Logic tasks. Logic tasks Peter lied
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Baal Island is inhabited only by humans and strange monkeys that cannot be distinguished from humans. Any of the inhabitants of the island speaks either only the truth, or only a lie. Who are the next two? A: “B lying monkey. I am human." B: "A told the truth." Task #1
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SOLUTION: The double statement used by A is true only if both parts of it are true. Suppose B is an honest person, in which case A is also honest (that's what B says), so B is a knave, as A claims, which contradicts our assumption. Therefore B is a knave. Knowing this very well, B said that A was also a liar. Thus, A's first statement is a lie, and B is not a lying monkey. However, B, as we have already found out, is definitely a liar, which means that B is not a monkey. B is a dishonest person. The second statement A shows us that A is a monkey. Therefore, A is a lying monkey.
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Task №2 Three goddesses sat in an ancient Indian temple: Truth, Falsehood and Wisdom. Truth only tells the truth, Lies always lie, and Wisdom can tell the truth or lie. The pilgrim asked the goddess on the left: “Who is sitting next to you?” "True," she replied. Then he asked the middle one: “Who are you?” "Wisdom," she replied. Finally he asked the one on the right, “Who is your neighbor?” "False," replied the goddess. And after that, the pilgrim knew exactly who was who.
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Solution: Let's designate each goddess with a certain letter. We have the following statements at our disposal: 1. A says that B is True. 2. B says she is Wisdom. 3. C says that B is False. The first sentence tells us that A is not True. The second sentence was also not told by the Truth, therefore Truth is C. Whence it is clear that the last sentence is true: B is False, and A is Wisdom.
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Task number 3 There are three coins on the table: gold, silver and copper. If you say a statement that turns out to be true, you will be given a coin. Nothing will be given to you for lying. What do you have to say to get a gold coin?
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Solution: "You will give me neither a copper nor a silver coin." If this statement is true, then they will give me a gold coin. If my statement is false, then the reverse statement must be true, namely: "You will give me either a copper or a silver coin." But then this contradicts the conditions of the task - they should not give coins for a lie. Therefore, the original statement is true.
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Task number 4 You arrived at a fork in the two roads. One of them leads to the False City, where there is a general store for the clues of the Universe, which are released for free. Another road leads to Pravdograd, where there is a gas station. Residents of False City always lie, and residents of Pravdograd always tell the truth and nothing but the truth. At the fork, one representative from each of the two cities is on duty. You don't know which one is from where. How to find out which road leads to Pravdograd if you are allowed to ask only one question to only one representative?
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Solution: There are several options for such questions. Indirect question: “Hey you! What will that person say if I ask him where this road leads? The answer to such a question will always contradict where the road actually leads. Trick question: “Hey you! That person who is on duty at the road leading to Pravdograd, is he from there? The answer will be positive only in two cases: either this is a resident of Pravdograd, standing on the road to Pravdograd, or a resident of False City, standing on the same road. In both cases, you can be sure that with an affirmative answer, this road will really lead you to Pravdograd. In the same way, a negative question can be formulated. Or another tricky question: “Hey you! What would you say if I asked you...?”. A resident of Pravdograd will always answer the truth, and a resident of Lzhegrad will lie. However, due to the wording of the question, the liar will have to lie twice, that is, to tell the truth.
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Task №5 Peter lied from Monday to Wednesday and told the truth on other days, and Ivan lied from Thursday to Saturday and told the truth on other days. One day they said in the same way: "Yesterday was one of the days when I lie." What day did they say this?
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Solution: It was Thursday. On this day, Peter truthfully said that yesterday (i.e. on Wednesday) he lied, and Ivan lied about the fact that yesterday (i.e. on Wednesday) he lied, because according to the condition on Wednesday he tells the truth.
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Task number 6 Lady Cat said: “I am the most beautiful. Mary is not the most beautiful." Jane said, “Kat is not the prettiest. I am the most beautiful." And Mary just said, "I'm the most beautiful." The white knight suggested that all the statements of the most beautiful of the girls are true, and all the statements of the other ladies are false. Based on this, determine the most beautiful of the ladies.
DID EISENHAUER LIE?
This episode, narrated by prominent American military and political figure Dwyde Eisenhower, has been frequently quoted in recent years. So, in his documentary about the Great Patriotic War, he was beaten by the popular television master Yevgeny Kiselev. In his largely controversial book, "Unknown Zhukov: a portrait without retouching", he is cited as an example by the writer Boris Sokolov (By the way, in 2001, in one of the central newspapers, I had to read in an article dedicated to Marshal Zhukov about the same episode, but without reference to the source, as a matter of course. They say, the marshal was controversial, although he was talented. But on the mined fields, before launching equipment on them, he drove infantry forward, etc. see above.). Here is this passage: “I was very struck by the Russian method of overcoming minefields, which Zhukov told about,” Eisenhower wrote in his book Crusade to Europe. “German minefields, covered by fire, were a serious tactical obstacle and caused significant losses and delay It was difficult to break through them, although our specialists used various mechanical devices to safely undermine them.Marshal Zhukov told me about his practice, which, roughly speaking, boiled down to the following: "When we approach a minefield, our infantry conducts attack as if the minefield does not exist. The losses suffered by troops from anti-personnel mines are considered to be only equal to those that we would suffer from artillery and machine-gun fire if the Germans covered the area not only with minefields, but with a significant number of troops. Attacking infantry does not detonate anti-tank mines. When it reaches the far end of the field, a passage is formed through which sappers go and remove anti-tank mines so that the equipment can be launched. "I vividly imagined what would happen if any American or British commander adopted such tactics, and more vividly imagined what people in any of our divisions would say if they tried to make the practice of this kind part of their military doctrine.
These words of a major military leader of the Second World War, and later one of the presidents of the United States of America, of course, would be impossible to read without horror if they corresponded to the truth. But let's try to figure out whether the above is true without unnecessary emotions.
In the film directed by Yevgeny Matveev "Fate" there is an episode: SS men under the barrels of machine guns force our captured soldiers to drag harrows through a minefield. In this case, the Nazis, or the authors of the film, understood that simply chasing prisoners without technical means, i.e. harrows, would be an ineffective occupation - some of the mines would certainly be missed and remain in the same combat state. Consequently, a simple attack to clear the fields (if you still imagine that such a thing took place) would be even less effective. After all, people are not robots - they would definitely start looking for loopholes (a wider jump, running along already laid tracks in front of the runner). This would nullify all the "strategic" plans of the commanders.
In conversations with veterans of the Great Patriotic War, I had to make sure more than once that none of them, who came out alive from the bloodiest battles, who lost hundreds and thousands of their comrades, had never heard of anything like that. But, apparently, we are talking about the massive use of such a strategy. Therefore, witnesses should have remained (at least one of those who ran to the edge of the field!). By the way, none of those who quoted the American marshal cited any other evidence as an example (In Sokolov's book, however, there is an excerpt from a letter from a German soldier, but it is written very indistinctly and is not very convincing). Also distrustfully reacted to the bike, told by the famous American marshal, as a matter completely meaningless from a technical point of view, and explosives experts with whom I had to talk.
Another thing is also curious, Georgy Konstantinovich, allegedly talking about the advantages of this "the best way to overcome minefields," had in mind the military operations of the Red Army in Europe. That is, those operations when the country had already overcome the crisis of the lack of modern weapons, when the Red Army learned to use these weapons, and when, finally, this army became especially in need of human resources. This is evidenced even by the fact that by the year 44, 17-year-old boys began to be drafted into the army, who died in the very first battles. And then, thanks to the victories in Europe, many of those 17-year-olds who survived were recalled back to the rear in order to protect them from further extermination. That is, there is no need to talk about the endless human resources of the Soviet Union - this is another myth invented in the West. (It must also be borne in mind that the Second World War was a war between two economies and significant human resources had to be kept in the rear in production.)
Meanwhile, from the time when the Red Army stopped retreating, barrage detachments ceased to be used (which, by the way, in various versions and at different times, existed in other armies of the world), and even penal companies in the attack no one fired in the back did not customize.
Of course, it is excusable for Americans to imagine Soviet soldiers as such zombies deprived of their own will, capable of good will, lining up in close ranks and typing a step (only in this way, if you obey logic, you can be guaranteed to clear the minefield of explosive devices), under enemy fire, carry out the order of your direct commander, who immediately, in accordance with the charter, is obliged to step ahead. To imagine this, I repeat, is forgivable for Americans (in modern Hollywood films you can see thousands of absurdities about our past and present), but perhaps we, Russians, should not take on faith any heresy that is published today in various dubious publications?
However, the question arises: how, in this case, did the infantry pass through minefields during attacks? The answer to it is given by the American military themselves, veterans of the Second World War. During the landing operation on the coast of Normandy, which marked the opening of the Second Front, which was directly commanded by Eisenhower, the Allies just encountered the very minefields and wire fences that one of the best top commanders of the German army of that time, Erwin Rommel, took care of with German pedantry . To the credit of the allies, these barriers could not become a serious obstacle to the landing. They acted with minefields ingeniously and simply (the technology, by the way, was worked out back in the First World War) - corridors were made in them with the help of aerial bombs and heavy artillery. By the way, mines are destroyed by detonation even today - the Americans used super-heavy bombs to destroy mines during the famous "Desert Storm" in 1991, and even in 2004 during the occupation of Iraq. And by 1944, the Red Army had an advantage over the German in artillery by about 20:1. And Zhukov, if only to save time and money, would certainly have preferred in this case artillery shelling in squares to the masses of infantry, whose numerical advantage over the German was not so overwhelming.
