches the water surface. The ocean is very calm today, but the tide begins, which raises
there were two numbers, and water for each hour by 15 cm. How long will the third step of the rope ladder be covered with water?
28. Dial
a) Divide the clock face with two straight lines into three parts so that, adding the numbers, in each part, you get the same sums.
b) Can this dial be divided into 6 parts so that in each part the find-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 record of the number four (IV) there were four sticks (IIII). The cracks on the dial divided it into 4 parts, as shown in fig. 17. The sums of the numbers in each part turned out to be different: in one - 21, in the other - 20, in the third - 20, in the fourth - 17.
I noticed that with a slightly different location of the cracks, the sum of the numbers in each of the four parts of the dial would be 20. With the new location of the cracks, they may not pass through the center of the dial. Redraw the dial in your notebook and find this new crack location.
Figure: 17. Cracks divided the dial into 4 parts.
30. Amazing Clock (Chinese Puzzle)
Once, a watchmaker was urgently asked to visit a house.
“I’m sick,” the watchmaker replied, “and I cannot go. But if the fix is \u200b\u200bnot difficult, I'll 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 treated the work very diligently, and when he finished examining the watch, it was already dark. Considering the work completed, he hastily put on the matched hands and put them on his watch: the large hand on the number 12, and the small hand on the number 6 (it was exactly 6 pm).
But soon after the apprentice returned to the workshop to inform the foreman that the work was done, the phone rang. The boy picked up the phone and heard the angry voice of the customer:
- You have corrected the clock badly, it shows the time incorrectly.
The master's apprentice, surprised by this message, hurried to the customer. When he arrived, the watch he had repaired showed the beginning of nine. The apprentice took out his pocket watch and handed it to the angry master of the house:
- Check, please. Your watch is not behind a second.
The stunned customer was forced to agree that his watch at the moment was indeed showing the correct time.
But the next morning the customer called again and said that the hands of the clock had obviously gone crazy and were walking around the dial as they liked. The master's apprentice ran to the customer. The clock showed the beginning of seven. He checked the time on his watch and got angry:
- You are laughing at me! Your watch is accurate!
The clock did show 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 reason for such incredible incidents.
Have you not guessed what the matter is?
31. Three in a row
Place 9 buttons in the shape of a square on the table, 3 buttons on each side and one in the center (fig. 18). Note that if two or more buttons are located along a straight line, then we will always call this arrangement "side by side". So, AB and CD are rows, each of which has 3 buttons, and EF is a row containing two buttons.
Figure: 18. How many rows are there?
Determine how many rows there are in the picture with 3 buttons each and how many such rows, each of which has only 2 buttons.
Now remove any 3 buttons and place the remaining 6 in 3 rows so that there are 3 buttons in each row.
32. Ten rows
It is not hard to guess how to arrange 16 checkers in 10 rows with 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 piece of blank paper, draw the shape shown in fig. 19, at the same time increasing its size by 2-3 times, and prepare 17 coins of the following denomination:
20 kopecks each - 5 pieces,
15 kopecks each - 3 pieces,
10 kopecks each - 3 pieces,
5 kopecks - 6 pieces.
Figure: 19. Place the coins over the squares of this shape.
Place the prepared coins on the squares of the drawn figure so that the sum of kopecks along each straight line shown in the figure is 55.
34. 1 to 19
In nineteen circles, rice. 20 is required to arrange 19 so that the sum of the numbers in any three circles lying on one straight line equals 30.
35. Fast, but careful
Solve the following 4 tasks "at speed" - who will give the correct answer faster:
Problem 1. At noon, a bus with passengers leaves Moscow for Tula. An hour later, a cyclist leaves Tula for Moscow and travels along the same highway, but, of course, much slower than a bus.
When the passengers of the bus and the cyclist meet, which of them will be further from Moscow?
Task 2. What is more expensive: a kilogram of dimes or half a kilogram of two-dimes?