So, a professional military man would never take the words of the Soviet Marshal on faith if they were actually uttered. Why, then, was Eisenhower cunning in his book? Perhaps the American was simply jealous of the successes of his Russian colleague and was looking for a reason to justify himself to his fellow citizens for the much smaller achievements of the armies he led. In addition, Eisenhower already at that time saw himself as a future politician (as he himself testifies in his book) and, naturally, sought to gain popularity among voters as a politician. And what is the meaning of the word spoken by a politician who wants to be elected - the Russians have already had the opportunity to make sure more than once. So Eisenhower bought his electorate cheaply with this "Russian horror story." Say, we, the Americans, lagged behind the pace of the offensive of the Soviet troops in the Second World War because the minefields were cleared with the help of technology. And if they did it like the Russians (that's the secret of success!), then not only in Berlin, they would have been in Moscow long ago!
But perhaps this is not the whole truth. The most interesting thing is that G.K. Zhukov could really tell this "terrible story" to Eisenhower. He, in turn, could "buy" a naive American (after all, it is known that guests from overseas often do not catch our domestic humor). And judging by the notes of eyewitnesses, Georgy Konstantinovich was a master at such jokes, apparently hiding his irritation behind them at times. When, under Khrushchev, he was massacred at one of the meetings of the Politburo, accusing him of Bonapartism, he answered not without a challenge: "Bonaparte lost the war, but I won!" When one of the Soviet newspapers already in the post-war years asked a number of military marshals, is it possible to get this highest military rank in peacetime? He alone answered in the affirmative that yes, if you study a lot and, among other things, pay more attention to Marxism (they say that at that time they were already trying to assign the marshal rank to Khrushchev). What is this if not a hidden mockery? And, to the generally idle question of an American, when any operation, including those carried out by the Red Army in order to divert forces from the front in the West, cost hundreds of thousands of lives, you see, the evil irony was quite appropriate.
So, perhaps, from a misunderstood joke, an unsubstantiated statement was born, which suddenly pops up in one or another publication dedicated to our outstanding commander. Having broken the backbone of the best army in the world, which until the year 43 was the German Army, the Red Army, at that time, undoubtedly acquired the qualities of the best. The Americans and the British did not have such rich experience in combat operations in the field. Our military equipment (especially ground-based) surpassed all foreign analogues in many respects. After the Battle of Kursk-Oryol, Soviet generals fought with fewer losses than their opponents.
Of course, the losses, especially in the initial period of the war, were huge. They were there later - probably, the youth and poor training of so many of our commanders and privates affected. But even that war was incredibly cruel. It was a war not of armies, but of countries and peoples. In its second period, starting with Stalingrad, the Germans also suffered completely senseless and unjustified losses. The Americans and the British, fighting on foreign territory, were unaware of such fury, where they spare neither themselves nor the enemy. From today's standpoint it is not possible to give a completely objective assessment of those events. And before condemning the past, let's look back at ourselves today. Is it not in our days that conscript boys were sent to die in Chechnya? Let's look back and see how indifferent we are to our compatriots today.
- How old is your father? the boy is asked.
“As much as I do,” he replies calmly.
- How is this possible?
- Very simple: my father became my father only when I was born, because before my birth he was not my father, which means that my father is the same age as me.
Is this reasoning correct? If not, what is wrong with it?
77. There are 24 kilograms of nails in a bag. How is it possible to measure 9 kilograms of nails on a pan balance without weights?
78. Peter lied from Monday to Wednesday and told the truth on other days, while Ivan lied from Thursday to Saturday and told the truth on other days. One day they said in the same way: "Yesterday was one of the days when I lie." What day was yesterday?
79. The three-digit number was written in numbers, and then in words. It turned out that all the numbers in this number are different and increase from left to right, and all words begin with the same letter. What is this number?
80. In an equality made up of matches:
X I I I \u003d V I I–V I,
an error has been made. How should one match be shifted in order for the equality to become true?
81. How many times will a three-digit number increase if the same number is assigned to it?
82. If there were no time, there would be no day. If there were no day, it would always be night. But if it were always night, there would be time. Therefore, if there were no time, there would be. What is the reason for this misunderstanding?
83. Each of the two baskets contains 12 apples. Nastya took a few apples from the first basket, and Masha took from the second as many as were left in the first. How many apples are left in the two baskets together?
84. One farmer has 8 pigs: 3 pink, 4 brown and 1 black. How many pigs can say that in this small herd there is at least one more pig of the same color as her own?
85. The only son of the shoemaker's father is a carpenter. Who is the cobbler to the carpenter?
86. If 1 worker can build a house in 5 days, then 5 workers can build it in 1 day. Therefore, if 1 ship crosses the Atlantic Ocean in 5 days, then 5 ships will cross it in 1 day. Is this statement correct? If not, what is the error in it?
87. Returning from school, Petya and Sasha went to the store, where they saw large scales.
“Let's weigh our portfolios,” suggested Petya.
The scales showed that Petya's portfolio weighed 2 kilograms, while Sasha's portfolio weighed 3 kilograms. When the boys weighed the two briefcases together, the scales showed 6 kilograms.
- How so? Petya was surprised. Because 2 plus 3 does not equal 6.
- Can't you see? Sasha answered him. - The arrow has shifted on the scales.
What is the real weight of portfolios?
88. How to place 6 circles on the plane in such a way that you get 3 rows of 3 circles in each row?
89. After seven washes, the length, width and height of the bar of soap had halved. How many washes will the remaining piece last?
90. How to cut off 1/2 m from a piece of matter in 2/3 m without the help of any measuring instruments?
91. It is often said that one must be born a composer (or an artist, or a writer, or a scientist). Is this true? Is it really necessary to be born as a composer (artist, writer, scientist)?
92. You don't have to have eyes to see. We see without the right eye. We also see without the left. And since we have no other eyes besides the left and right eyes, it turns out that neither eye is necessary for vision. Is this statement correct? If not, what is wrong with it?
93. The parrot has lived less than 100 years and can only answer yes and no questions. How many questions does he need to ask to find out his age?
94. How many cubes are shown in fig. 51?
95. Three calves - how many legs?
96. One man who fell into captivity recounts the following: “My dungeon was in the upper part of the castle. After many days of effort, I managed to break one of the bars in the narrow window. It was possible to crawl through the resulting hole, but the distance to the ground was too great to simply jump down. In the corner of the dungeon, I found a rope forgotten by someone. However, it turned out to be too short to be able to go down it. Then I remembered how one wise man lengthened a blanket too short for him, cutting off part of it from below and sewing it on top. So I hastened to split the rope in half and re-tie the two resulting parts. Then it became long enough, and I safely went down it. How did the narrator manage to do this?
97. The interlocutor asks you to think of any three-digit number, and then offers to write down its numbers in reverse order to get another three-digit number. For example, 528–825, 439–934, etc. Then he asks to subtract the smaller number from the larger number and tell him the last digit of the difference. After that, he names the difference. How he does it?
98. Seven walked - they found seven rubles. If not for seven, but for three, would you find a lot?
99. Divide the drawing, consisting of seven circles, with three straight lines into seven parts so that there is one circle in each part (Fig. 52).
100. The globe was pulled together by a hoop along the equator. Then the length of the hoop was increased by 10 meters. At the same time, a small gap formed between the surface of the globe and the hoop. Can a person get through this gap? The length of the earth's equator is approximately 40,000 kilometers.
1. One coin must be pulled out from the first bag, two from the second, three from the third, and so on (all 10 coins from the tenth bag). Next, you should once weigh all these coins together. If there were no fake coins among them, i.e., they would all weigh 10 grams, then their total weight would be 550 grams. But since there are counterfeit coins (11 grams each) among the weighed coins, their total weight will be more than 550 grams. Moreover, if it turns out to be 551 grams, then the fake coins are in the first bag, because we took one coin from it, which gave one extra gram. If the total weight is 552 grams, then the counterfeit coins are in the second bag, because we took two coins from it. If the total weight is 553 grams, then the counterfeit coins are in the third bag, and so on. Thus, with only one weighing, it is possible to determine exactly which bag contains the counterfeit coins.
2. It is necessary to take cookies from a jar with the inscription "Oatmeal cookies" (you can use any other). Since the jar is labeled incorrectly, it will be shortbread or chocolate. Let's say you got a shortbread. After that, you need to swap the labels "Oatmeal Cookies" and "Shortbread Cookies". And since, according to the condition, all the labels are mixed up, now there is oatmeal in the jar with the inscription “Chocolate cookies”, and there is chocolate in the jar with the inscription “Oatmeal cookies”, which means that these two labels must also be swapped.
3. Only three socks need to be taken out of the closet. In this case, only 4 options are possible: all three socks are white; all three socks are black; two socks are white, one is black; two socks are black, one is white. In each of these combinations there is one matching pair - white or black.
4. The clock will strike 12 hours in 66 seconds. When the clock strikes 6 o'clock, there are 5 intervals from the first strike to the last. The interval is 6 seconds (1/5 of 30). When the clock strikes 12 o'clock, there are 11 intervals from the first strike to the last. Since the length of the interval is 6 seconds, it takes 66 seconds for the clock to break through 12 hours: 11 6 = 66.
5. The pond will be half covered with lily leaves on the 99th day. According to the condition, the number of leaves doubles every day, and if on the 99th day the pond is half covered with leaves, then the next day the second half of the pond will be covered with lily leaves, i.e., the pond will be completely covered with them after 100 days.
6. The path traveled to the fifth floor (4 spans) by a passenger elevator is twice as long as the path traveled to the third floor (2 spans) by a freight elevator. Since the passenger elevator goes 2 times faster than the freight elevator, they will pass their paths at the same time.
7. To solve this problem, you need to write an equation. The number of geese in a flock is X. “Now, if there were as many of us as there are now (i.e. X), - said the geese, - and so much more (i.e. X), and even half as much (i.e. 1/2 X), and even a quarter-so much (i.e. 1/4 X), and even you (i.e. 1 goose), then we would have been 100 geese. It turns out the following equation:
Let's add on the left side of the equation:
So, there were 36 geese in the flock.
8. The error lies in squaring each part of the equation -2 = 2. The appearance is created that the same operation is performed on each part of the equality (squaring), but in fact, different operations are performed on each part of the equality, because we multiply the left side by -2, and multiply the right side by 2.
9. The statement that the atomic nucleus is 2 times smaller than the atom itself is, of course, incorrect: after all, 10-12 cm is less than 10-6 cm not 2 times, but a million times.
10. The plane in flight "holds" on the air, so it is impossible to fly by plane to the Moon, because there is no air in outer space.
11. The needle is made of steel and the coin is made of copper. Steel is much harder than copper, and therefore it is quite possible to pierce a coin with a needle. It's impossible to do it manually. If you try to hammer the needle into the coin with a hammer, then nothing will work either: the area of \u200b\u200bthe sharp end of the needle is so small that its tip will, vibrating, slide along the surface of the coin. In order for the needle to be stable, it is necessary to drive it with a hammer into a coin through a piece of soap, paraffin or wood: this material will give the needle an unchanged and necessary direction, in which case it will freely pass through the copper coin.
12. More than a thousand pins can be placed in a glass. In this case, not a drop of water will spill out of it, but a small water bulge, a "slide" will form above the edges of the glass. According to the law of Archimedes, a body immersed in water displaces a volume of water equal to the volume of the body. The volume of one pin is so small that the volume of the water "slide" above the surface of the glass is equal to the volume of more than a thousand pins.
13. The portrait depicts the son of Ivanov. To solve the problem, you can make a simple scheme:
14. It is necessary to turn to any of the warriors with the following question: “If I ask you if this exit leads to freedom, will you answer me“ yes ”?” With such a formulation of the question, the warrior who lies all the time will be forced to tell the truth. Suppose you, pointing him to the exit to freedom, say: “If I ask you, does this exit lead to freedom, will you answer me yes?” In this case, it will be true if he answers “no”, but he needs to lie, and therefore he is forced to say “yes”.
15. The thief tied the lower ends of the ropes together. On one of them, he climbed to the ceiling, cut the second rope at a distance of about 30 centimeters from the ceiling and let it fall down. From a piece of the second rope, left hanging, he tied a loop. Then, seizing the loop, he cut the first rope and put it through the loop.
After that, he climbed down the double rope and pulled the rope out of the noose.
16. If the taxi driver is deaf, how did he understand where to take the girl? And one more thing: how did he understand that she was saying anything at all?
17. The water will never reach the porthole because the liner rises with the water.
18. He reasoned thus: “Each of us may think that his own face is clean. B. is sure that his face is clean, and laughs at C's dirty forehead. But if B. saw that my face was clean, he would be surprised by V.'s laughter, since in this case V. would not have a reason to laugh . However, B. is not surprised, so he may think that V. is laughing at me. Therefore, my face is dirty."
19. You need to move the top match, forming a tiny square in the center of the figure.
20. A point on the path that the traveler passes at the same time of day both during the ascent and during the descent exists ( BUT). This can be easily verified using the following diagram (Fig. 53).
Axis X - is the time of day, and the axis y - is the lift height. The curved lines are the ascent and descent graphs, respectively. The point of their intersection is exactly the one that the traveler passes at the same time of day both on the ascent and on the descent.
21. The statues should be arranged as follows (Fig. 54).
22. See fig. 55.
23. The exchange is beneficial to the mathematician and disadvantageous to the merchant, since the amount of money that the merchant pays to the mathematician, even if negligible at the beginning, increases exponentially, and the money that the mathematician pays to the merchant increases in arithmetic progression. After 30 days, the mathematician will give the merchant about 50,000 rubles, and the merchant will owe the mathematician more than 10,000,000 rubles.
24. New Year's Eve and earlier (i.e., according to the old style) were celebrated on January 1. However, the old January 1 (Old New Year) now, i.e., according to the new style, falls on January 14, so there is no contradiction or misunderstanding here. In the condition of the problem, an appearance of contradiction is created due to the fact that different concepts are mixed in the same words: New Year according to the new style and New Year according to the old style. Indeed, New Year's Eve in the old style would fall on December 19th, and New Year's Day in the old style in the new style would fall on January 14th.
25. See fig. 56.
26. See fig. 57.
27. The person on the left, be he Truthful, to the question "Who is standing next to you?" could not answer what he answered - "Truth-lover." So, on the left is not the Truth-lover.
But the Truth-lover is not in the center, because, being a Truth-lover, to the question “Who are you?” he could not have answered the way he answered - "Diplomat".
This means that the Truth-lover is on the right, and, consequently, next to him, that is, in the center, is the Liar, and the Diplomat is on the left.
28. The sequence of transfusions is presented in the following table, where I is a bucket of 10 liters; II - a bucket with a volume of 7 liters; III - a bucket of 3 liters.
Thus, to divide 10 liters of wine in half, using two empty buckets of 7 liters and 3 liters, you can use 10 transfusions.
29. Katya will arrive at the train first, and Andrey will most likely miss the train, as he will arrive at the station by the time his clock reads 8:05. And in fact it will be 10 minutes later - at 8 hours and 15 minutes. Katya will try to arrive by her clock by 7:50, but in fact it will be 7:45 then.
30. To solve this problem, you need to write an equation. But first, based on the confusing answer of the dinosaur, the following scheme should be built (we will take the age of the turtle in the past as X):
So, in the diagram we see that now the dinosaur is really 10 times more years old than the turtle was when the dinosaur was as old as the turtle is now. Since the difference in age in the past and in the present remains the same, we make equation 110 - X = 10X – 110.
Let's transform it:
110 + 110 = 10X + X ,
220 = 11X ,
X = 220: 11 = 20.
Therefore, the turtle was 20 years old in the past, the dinosaur is now 10 times older, i.e. 200 years old.
31. The sum of the diameters of small semicircles ( AC) + (CD) + (D.B.) is equal to the diameter of the great semicircle AB, but due to the fact that the length of the semicircle is equal to half the product of the number π per diameter, the distances covered by the cars will be exactly the same. Consequently, the backlog of the police car from the hijacker will not decrease, and the pursuit in this area will not be successful.
32. To solve this problem, you need to draw up a simple scheme (let's denote Katya's current age as X):
It follows from the diagram that the oldest is Katya, followed by Olya and Nastya by age.
33. All the truthful ones correctly claimed that everything they wrote was true, but all the liars falsely claimed that everything they wrote was true. Thus, all 35 essays turned out to be with a statement about the veracity of what was written.
34. Each person has 2 parents, 4 grandparents, 8 great grandparents, 16 great great grandparents. Let's find out how many great-great-grandparents and great-great-grandfathers all of the great-great-grandparents of each of us had: 16 16 \u003d 256. This result is obtained, of course, if we exclude cases of incest, that is, marriages between different relatives.
If we take into account that one generation is about 25 years, then eight generations (which were discussed in the condition of the problem) correspond to 200 years, that is, 200 years ago, every 256 people on Earth were relatives of each of us. For 400 years, the number of our ancestors will be: 256 256 = 65,536 people, i.e. 400 years ago, each of us had 65,536 relatives living on the planet. If we “unscrew” history 1000 years ago, it turns out that the entire population of the Earth at that time was relatives of each of us. So, indeed, all people are brothers.
35. You can try, using the inertia of the bottle, with a sharp movement to pull the handkerchief out from under it.
But, most likely, nothing will work out: the position of the bottle is too unstable. However, remember that the friction force decreases with vibrations. With the fist of one hand, you should evenly and gently knock on the table near the bottle, and with the other hand, gently pull the handkerchief. At a certain frequency and force of blows on the table, the handkerchief will begin to slide smoothly out from under the bottle. At the same time, it is important to pay attention to the fact that there is not a very large edge at the edge of the scarf: it, as a rule, knocks down the bottle at the last moment. Therefore, it is better that the scarf is generally without an edge.
36. With a single dash, one of the plus signs will turn into the number four, resulting in equality:
Here is this dash: → 5 "+ 5 + 5 = 550.
37. In this reasoning, various mathematical operations are mixed in the same words: division by two and multiplication by two. It is on this confusion that the catch is based in the form of an outwardly correct proof of a false thought.
38. See fig. 58.
39. Room for an apartment.
40. It is impossible, because after 72 hours, i.e. after three days, it will again be 12 o'clock at night, and the sun does not shine at night (unless, of course, it happens beyond the Arctic Circle on a polar day).
41. The hostess has 25 rubles, the boy has 2 rubles. Only 27 rubles, which means that the 2 rubles that the boy received are included in 27 rubles. And in the condition of the problem, 2 rubles are added to 27 rubles, which the boy has, and therefore 29 rubles are obtained. It is necessary not to add 2 rubles to 27 rubles, but to subtract.
42. 1 l is equal to 1 dm3. Consequently, 1,000,000 dm3 of water, or 1000 m3 of water, were poured into the pool (since 1 m is equal to 10 dm). Knowing the area of the pool (1 ha = 10,000 m2) and the volume of water poured into it, it is easy to calculate its depth:
It is impossible to swim in a pool 10 centimeters deep.