Problem 3. At 6 o'clock the wall clock struck 6 beats. On my pocket watch, I noticed that the time elapsed from the first blow to the sixth was exactly 30 seconds.
If it took the clock 30 seconds to strike 6 times, how long will the clock continue striking at noon or midnight when the clock strikes 12 times?
Problem 4. Three swallows flew from one point. When will they be on the same plane?
Now check your decisions with calm reasoning and look in the section "Answers".
- Well, how? Have you fallen into those little traps that are contained in these simple tasks?
Such tasks are so attractive because they sharpen the attention and teach you to be careful in the usual train of thought.
all integers from 1 to
Figure: 20. Place the numbers from 1 to 19 in circles.
36. Figured cancer
Figured crayfish, shown in Fig. 21, composed of 17 pieces.
Put together from the pieces of this cancer two shapes at once: a circle and a square next to it.
37. The cost of the book
They paid 1 ruble for the book and half the cost of the book. How much does a book cost?
38. Restless Fly
On the Moscow - Simferopol highway, two athletes simultaneously began a training bike ride towards each other.
At that moment, when there were only 300 km between the cyclists, the fly became very interested in the mileage. Falling off the shoulder of one cyclist and ahead of him, she rushed towards another. After meeting the second cyclist and making sure everything was safe, she immediately turned back. The fly flew to the first athlete and again turned to the second.
So she flew between the approaching cyclists until the cyclists met. Then the fly calmed down and sat on one of them.
The fly flew between cyclists at a speed of 100 km per hour, and all this time the cyclists traveled at a speed of 50 km per hour.
How many kilometers did the fly fly?
39. In less than 50 years
Will there be such a year in this century that if you write it down in numbers and turn the piece of paper upside down, then the number that appears on the turned piece of 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 to leave, and said that the money was in an envelope on the desk. The girl, glancing at the envelope, saw the number 98 written on it, took out the money and, without counting it, put it in
bag, and the envelope was crumpled and thrown away.
She bought 90 rubles worth of things in the store, and when she wanted to pay off, it turned out that not only did she not have eight rubles left, as she had expected, but even four rubles were missing.
At home, she told dad about this and asked if he was mistaken when he counted the money. The father replied that he counted the money correctly, but she herself was mistaken 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 just two pieces of paper, make sure that the sums of the numbers in both columns are the same.
Figure: 22. Equalize unequal amounts.
41. How old am I?
When my father was 31, I was 8, and now my father is twice my age. How old am I now?
42. Rate "at a glance"
Here are two columns of numbers:
123456789 1
12345678 21
1234567 321
123456 4321
12345 54321
1234 654321
123 7654321
12 87654321
1 987654321
Look closely: the numbers in the second column are formed from the same numbers as the numbers in the first column, but with the opposite order of their arrangement. (For clarity, the zeros in the left column have been omitted.)
Which column will give the best result when added?
First, compare these sums "at a glance", that is, without making the addition yet, try to determine whether they should be the same or one should be larger than the other, and then check the addition.
43. Speed \u200b\u200bfolding
Eight six-digit terms (...) are chosen so that, by reasonably grouping them, you can “mentally” find the sum in 8 seconds. Can you sustain this speed?
There are directions in the "Answers" section, but ... you will be looking for them longer.
And show your friends two tricks, which you can also jokingly call “speed folding”.
First trick. Say, “Without showing me, write in a column as many multi-digit numbers as you like. Then I will approach], very quickly I will write as many more numbers and instantly add them all. "
Let's say friends wrote:
7621
3057
2794
4518
And you write down such numbers, each of which complements to 9999, one after the other all the written numbers. These numbers will be:
5481
7205
6942
2378
Indeed: (...)
Now it's not hard 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 4 extra units. It turns out:
10,000 X 4 - 4 \u003d 40,000 - 4 \u003d 39,996.
Here is the whole secret of the trick!