43. To compare these values, it is necessary to bring the square root and the cubic root to the root of the same degree. It could be a sixth root. The root expressions will change accordingly. It turns out
The sixth root of nine is slightly larger than the sixth root of eight, so
more than
44. We denote the cost of the line as X. Then one boy has money ( X- 24) kopecks, and the other ( X- 2) kopecks. When adding up their money, they still couldn't buy the ruler. Let's make a simple inequality:
(x – 24) + (x – 2) < x.
Let's transform it:
x – 24 + X – 2 < X ,
2X – 26 < X ,
2x - x < 26,
X < 26.
So, the ruler costs less than 26 kopecks, but more than 24 kopecks, since, according to the condition, one boy does not have enough 24 kopecks to reach its value. The ruler costs 25 kopecks.
45. It is necessary to ask any deputy: "Are you a conservative?" If he answered “yes”, then today is an even number, and if “no”, then it is odd. On even numbers, conservatives will say a true yes, and liberals, lying, will also say yes. On odd numbers, on the other hand, conservatives answering a question will say no, but liberals, who speak only the truth these days, will also say no.
46. At first glance, it seems that a bottle costs 1 ruble, and a cork costs 10 kopecks, but then a bottle is 90 kopecks more expensive than a cork, and not 1 ruble, as by convention. In fact, a bottle costs 1 ruble 05 kopecks, and a cork costs 5 kopecks.
47. It may seem that Olya walks 30 steps - 2 times less than Katya (since she lives 2 times lower). Actually it is not. When Katya goes up to the fourth floor, she overcomes 3 flights of stairs between floors. This means that there are 20 steps between two floors: 60: 3 = 20. Olya climbs from the first floor to the second, therefore, she overcomes 20 steps.
48. This is the number 91, which, when turned upside down, becomes 16. In doing so, it decreases by 75 (because 91–16 = 75). When solving this problem, it must be taken into account that when a number is turned over, its digits not only turn over, but also change places.
49. There will be 128 holes on the unfolded sheet. It must be taken into account that with each folding of the sheet, the number of holes doubles.
50. Three people: grandfather, father and son - these are two fathers and two sons - caught three birds with one stone, each one.
51. The effect of this trick problem is that increasing any three-digit number to a six-digit number by duplicating it is equivalent to multiplying this three-digit number by 1001. In addition, the product of the numbers 13, 11 and 7 is also 1001. Therefore, if the resulting six-digit number is divided by any sequences for these three numbers (13, 11, 7), then you get the original three-digit number.
52. See fig. 59.
53. 90 schoolchildren speak one language or another, since, according to the condition, 10 people have not mastered a single language. Of these 90 people, 15 did not pass German, since 75 passed it by condition, and 7 people did not pass English, since 83 passed it by condition. This means that there are 22 people who did not pass one of the exams (since 15 + 7 = 22).
68 schoolchildren mastered two languages (90–22 = 68).
54. Any dish of the correct cylindrical shape, when viewed from the side, is a rectangle. As you know, the diagonal of a rectangle divides it into two equal parts. Similarly, a cylinder is bisected by an ellipse. Water must be drained from a cylindrical dish filled with water until the surface of the water on one side reaches the corner of the dish, where its bottom meets the wall, and on the other side, the edge of the dish through which it is poured. In this case, exactly half of the water will remain in the dish (Fig. 60).
55. It may seem that during the specified period, the hands of the clock will coincide only 3 times: at 12 o'clock in the afternoon, then at 24 o'clock on the same day and at 12 o'clock the next day. In fact, the hour and minute hands coincide every hour 1 time (when the minute hand overtakes the hour hand). From 6 o'clock in the morning of one day to 10 o'clock in the evening of another day, 40 hours pass - which means that during this time the hour and minute hands must coincide 40 times. But 3 hours out of those 40 hours are an exception: 12 hours of one day, 24 hours of the same day, and 12 hours of another day. Imagine that at 12 o'clock the hands coincided, the next time the minute hand catches up with the hour hand not at the first hour, but at the beginning of the second, i.e. from 12 o'clock to 1 o'clock (it doesn't matter - day or night), the hands do not coincide. Therefore, the hour and minute hands from 6 o'clock in the morning of one day to 10 o'clock in the evening of the next day will coincide 37 times.
56. Let's take the speed of the ship as X, and the speed of the river y. Since the ship floats from Nizhny Novgorod to Astrakhan, its own speed and the speed of the river add up, i.e., it floats to Astrakhan at a speed ( x + y). On the way back, the ship sails against the current, i.e. at a speed ( x - y). As you know, distance is equal to the product of speed and time. Knowing that the ship made the same journey in 5 and 7 days, we can make an equation:
5(x + y) = 7(x - y).
Let's transform it:
5x + 5 y= 7X - 7y,
7y + 5y= 7X - 5X,
12y= 2X,
6y = x.
As you can see, the own speed of the ship is 6 times the speed of the river. So, downstream (from Nizhny Novgorod to Astrakhan) he swims at a speed 7 times greater than the speed of the river, because in this case the speeds of the ship and the river add up. Since the raft floats only with the flow, its speed is equal to the speed of the river, which means that it is 7 times less than the speed of the ship on the way to Astrakhan. Consequently, the raft will spend 7 times more time on the same path than the ship:
The raft will cover the distance from Nizhny Novgorod to Astrakhan in 35 days.
57. You can immediately answer that 12 hens will lay 12 eggs in 12 days. However, it is not. If three hens lay three eggs in three days, then one hen lays one egg in the same three days. Therefore, in 12 days she will lay 12: 3 = 4 eggs. If there are 12 hens, then in 12 days they will lay 12 4 = 48 eggs.
58. 111 – 11 = 100.
59. Of course, this reasoning is wrong. The appearance of its correctness and persuasiveness is created due to the fact that it almost imperceptibly mixes and replaces the concepts of “day” and “day”, or rather, “working day”. And these are completely different concepts, because a day is 24 hours, and a working day is 8 hours. There are 365 days in a year, and this is the time in which we work, rest, and sleep. In the argument, the concept of “365 days” is replaced by the concept of “365 days”, and it is assumed that all these days (and in fact - a day) are occupied only with work. Further, from these "365 days" the time spent on sleep, rest, etc. is subtracted, and this time must be subtracted not from days (moreover, working days), but from days. Then the number of days (working) will remain the same, and there will be no misunderstanding.
60. It is necessary to take the second filled glass on the left and pour it into the second empty glass on the right, then the filled and empty glasses will alternate (Fig. 61).
61. The reasoning is wrong. It is possible to say that more workers will be able to build a house much faster only within whole days, that is, if we measure the time of work in days. If we measure this time in hours, and even more so in minutes and seconds, then this pattern (more workers - faster work) does not work. The reasoning error lies in the fact that it mixes different concepts denoting different time intervals. The concept of "day" is almost imperceptibly replaced by the concepts of "hour", "minute", "second", due to which the appearance of the correctness of this reasoning is created.
62. That word is "wrong". It's always written that way - "wrong". The effect of this joke problem is that it uses the word "wrong" in two different senses.
63. The parrot can indeed repeat every word it hears, but it is deaf and does not hear a single word.
64. Of course, a match, because without it you cannot light a candle or a kerosene lamp. The question of the task is ambiguous, because it can be understood either as a choice between a candle and a kerosene lamp, or as a sequence in lighting something (first a match, and only from it - everything else).
65. It may seem that Peter will sleep for 14 hours, but in fact he will only be able to sleep for 2 hours, because the alarm clock will go off at 9 pm. A simple mechanical alarm clock does not distinguish between day and night and always rings at the time it was set. If it were a computer-type electronic alarm clock that could be programmed, then Peter would be able to sleep from 7 pm to 9 am.
66. The logical regularity that the denial of the truth is a lie, and the denial of a lie is the truth, is valid only when it comes to the same subject. In this case, we should be talking about the same proposal. If this were the case, then one statement would necessarily be true and the other false, or vice versa. But in the problem we are talking about two different sentences. Therefore, it is not surprising that they are both false.
67. The sum of eight digits, equal to two, can be obtained if one of these digits is a two, and the rest are zeros. There is only one such eight-digit number. This is 20,000,000. But the sum of eight digits, equal to two, can also be obtained if two of these digits are ones and the rest are zeros. There are seven such eight-digit numbers: 11,000,000, 10,100,000, 10,010,000, 10,001,000, 10,000,100, 10,000,010, 10,000,001.
So, there are eight eight-digit numbers, the sum of the digits of which is equal to two.
68. The perimeter of a figure is the sum of the lengths of all its sides. This figure has 12 sides. If its perimeter is 6, then one side is 6: 12 = 0.5. The figure consists of 5 identical squares, with a side of 0.5.
The area of one square is 0.5 0.5 = 0.25. Therefore, the area of the entire figure is 0.25 5 = 1.25.
69. Difficulty in solving may arise due to an unusually formulated condition of the problem. The task itself is very simple. All that is required is to write down mathematically what is expressed in words in it, that is, to unravel its verbal condition. The sum of the squares of 2 and 3 is 22 + 32. The cube of the sum of the squares of 2 and 3 is (22 + 32)3. The sum of the cubes of these numbers is 23 + 33. The square of this sum is (23 + 33)2. We need to find the difference between the first and second:
(22 + Z2)3 - (23 + Z3)2 = (4 + 9)3 - (8 + 27)2 = 133 - 352 = 2197–1225 = 972.
70. This number is 2. Half of this number is 1, and half of half of this number (i.e. one) is equal to 0.5, i.e. also half.
71. The reasoning is wrong. It is not certain that Sasha Ivanov will eventually visit Mars. The external correctness of this reasoning is created by the use in it of one word Human in two different senses: in the broad (abstract representative of humanity) and in the narrow (concrete, given, this particular person).
72. As we can see by the condition, to obtain orange paint, 3 times more yellow paint is required than red paint: 6: 2 = 3. This means that from the available amount of yellow and red paints, you need to take 3 times more yellow paint than red, i.e. .3 grams of yellow and 1 gram of red. You can get 4 grams of orange paint.
73. See fig. 62.
You can also remove the other 2 matches.
74. You need to put a comma: 5< 5, 6 < 6.
75. First you need to find out what is the total age of all team players: 22 11 = 242. Let's take the age of the retired player as X. After he was eliminated, the total age of the team's players became 242 - X. Since there are 10 players and their average age is known (21), we can write the following equation:
(242 – X): 10 = 21,
242 – x = 210,
x = 242–210 = 32.
The retired player is 32 years old.
76. The reasoning is, of course, wrong. The effect of its external correctness is achieved through the use of the concept of "age of the father" in two different senses: the age of the father as the age of the person who is this father, and the age of the father as the number of years of paternity. By the way, in the second sense, the concept age, usually not used: usually under the phrase father's age the age of this person is understood, and nothing else.
77. First you need to divide 24 kilograms of nails into two equal parts of 12 kilograms each, balancing them on the scales. Then also divide 12 kilograms of nails into two equal parts of 6 kilograms each. After that, set aside one part, and divide the other in the same way into parts of 3 kilograms. Finally, add these 3 kilograms to the six-kilogram part of the nails. The result is 9 kilograms of nails.
78. It was Thursday. On this day, Peter truthfully said that yesterday (that is, on Wednesday) he lied, and Ivan lied about the fact that yesterday (that is, on Wednesday) he lied, because according to the condition on Wednesday he tells the truth.
79. This number is 147.
Current page: 2 (total book has 5 pages) [accessible reading excerpt: 1 pages]
120. To obtain orange paint, mix 6 parts of yellow paint with 2 parts of red. There are 3 gr. yellow paint and 3 gr. red. How many grams of orange paint can be obtained in this case?
121. When asked how old he was, Vadim replied that in 13 years he would be four times as old as 2 years ago. How old is he?
122. 4 squares are made out of 12 matches. How should two matches be removed to leave 2 squares?
123. What sign must be placed between the numbers 5 and 6 so that the resulting number is greater than 5 but less than 6?
5 < 5? 6 < 6
124. There are 11 players on a football team. Their average age is 22 years. During the match, one of the players dropped out. At the same time, the average age of the team became equal to 21 years. How old is the eliminated player?
125 – How old is your father? the boy is asked.
“As much as I do,” he replies calmly.
- How is this possible?
- Very simple: my father became my father only when I was born, because before my birth he was not my father, then my father is the same age as me.
Is this reasoning correct? If not, what is wrong with it?
126. There are 24 kg of nails in a bag. How can you measure 9 kg of nails on a pan balance without weights?
127. Peter lied from Monday to Wednesday and told the truth on other days, and Ivan lied from Thursday to Saturday and told the truth on other days. One day they said in the same way: "Yesterday was one of the days when I lie." What day was yesterday?
128. A three-digit number was written in numbers, and then in words. It turned out that all the numbers in this number are different and increase from left to right, and all words begin with the same letter. What is this number?
129. A mistake was made in the equality made up of matches. How should one match be shifted in order for the equality to become true?
130. How many times will a three-digit number increase if the same number is added to it?
131. If there were no time, there would not be a single day. If there were no day, it would always be night. But if it were always night, there would be time. Therefore, if there were no time, there would be. What is the reason for this misunderstanding?
132. There are 12 apples in each of two baskets. Nastya took a few apples from the first basket, and Masha took from the second as many as were left in the first. How many apples are left in the two baskets together?
133. One farmer has eight pigs: three pink, four brown and one black. How many pigs can say that in this small herd there is at least one more pig of the same color as her own? (The task is a joke).
134. There are two identical buckets filled with water on two scales. The water level in them is the same. A wooden block floats in one bucket. Will the scales be in balance?
135. If one worker can build a house in 5 days, then 5 workers will build it in one day. Therefore, if one ship crosses the Atlantic Ocean in 5 days, then 5 ships will cross it in one day. Is this statement correct? If not, what is the error in it?
136. Returning from school, Petya and Sasha went to the store, where they saw a large scale.
“Let's weigh our portfolios,” suggested Petya.
The scales showed that Petya's portfolio weighed 2 kg, while Sasha's portfolio weighed 3 kg. When the boys weighed the two briefcases together, the scales showed 6 kg.
“How is it,” Petya was surprised, “because 2 + 3 is not equal to 6.
- Can't you see? - Sasha answered him, - the scale has shifted the arrow.
What is the real weight of portfolios?
137. How to place six circles on a plane in such a way that there are three rows of three circles in each row?
138. After seven washes, the length, width and height of a bar of soap has halved. How many washes will the remaining piece last?
139. How to cut off half a meter from a piece of matter 2/3 m without the help of any measuring instruments?
140. 13 identical sticks are drawn on a rectangular sheet of paper at an equal distance from each other (see figure). The rectangle is cut along the straight line AB passing through the upper end of the first stick and through the lower end of the last. After that, both halves are shifted as shown in the figure. Surprisingly, instead of 13 sticks there will be 12. Where and how did one stick disappear?
141. It is often said that one must be born a composer, or an artist, or a writer, or a scientist. Is this true? Is it really necessary to be born as a composer (artist, writer, scientist)? (The task is a joke).
142. In order to see, it is not at all necessary to have eyes. We see without the right eye. We also see without the left. And since we have no other eyes besides the left and right eyes, it turns out that neither eye is necessary for vision. Is this statement correct? If not, what is wrong with it?
143. A parrot has lived less than 100 years and can only answer yes and no questions. How many questions does he need to ask to find out his age?
144. How many cubes are shown in this picture?
145. Three calves - how many legs? (The task is a joke).
146. One person who fell into captivity tells the following. “My dungeon was at the top of the castle. After many days of effort, I managed to break one of the bars in the narrow window. It was possible to crawl into the hole that had formed, but the distance to the ground left no hope of simply jumping down. In the corner of the dungeon, I found a rope forgotten by someone. However, it turned out to be too short to be able to go down it. Then I remembered how one wise man lengthened a blanket too short for him, cutting off part of it from below and sewing it on top. So I hastened to split the rope in half and re-tie the two resulting parts. Then it became long enough, and I safely went down it. How did the narrator manage to do this?
147. The interlocutor asks you to think of any three-digit number, and then offers to write down its numbers in reverse order to get another three-digit number. For example, 528–825, 439–934, etc. Then he asks to subtract the smaller number from the larger number and tell him the last digit of the difference. After that, he names the difference. How he does it?
148. Seven walked - they found seven rubles. If not for seven, but for three, would you find a lot? (The task is a joke).
149. How to divide a drawing consisting of seven circles by three straight lines into seven parts in such a way that each part contains one circle?
150. The globe was pulled together by a hoop along the equator. Then the length of the hoop was increased by 10 m. At the same time, a small gap formed between the surface of the globe and the hoop.
Can a person get through this gap? (The length of the earth's equator is approximately 40,000 km).
151. A tailor has a piece of cloth 16 meters long, from which he cuts 2 meters daily. After how many days will he cut the last piece?
152. Four equal squares are built from 12 matches. How to shift three matches in such a way that you get three equal squares?
153. A wheel with blades is installed near the bottom of the river, and it can rotate freely. If the river flows from left to right, in which direction will the wheel turn? (See picture).
154. In a communal apartment, resident Ivanov put 3 logs of his firewood into a common stove, and resident Sidorov put 5 logs. The tenant Petrov, who did not have his own firewood, received permission from both neighbors to cook his dinner on a common fire. In reimbursement of expenses, he paid the neighbors 8 rubles. How should they share this payment among themselves?
155. It is well known to everyone that a stone thrown into calm water (puddles, ponds, lakes) gives rise to circles diverging in different directions on its surface. But what will this phenomenon be in moving or flowing water? Will the waves from a stone thrown into the water of a fast river be circular, or will they stretch in the direction of the current and take the form of ellipses?
156. What number (not counting zero) is divisible by all numbers without a remainder?
157. How can 24 people be arranged in six rows so that each row consists of 5 people?
158. The father is 32 years old, and the son is 7 years old. In how many years will the father be six times as old as the son?
159. If you have 10 pairs of gray socks and 10 pairs of black socks mixed up in your closet, then in complete darkness, by touch, only three socks need to be removed from the closet in order to get a matching pair with a guarantee. If you have 10 pairs of gray gloves and 10 pairs of black gloves mixed in your closet, how many gloves do you need to take out of the closet in complete darkness, by touch, in order to guarantee a matching pair?
160. As you know, all physical bodies consist of molecules, and molecules consist of atoms, which are unimaginably small particles (if you mentally divide a millimeter on your ruler into a million parts, then one millionth of a millimeter will be the approximate size of an atom). Now imagine that a notebook page is torn in half, then one of the halves is again divided in half, then one of the quarters is again divided in two, etc. How many times will it be necessary to divide the notebook page in this way to make it the size of an atom? (Assume that a notebook page weighs 1 g, and the weight of an atom is 10 -24 g).
161. Building bricks weigh 4 kg. How much does a toy brick made of the same material weigh if all its dimensions are half the size?