Second trick. Write one under the other any 2 numbers of any size. I'll add the third and instantly, from left to right, write the sum of all three numbers.
Let's say you wrote:
72 603 294
51 273 081
For example, I will add the following number: 48 726 918 and immediately tell you the amount.
What number should be attributed and how in this case to quickly find the amount, 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 other in his left. You can easily guess which hand he has which coin.
Suggest that he triple the number of kopecks in the coin in his right hand and double the number of kopecks in the coin in his left hand. Let him add up the results obtained and tell you only the resulting amount.
If the named amount is even, then in the right hand there are 2 kopecks, if it is odd, then 2 kopecks in the left hand.
Explain why this always works, and think of how you can diversify this trick.
45. How many are there?
The boy has as many sisters as there are brothers, and his sister has half as many sisters as brothers.
How many brothers and sisters are there in this family?
46. \u200b\u200bThe same numbers
Using only addition, write 28 down with five twos and 1000 with eight eights.
47. One hundred
Using any arithmetic operations, make up the number 100 either from five ones or from five fives, and 100 from five fives can be made in two ways.
48. Arithmetic duel
At one time there was such a custom in the math circle of our school. To each new member of the circle, the chairman of the circle offered a simple problem - a kind of mathematical nut. If you solve the problem, you immediately become a member of the circle, and if you fail to cope with the nut, then you can attend the circle as a volunteer.
I remember once our chairman suggested to one newcomer Vitya the following problem: (...)
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).
There are many more different solutions to this problem:
Make up the number 20 by adding exactly eight odd numbers, among which it is also allowed to have the same terms.
Find all the different solutions to this problem and find out how many of them will be such sums that contain the largest number of unequal terms?
A little advice. If you choose the numbers at random, then in this case, you will come across several solutions, but haphazard trials will not give you confidence that you have exhausted all the solutions. If you introduce some order, a system into the “trial method”, then none of the possible solutions will escape you.
50. How many routes?
From a letter from schoolchildren: “Studying in a math circle, we drew a plan of sixteen quarters of our city. On the attached plan diagram (Fig. 23), all quarters are conventionally depicted by the same squares.
We were interested in this question:
How many different routes can you map from point A to point C if you move along the streets of our
Figure: 23. How many routes are there from L to C?
cities only forward and to the right, to the right and forward? Routes may coincide in their individual parts (see the dotted lines on the plan diagram).
We got the impression that this is not an easy task. Did we solve it correctly if we counted 70 different routes? "
What should be answered to this letter?
52. Different actions, one result
If between two twos the addition sign is replaced by the multiplication sign, then the result will not change. Indeed: 2+ 2 \u003d 2X2. It is not difficult to find and. 3 numbers with the same property, namely: 1 + 2 + 3 \u003d \u003d 1X2X3. There are also 4 single-digit numbers, which, when added or multiplied by each other, give the same result.
Who will pick these numbers faster? Is it done? Continue the competition! Find 5, then 6, then 7, etc. single-digit numbers that have the same property. Keep in mind, however, that starting with a group of 5 numbers, the answers may vary.
53. Ninety nine and one hundred
How many plus signs (+) must be placed between the digits of 987 654 321 to add up to 99?
There are two possible solutions. 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). A schoolgirl from Kemerovo claims that two solutions are also possible here.
54. Dismountable chess board
The cheerful chess player cut his cardboard chessboard into 14 pieces, as shown in fig. 25. The result is a collapsible chessboard. To his comrades who came to him to play chess, he first suggested a puzzle: to make a chessboard out of these 14 parts. Cut out the same figures from checkered paper and see for yourself - it is difficult or easy to make a chessboard out of them.
60. Puzzled chauffeur
What did the driver think when he looked at the meter of his car's speedometer (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 read the same both from left to right and from right to left:
15951.
- Interesting! .. - muttered the driver. - Now, probably not soon, another number with the same feature will appear on the counter.
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
A brigade of excellent quality, consisting of a foreman - an old, experienced worker - and 9 young workers who had just graduated from a vocational school, took part in the fulfillment of an urgent order for the manufacture of measuring instruments for the Tsimlyansk hydroelectric complex.
During the day, each of the young workers assembled 15 instruments, and the foreman - 9 more instruments than the average of each of the 10 members of the brigade.
How many measuring instruments were installed by the team in one working day?
62. Bread delivery on time
Starting the delivery of grain to the state, the collective farm management decided to deliver a trainload of grain to the city at exactly 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 kolkhoz to the city, and at what speed do you have to travel to arrive just in time?
63. On the suburban train
In an electric train carriage, two schoolgirl friends were traveling from the city to their dacha.
- I notice, - said one of my friends, - that return suburban trains meet us every 5 minutes. How many suburban trains do you think arrive in a city in one hour if the train speeds in both directions are the same?
“Of course, 12, since 60: 5 \u003d 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. Scary Dream of a Football Fan
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 would hit a soccer ball against a wall and then catch 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 frantically across the smooth floor of the room, trying to crush a small celluloid ball. The poor ball in despair rushed from side to side, exhausted and unable to jump.
Could he, without looking up from the floor, still take refuge somewhere from the pursuit of the cast-iron ball?
Figure: 30. The ball tried to crush the ball.
To solve the problems of the second section, familiarity with actions on simple and decimal fractions is required.
The reader who has not yet studied fractions can temporarily skip the tasks in this section and move on to the next chapters.
66. Clock
Traveling around our great and wonderful homeland, I found myself in places where the difference in air temperatures between day and night was so great that when I was outdoors day and night, it began to affect the course of the clock. I noticed that the changes in temperature during the day led the clock forward by a minute, and overnight lagged behind by a minute.
On the morning of May 1, the clock was still showing the correct time. By what date will they go forward 5 minutes?
67. Ladder
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 flights between the floors have the same number of steps?
68. Puzzle
What sign should be placed between the numbers 2 and 3 written side by side to get a number greater than two but less than three?
69. Interesting fractions
If you add its denominator to the numerator and denominator of the fraction 1/3, then the fraction will double.
Find a fraction that would increase from the addition of the denominator to its numerator and denominator: a) three times, b) four times.
(Those with algebra can generalize the problem and solve it using an equation.)
70; What number?
Half past two. What is this number?
71. The way of the student
Borya makes a long way to school every morning.
At a distance on the way from home to school there is an MTS building with an electric clock on the facade, and at a distance along the entire path there is a railway station. When he passed the MTS, the clock was usually 7:30, and when he reached the station, the clock showed 25 minutes to 8 o'clock.
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. In how many seconds will it reach the twelfth flag at a constant speed? Don't get screwed!
73. Did you guess?
Ostap was returning home from Kiev. The first half of the journey he traveled by train 15 times faster than if he walked. However, the second half of the journey he had to travel on oxen - 2 times slower than if he walked.
Did Ostap win any time compared to walking?
74. Alarm clock
The alarm clock is 4 minutes behind. in hour; It was delivered exactly 3.5 hours ago. Now the clock showing the exact time is exactly 12.
In how many minutes will the alarm clock also be 12?
75. Instead of small shares, large
There is a very exciting profession in engineering plants; it is called - markers. The scribe marks on the workpiece those lines along which this workpiece should be processed in order to give it the required shape.
The marketer 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 plates to the scribe and asked him, if possible, to mark the plates so that they did not have to crush any of them into very small pieces. This means that the simplest solution - to cut each plate into 12 equal parts - was not suitable, as this resulted in many small shares. How to be?
Is it possible to divide these records into larger parts? The scribe thought, made some arithmetic calculations with fractions, and nevertheless found the most economical way of dividing these plates.