162. Is it possible to determine its height from a photograph of a tower? If possible, how to do it? (The photograph, of course, must be professional, that is, not distorting the true proportions of the objects depicted on it).
163. How can one write the greatest possible number with four units, but at the same time not use any action signs?
164. It is sometimes said that a three-legged table never swings, even if its legs are of unequal length. Is this statement correct?
165. When we are on the open sea, we can observe the horizon line everywhere around us. How is it located: at the level of our eyes, above or below it?
166. What is the smallest positive integer that can be written with two digits without using any action signs?
167. What size will an angle of 2º appear when viewed through a magnifying glass four times?
168. The globe is tied along the equator with steel wire. If it is cooled by 1º, it will shorten and crash into the ground. How big will this recess be? (Cooling by 1º, the steel wire is shortened by 1/100,000 of its length; the length of the earth's equator is ≈ 40,000 km).
169. How is it possible to determine the magnitude of an acute angle (on a drawing) without making any measurements?
170. How to express the number 1000 with eight identical digits? (You can use action signs).
171. One father gave his son 500 rubles, and another gave 400 rubles to his son. However, it turned out that both sons together increased the amount of their money by only 500 rubles. How is this possible?
172. Which of the two rectangular boxes with a square base is more spacious - the right one, the wide one, or the left one, which is three times as high, but twice narrower than the right one? (See picture).
173. Can you find three successive (following one after another in the natural series of numbers) numbers that differ in such a property that the square of the middle number is one greater than the product of the other two extreme numbers.
174. A cherry stone is surrounded by a layer of pulp, which has the same thickness as the stone itself. How many times is the volume of the pulp of a cherry greater than the volume of its stone?
175. It is well known to everyone that the moon and sun, observed near the horizon, have a much greater magnitude than when they hang high in the sky, being at the zenith. This is due to the fact that when we observe the moon or the sun at the horizon, they are closer to the earth and therefore look larger. Is this reasoning correct?
176. Wanting to check whether the cut piece of matter has the shape of a square, you bend it diagonally and make sure that the edges of this piece of matter coincide. Is such a check sufficient?
177. How can one express one, while using all ten digits and signs of mathematical operations?
178. The interlocutor invites you to think of a certain number, then do some sequence of mathematical operations with it and tell him the result, after which he calls the conceived number. How he does it?
179. It is very easy to express the number 24 with three eights: 8 + 8 + 8, and the number 30 with three fives: 5 × 5 + 5. Is it possible to express the numbers 24 and 30 with three other identical digits (not eights and not fives, respectively), with this using the signs of mathematical operations?
180. How to write down as many numbers as possible with any three digits without using any signs of action?
181. Suppose you need to make a bookshelf 1 m long and 20 cm wide, but you have a board that is shorter but wider - 75 cm long and 30 cm wide. From it, of course, you can make a board of the required dimensions by sawing along a strip 10 cm wide and, sawing it into three equal parts of 25 cm each, build up the board with two of them by gluing (see figure).
Such a solution to the problem is uneconomical in terms of the number of operations (three sawing and three gluing), and, in addition, the bookshelf would be too fragile in the place where the small planks are glued to the main board.
From an existing board 75 cm long and 30 cm wide, how to make a bookshelf of the required dimensions with greater strength using fewer operations?
182. How is it possible to construct a right angle without making any measurements with the help of special tools?
183. The interlocutor invites you to think of any two-digit number and duplicate it twice so that you get a six-digit number. For example, 27 - 272727 or 78 - 787878. Then, without knowing, of course, your six-digit number, he suggests that you divide it by 37 and guarantees that the division will pass without a remainder. You are dividing, and indeed there is no remainder. Then he suggests dividing the resulting result by 13 and again assures you that there will be no remainder. You divide and again without a trace. Then, in the same way, he asks you to divide the result by 7 and after that by another 3. The final division again does not give a remainder and, moreover, you get the two-digit number you conceived, which the interlocutor did not know. How does he do this amazing, at first glance, trick?
184. A huge cigarette is displayed in the window of a tobacco shop, which is 20 times longer and 20 times thicker than an ordinary cigarette. If it takes half a gram of tobacco to stuff an ordinary cigarette, how much tobacco is needed to stuff it into a cigarette displayed in a shop window?
185. How to divide the clock face (see figure) into six parts (of any shape), so that the sum of the numbers available in each section is the same.
186. Before you are three cubic boxes. The first of them has a rib measuring 6 cm, the second - 8 cm, and the third - 9 cm. Which is larger: the volume of the first two boxes combined or the volume of the third box?
187. Approximately how many times is a two-meter giant heavier than a one-meter dwarf?
188. How, without using measuring instruments, to determine the magnitude of the angle formed by the hour and minute hands when the clock shows seven o'clock?
189. From four matches, an image of a scoop is assembled, in which there is garbage. How to shift two matches so that there is no garbage in the scoop, or rather, so that it is outside the scoop?
190. An airplane covers the distance from one city to another in 1 hour and 20 minutes. However, it takes only 80 minutes for the return flight. How can this be explained? (The task is a joke).
191. Two watermelons of different sizes are sold on the market. One of them is one and a half times wider than the other, and it costs twice as much as it. Which of these watermelons is more profitable to buy and why?
192. Let us prove that there are no uninteresting people. Let's argue on the contrary: let's say there are uninteresting people. Let's mentally collect them together and single out among them the largest in height, or the smallest in weight, or some other "most ...". This person, who stands out among others, will undoubtedly be interesting for his non-standard, therefore he cannot be called uninteresting and must be excluded from the group of uninteresting people. Further, among the remaining uninteresting people, we again single out some “most ...” and exclude him. And so on until there is only one person left, who can no longer be compared with anyone. But that's what makes him interesting. Thus, uninteresting people do not exist. Is this reasoning correct? If not, what is wrong with it?
193. Having taken off from St. Petersburg, the helicopter flew strictly to the north for 500 km, then turned to the east and flew another 500 km, then, turning to the south, flew another 500 km, and finally, turning to the west, flew the last 500 km. During the flight, the helicopter was at the same height. Where did he land: in the same place where he flew out or to the north (south, west, east) of this place?
194. What will be the height of a column made up of all millimeter cubes enclosed in one cubic meter?
195. Hour and minute hands are located at the same distance from the number VI. At what time could this happen?
196. A figure of a cross is built from 12 matches, the area of which is equal to five "match" squares. How, without the help of measuring instruments, to shift the matches in such a way that the new figure covers an area equal to only four match squares?
197. How to increase the distance between two points three times if there is no ruler at hand, but only a compass?
198. The first mug is twice as high as the second, but the second is twice as wide as the first. Which of these mugs has more capacity?
199. The interlocutor asks you to think of any three-digit number, after which he instantly multiplies it by 999. For example, you think of the number 147, but after a moment the interlocutor tells you the result of multiplying this number by 999, namely 146 853. You check on paper or calculator - everything is correct, it will really be 146,853. You ask him to repeat this operation, giving him another three-digit number, for example, 276. He also quickly multiplies it by 999 and tells you the result - 275,724. You check - everything is correct. With the same ease and speed, the interlocutor multiplies any three-digit numbers offered to him by 999, never making a mistake and explaining this with his “mathematical abilities”. You, of course, guess that the point here is not in abilities, but in something else. What is the secret of lightning-fast multiplication of any three-digit number by 999?
200. A snail decided to climb a tree that is 15 meters high. Every day she climbed 5 meters, but every night, while sleeping, she went down 4 meters. In how many days after the start of her journey will she reach the top of the tree?
Answers and comments
1. Of course, there is such a place on the globe. This is the geographic south pole. Whichever way you go from it, there will be only one direction - to the north, because the north is everywhere around it. Therefore, a compass needle placed on the south pole will point north at both ends. In the same way, a compass needle placed at the geographic north pole of the Earth will point south with both ends.
2. One of the five people must pick up their apple along with the basket. The effect of this not very serious task is based on the ambiguity of the expression "the apple is left in the basket." After all, it can be understood both in the sense that no one got it, and in the fact that it simply did not leave the place of its original stay, and these are completely different things.
3. This can be done in various ways:
4. The peasant must, after transporting the goat, return and take the wolf, which he also transports to the other side. After that, he leaves it there, and picks up the goat and takes it back. Here he leaves the goat and transports cabbage to the wolf, after which he returns and, finally, transports the goat to the other side.
5. One coin must be pulled out from the first bag, two from the second, three from the third, etc. (all ten coins from the tenth bag). Then all these coins should be weighed together once. If there were no counterfeit coins among them, i.e., they would all weigh 10 grams each, then their total weight would be 550 grams. But since there are counterfeit coins (11 grams each) among the weighed coins, their total weight will be more than 550 grams. Moreover, if it turns out to be 551 grams, then the fake coins are in the first bag, because we took one coin from it, which gave an extra one gram. If the total weight is 552 grams, then the counterfeit coins are in the second bag, because we took two coins from it. If the total weight is 553 grams, then the counterfeit coins are in the third bag, and so on. Thus, with only one weighing, it is possible to determine exactly which bag contains the counterfeit coins.
6. You need to take cookies from a jar with the inscription "Oatmeal cookies" (you can - from any other). Since the jar is labeled incorrectly, it will be shortbread or chocolate. Let's say you got a shortbread. After that, you need to swap the labels "Oatmeal Cookies" and "Shortbread Cookies". And since, according to the condition, all the labels are mixed up, now there is oatmeal in the jar with the inscription “Chocolate cookies”, and there is chocolate in the jar with the inscription “Oatmeal cookies”, so these two labels must also be swapped.
7. At first glance, it may seem that a person will take the last pill in an hour and a half, because this is exactly three times for half an hour. In fact, he will drink the last pill not in an hour and a half, but in an hour. Imagine that he drinks the first pill. Half an hour passes. He takes the second pill. Another half hour passes. He takes his third pill. Therefore, the person will drink the last pill an hour after the start of treatment.
8. The number 66 just needs to be turned upside down. It will turn out 99, and this is 66, increased by one and a half times.
9. Peter started his watch and before leaving he memorized their reading, which, for example, is equal to a. Arriving at a friend, he immediately learned from him the time, which is equal to b. Before leaving, he again remembered the time on the clock of a friend, which this time was with. Arriving home, Peter noticed that his watch was showing d. Difference (d-a) This is the time of his absence from home. Difference (c-b) is the time he spends at a party. Difference between first and second time (d - a) - (c - b) is the time spent on the road. Half of this time
was spent on the return trip. When Peter went home, the clock of his acquaintance, as already mentioned, showed with. If we add the time spent on the way back to the time spent going home, i.e., to with, then you get the exact reading of Peter's clock when he returns home:
10. It is necessary to cut all 5 links of one piece and use them to connect the remaining 5 pieces. In this case, the total cost of the work will be 1 ruble 30 kopecks, which is 20 kopecks cheaper than the cost of a new chain.
11. At first glance, the question of the problem looks meaningless, since it seems beyond doubt that all points of the wheel move at the same speed. This is true for the movement of all points of the wheel around its center. But in the question of the problem, we are talking about their movement in the direction of the forward movement of the wheel. In this case, it turns out that the points of the wheel located in its upper part move in the same direction as the wheel, and the points located in its lower part move in the opposite direction (see figure). Therefore, the speed of the upper points of the wheel is added to the speed of the wheel, and the speed of its lower points is subtracted from it. Thus, in the direction of the forward movement of the wheel, its upper points move faster and the lower ones slower.
12. At first glance, it seems that such reasoning is absolutely correct: if one glass is poured from a full samovar in half a minute, then all 30 glasses will pour out of it in 15 minutes. But this is true only in a mathematical sense, and in this case we are talking about a physical phenomenon with its own laws. Moreover, even if you do not know anything about them, it is still quite clear (even on the basis of everyday life experience) that freely flowing (from anywhere) water does not pour out at the same speed, not evenly. At first, when a certain reservoir is full of water, its pressure is great, and it flows out faster. As the container empties, the pressure of the water in it drops, and it begins to flow more slowly. Thus, the first glasses of water are poured out of the samovar under high pressure, and the rest under less pressure, so at first the glasses are filled faster and then more slowly. Consequently, all 30 glasses will pour out of the samovar with a continuously open tap not in 15 minutes, but in a longer period of time.
13. It may seem that a 60-tooth harrow will loosen the ground deeper. However, it is not. Recall that the larger the area of support of a body, the less pressure it exerts on the surface under this body. (For this reason, for example, a person walking on a snowdrift falls into it with each foot, and a skier does not fall through, sliding freely on its surface). A 60-tooth harrow has a larger footprint than a 20-tooth harrow, meaning that 60 tines push the ground with less force than 20 tines. This means that a harrow with 20 teeth will loosen the ground deeper. (See also problem 26).
14. If you draw a horseshoe in the form of an arcuate line, then you will not be able to cut it with two straight lines into more than five parts. If you draw a horseshoe the way it really is, that is, having a width, then the task (maybe not on the first try) is doable.
15. The owner of the house sawed a silver bar in three places, dividing it into 4 pieces, the length of which was respectively 1, 2, 4 and 8 decimeters. On the first day, he gave the worker the shortest piece. On the second day, he took this piece from him and gave him a two-decimeter one. On the third day, he again gave him a one-decimeter piece. On the fourth day, the owner took away the one and two decimeter pieces from the worker and gave him a four decimeter piece in return, and so on.
16. First you need to weigh 16 coins, putting 8 pieces on each scale. If some bowl outweighs, then it contains a heavier coin. If the bowls balance, then the desired coin is among those 8 that were not weighed. Next, from the pile in which the heavy coin is located, you need to take 6 pieces and, breaking them into 3, weigh them again. If any of the scales outweigh, then among the 3 coins in it, there is the desired coin. If the bowls are balanced, then she is among the two not weighed. And finally, one must weigh either these two remaining coins on two scales, or any two of those three, among which is the heavier one. In the second case, if one of the scales outweighs, then the heavy coin is in it, and if equilibrium is established, then the desired coin is the remaining one.
17. Only three socks need to be taken out of the closet.
18. The clock strikes twelve in sixty-six seconds. When the clock strikes six, five intervals pass from the first strike to the last. The interval is six seconds (one fifth of thirty). When the clock strikes twelve, there are eleven intervals from the first strike to the last. Since the length of the interval is six seconds, it takes sixty-six seconds for the clock to strike twelve (11 × 6 = 66).
19. The pond will be half covered with lily leaves on day 99. According to the condition, the number of leaves doubles every day, and if on day 99 the pond is half covered with leaves, then the next day the second half of the pond will be covered with lily leaves, i.e. the pond will be completely covered with them after 100 days.
20. If one and a half hens lay one and a half eggs in a day and a half, then in the same time (i.e., in a day and a half) three hens will lay three eggs, and one hen - one egg. A hen that lays one and a half times better will lay one and a half eggs in the same time (in a day and a half), that is, one egg a day. This means that in 15 days (a decade and a half), this chicken will lay a dozen and a half eggs. Thus, the answer to the question posed is one chicken.
21. Rising to the fifth floor, the passenger elevator overcomes four spans, and the freight elevator passes two spans to the third floor. Thus, the path traveled by a passenger elevator is twice the distance traveled by a freight elevator. Since the passenger elevator goes twice as fast as the freight elevator, they will reach their floors at the same time.
22. To solve this problem, you need to make an equation.
The number of geese in a flock is x. “Now, if there were as many of us as now (i.e. x), - said the geese, - and so many more (i.e. x), and even half as much (i.e.), and even a quarter as many (i.e.), and even you (i.e., one goose), then we would have been 100 geese. It turns out: .
Let's add on the left side of the equation:
36 geese flew in a flock.
24. To solve this problem, you need to make an equation. Let us denote the number of animals as x and the number of birds as y. There are 30 animals in the zoo, i.e. x + y = 30, and then x = 30 - y. There are a hundred legs in the zoo, i.e. 4 x + 2 y \u003d 100. Let us substitute the expression x \u003d 30 - y into this equality. We get: 4 (30 - y) + 2 y \u003d 100.
Let's convert: 120 - 4 y + 2 y \u003d 100 or 120 - 2 y \u003d 100, or 20 \u003d 2 y. So y = 10, i.e. there are 10 birds in the zoo. And the animals in the zoo: 30–10 = 20.
25. The error lies in squaring each part of the equation (-2 = 2). The appearance is created that the same operation is performed on each part of the equality (squaring), but in fact, different operations are performed on each part of the equality, because we multiply the left side by - 2, and multiply the right side by 2.
26. At first glance, it seems that lying undressed on a bare rocky surface, like on a soft feather bed, is completely impossible. However, it is not. Recall that the larger the area of support of a body on a certain surface, the less pressure it exerts on this surface. The feather bed seems soft to us, and the wooden floor is hard, because the area of contact of our body with the feather bed is much larger than with the floor, due to which the body puts much less pressure on the feather bed than on the floor. Therefore, if we arrange a bare rocky surface in such a way that the area of its contact with our body is as large as possible, then this surface will be as soft for us as a featherbed. To do this, it is possible to make protrusions and recesses in a rocky surface corresponding to the relief of that part of our body with which we will lie on this surface. But such a procedure, apparently, is not easy to accomplish. You can do it differently: lie down, undressed, on a viscous, not frozen clay or plaster, or cement, etc. surface for a few seconds and get up. At the same time, this surface will accurately reflect the relief of our body. When it hardens and becomes hard as a stone, you can lie down in the forms formed in it by our body. The area of contact of the body with the surface in this case will be large, its pressure on it, on the contrary, will be minimal, and you can lie on such a rocky surface in the same way as on a soft feather bed. (See also problem 13).
Task Conditions
1. Each of the 10 bags contains 10 coins. Each coin weighs 10 g. But in one bag all the coins are counterfeit - not 10 g each, but 11 g each. How, using only one-time weighing, can you determine which bag contains counterfeit coins (all bags are numbered from 1 to 10)? The bags can be opened and any number of coins can be pulled out of each.
2. On all three iron cans with cookies, the labels are mixed up: "Oatmeal cookies", "Shortbread cookies" and "Chocolate cookies". The jars are closed, and you can only take one cookie from one (any) jar, and then arrange the labels correctly. How to do it?
3. There are 22 blue socks and 35 black socks in your closet.
You need to take a pair of socks from the closet in complete darkness. How many socks do you need to take to be sure you get a matching pair?
4. It takes 30 seconds for an old clock to strike 6 o'clock. How many seconds does it take for the clock to strike 12 o'clock?
5. One lily leaf grows in the pond. Every day the number of leaves doubles. On what day will the pond be half covered with lily leaves if it is known that it will be completely covered with them in 100 days?
6. A passenger elevator rises to the fifth floor at twice the speed of a freight elevator that goes to the third floor.
Which of these two elevators will arrive first: freight to the third floor or passenger to the fifth, if they started from the first floor at the same time?