Subsequently, he easily crushed 5 plates to distribute them in equal shares between six parts, 13 plates for 12 parts, 13 plates for 36 parts, 26 for 21, etc.
What did the striper do?
76. Soap Bar
A bar of soap is put on one pan of the scales, on the other the same bar and another kg. Libra in balance.
How much does the bar weigh?
79. Misha's kittens
Misha will see an abandoned kitten somewhere, he will certainly pick it up and bring it home. He always brings up several kittens, and how many, he did not like to talk, so that they would not laugh at him.
Sometimes they would ask him:
- How many kittens do you have now?
“A little,” he will answer. - Three quarters of them, and even three quarters of one kitten.
The comrades thought he was just joking. Meanwhile, Misha asked them a problem that was not at all difficult to solve. Try it!
80. Average speed
Half the way, the horse walked empty at a speed of 12 km / h. The rest of the way she went with a cart, making 4 km / h.
What is the average speed, that is, with what constant speed would a horse need to move in order to use the same amount of time for the entire 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 go half of the way that he traveled asleep. How much of the entire journey has he traveled asleep?
82. How long is the train?
Two trains run 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 on the second train noticed that the first train was passing by him for 6 seconds. How long is the first train?
83. Cyclist
When the cyclist traveled 2/3 of the way, a tire burst.
He spent the rest of the way on foot twice as long as cycling.
How many times did the cyclist go faster than he walked?
84. Competition
Turners Volodya A. and Kostya B. - students of the metalworkers' vocational school, having received from the master the same outfit for the manufacture of a batch of parts, wanted to complete their tasks simultaneously and ahead of time.
After a while it turned out, however, that Kostya had done only half of what was left to do for Volodya, and Volodya had to do half of what he had already done.
How many times would Kostya have to increase his daily output in comparison with Volodya in order to have time to complete his outfit at the same time?
Chapter two
DIFFICULT PROVISIONS
87. Hecho Blacksmith Savvy
Traveling in Georgia last summer, we sometimes entertained ourselves by inventing all kinds of extraordinary stories inspired by some ancient monument.
Once we came to a lonely ancient tower. We examined her, sat down to rest. And there was a student of mathematics 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 to wife to a rich neighbor, and she fell in love with a simple guy, the blacksmith Khecho. Darijan and Khecho tried to escape to the mountains from captivity, but their servants Knyazevs caught them.
The prince got angry and decided to execute both of them the next day, but for the night he ordered them to be locked up in this high, gloomy, abandoned, unfinished tower, and along with them the servant Darijan, a teenage girl who helped them escape.
I was not at a loss in the Khecho tower, looked around, climbed the steps to the top of the tower, looked out the window - it was impossible to jump, you would crash. Then Khecho noticed a rope forgotten by the builders near the window, thrown over a rusty block, reinforced higher
window. Empty baskets were tied to the ends of the rope, and a basket to each end. Khecho remembered that with the help of these baskets the bricklayers lifted the brick up, and lowered the rubble down, moreover, if the weight of the load in one basket exceeded the weight of the load in the other by about 5-6 kg (in modern terms), then the basket lowered rather to the ground; another basket at this time was raised to the window.
Khecho determined by eye that Darijan weighs about 50 kg, the maid is no more than 40 kg. Khecho knew his weight - about 90 kg. In addition, he found a chain weighing 30 kg in the tower. Since each basket could accommodate a person and a chain or even 2 people, all three of them managed to descend to the ground, and they descended so that the weight of the lowering basket with a person never exceeded the weight of the lifting 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 mistress to 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 one direction, so that the white mouse is eaten last."
But with which mouse to start 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 the 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.
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1,2,3,4,5,6…
1,4,16…
45,39,33,27…
0,3,8,15,24…
112,56,28,14…
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6* : *7 = *
68:17 = 4
** +** =20
10+10= 20
* 2 -* = *
12- 4 = 8
*** +**=1**
101 +41+142
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