7. A goose is flying. Towards him is a flock of geese. "Hello, 100 geese," he tells them. They answer: “We are not 100 geese; Now, if there were as many of us as there are now, and even as many, and even half as many and a quarter as many, and even you, then there would be 100 of us geese.
How many geese fly in a flock?
8. Let us prove that 3 = 7. It is known that if the same operation is performed on each part of the equality, then the equality will remain unchanged. Let's subtract five from each part of our equality: 3 - 5 \u003d 7 - 5. It turns out: - 2 \u003d 2. Now let's square each part of the equality: (- 2) 2 \u003d 2 2. It turns out: 4 = 4, therefore: 3 = 7. Find an error in this reasoning.
9. As you know, in any atom there is a nucleus, the size of which is less than the size of the atom itself. If the size of the atomic nucleus is 10–12 cm, and the size of the entire atom is 10–6 cm, then the nucleus is 2 times smaller than the atom itself: 12: 6 = 2. Is this statement true?
If not, how many times smaller is an atomic nucleus than an atom?
10. Is it possible to fly to the moon by plane? It must be taken into account that the aircraft are equipped with jet engines, like space rockets, and operate on the same fuel as they do.
11. Is it possible to pierce a fifty-kopeck coin with a needle?
12. A standard glass (200 g) is filled to the brim with water. How many pins can be thrown into it so that not a drop of water spills out of the glass?
13. Ivanov has a portrait hanging in his office. Ivanov is asked: “Who is depicted in this portrait?” Ivanov confusedly answers:
"The father of the person depicted in the portrait is the only son of the father of the speaker." Who is in the portrait?
14. The missionary was captured by the savages, who put him in prison and said: “From here there are only two ways out - one to freedom, the other to death; two warriors will help you get out - one always tells the truth, the other always lies, but it is not known which of them is a liar and which is a truth lover; you can ask any of them only one question.” What question should be asked to get out to freedom?
15. Two ropes of rare silk hang in the monastery. They are attached to the middle of the ceiling at a distance of one meter from each other and reach the floor. The acrobat thief wants to steal as much rope as possible. The height of the ceiling is 20 m. The thief knows that if he jumps or falls from a height of more than 5 m, he will not be able to get out of the monastery. Since he does not have a ladder, he can only climb the rope. He found a way to steal both ropes almost entirely. How to do it?
16. The girl was riding in a taxi. On the way, she talked so much that the driver got nervous. He told her that he was very sorry, but he couldn't hear a word - because his hearing aid didn't work, he was deaf as a cork. The girl fell silent, but when they reached the place, she realized that the driver had played a joke on her. How did she guess?
17. You are in the cabin of an ocean liner at anchor. At midnight, the water was 4 m below the porthole and rose 0.5 m/h. If this speed doubles every hour, how long will it take the water to reach the porthole?
18. Three travelers lay down to rest in the shade of trees and fell asleep. While they slept, the pranksters smeared charcoal on their foreheads. Waking up and looking at each other, they began to laugh, and it seemed to each of them that the other two were laughing at each other.
Suddenly one of them stopped laughing as he realized that his own forehead was also dirty. How did he guess about it?
19. By moving only one of the four matches, make a square (Fig. 45). Matches cannot be bent or broken:
20. As the sun rose, the traveler began to climb the narrow, winding path to the top of the mountain. He walked faster and slower, stopping often to rest. Having traveled a long way, he reached the summit just before sunset. After spending the night at the top, at sunrise he set off on his return journey along the same path. He also descended at an uneven speed, repeatedly resting along the way, and by sunset he reached the foot of the mountain. It is clear that the average rate of descent exceeded the average rate of ascent. Is there such a point on the path that the traveler passed at the same time of day both during the ascent and during the descent?
21. The sculptor has 10 identical statues. He wants three statues on each of the four walls of the hall. How to place them?
22. Draw, without lifting the pencil from the paper, the following figures (Fig. 46):
23. One mathematician suggested such a deal to a merchant. The mathematician gives the merchant 100 rubles, and the merchant gives the math in exchange for 1 k.
Every next day, the mathematician gives the merchant 100 rubles. more than on the previous one, that is, on the second day he gives him 200 rubles, on the third - 300 rubles. and so on. And the merchant gives mathematics in return twice as much money as on the previous day, i.e. on the second day he gives him 2 k., on the third - 4 k., on the fourth - 8 k., on fifth - 16 k., etc.
They agreed to make such an exchange within 30 days. Who benefits from this exchange and why?
24. According to the old style, the anniversary of the October Revolution falls on October 25, and according to the new style - on November 7. Thus, all events according to the old style precede the same events according to the new style by 13 days. This means that if, according to the new style, the New Year falls on January 1, then according to the old style, it should fall on December 19. Why then do we celebrate the old New Year on January 14?
25. A drawing of a glass filled with wine was made from matches (Fig. 47). Rearrange two matches so that in the newly received picture the wine is outside the glass. When demonstrating the role of wine, a match can play:
26. How to arrange six cigarettes in such a way that they are all in contact with each other, that is, that each of them touches the other five?
27. Three people are standing in front of you. One of them is a Truth-lover (always tells the truth), another is a Liar (always lies), and the third is a Diplomat (sometimes tells the truth, sometimes lies). You do not know who is who and ask a question to the person who is standing on the left:
- Who is standing next to you?
“Truth,” he replies.
Then you ask the person in the center:
- Who are you?
“Diplomat,” he replies.
And finally, you ask the person on the right:
- Who is standing next to you?
“Liar,” he replies.
Who is on the left, who is on the right, who is in the center?
28. There are 10 liters of wine in a ten-liter bucket. You have two empty buckets at your disposal: one - 7 liters, and the other - 3 liters. How to use these buckets to divide 10 liters of wine into two identical parts of 5 liters by transfusions?
29. Andrei's watch is 10 minutes behind, but he is sure that they are 5 minutes ahead. He agreed with Katya to meet at 8:00 at the train to go out of town. Katya's watch is 5 minutes fast, but she thinks it is 10 minutes behind. Which one will be the first to get on the train?
30. A 110-year-old turtle asked a dinosaur, "How old are you?" The dinosaur, accustomed to expressing himself in a complicated and confusing way, replied: “I am now 10 times older than you were when I was the same age as you are now.” How old is the dinosaur?
31. The car thief stole a car while trying to get into the checkpoint B, however, was discovered by the police at the checkpoint A. Leaving the chase, he began to dodge, moving from A in B along the curve ACDB along the arcs of small semicircles as shown by the arrows (Fig. 48). The policemen chasing him started from A a moment later and, hoping to intercept the hijacker at the point B, set off along the arc of a great semicircle. Will they catch up with the hijacker at the point B, if their speeds are exactly the same (Fig. 48)?
32. Katya is twice as old as Nastya will be when Olya is as old as Katya is now. Who is the oldest and who is the youngest?
33. In one class, the students were divided into two groups. Some had to always tell only the truth, while others - only a lie. All students of the class wrote an essay on a free topic, and at the end of the essay, each student had to attribute one of the phrases: “Everything written here is true”, “Everything written here is a lie”. In total, there were 17 truth-tellers and 18 liars in the class. How many essays with a statement about the veracity of what was written did the teacher count when checking the work?
34. How many great-great-grandparents did all your great-great-grandparents have in total?
35. A handkerchief lies unfolded on the table. On it in the center stands an empty glass bottle with its neck down. How to pull a handkerchief out from under a bottle without touching it?
36. On the left side of the equality, you need to put only one dash (stick) in order for the equality to turn out to be true:
5 + 5 + 5 = 550.
37. Let us prove that three times two will not be six, but four.
Take a match, break it in half. It's one time two. Then take a half and break it in half. This is the second time twice. Then take the remaining half and break it in half too. This is the third time twice. It turned out four. Therefore, three times two is four, not six. Find the error in this reasoning.
38. How to connect nine dots to each other with four lines without lifting the pencil from the paper (Fig. 49)?
In a hardware store, a customer asked:
- How much does one cost?
“Twenty roubles,” answered the seller.
How much is twelve?
- Forty rubles.
- Okay, give me a hundred and twelve.
- Please, sixty rubles from you.
What did the visitor buy?
40. If it rains at 12 o'clock at night, can we expect that in 72 hours there will be sunny weather?
41. Three people paid 30 rubles for lunch. (each for 10 rubles). After they left, the hostess discovered that dinner cost not 30 rubles, but 25 rubles. and sent the boy in pursuit to return 5 p. Each of the travelers took 1 r., and 2 r. they left the boy. It turns out that each of them paid not 10 rubles, but 9 rubles. There were three of them: 9 3 = 27, and the boy had two more rubles: 27 + 2 = 29. Where did the ruble go?
42. 1,000,000 liters of water were poured into a pool of 1 ha. Can you swim in this pool?
43. Which is more: or?
44. One boy does not have enough to the cost of the ruler 24 k., and the other does not have enough to this cost 2 k. When they put their money together, they still could not buy the ruler. How much does a line cost?
45. In one parliament, the deputies were divided into conservatives and liberals. The conservatives spoke only the truth on even numbers, and only untruths on odd numbers. Liberals, on the other hand, told only the truth on odd numbers, and only lies on even numbers. How, with the help of one question posed to any deputy, it is possible to determine exactly what date today is: even or odd? Answers should be definite: "yes" or "no".
46. A bottle with a cork costs 1 p. 10 k. A bottle is more expensive than a cork by 1 p. How much is the bottle and how much is the cork?
47. Katya lives on the fourth floor, and Olya lives on the second. Rising to the fourth floor, Katya overcomes 60 steps. How many steps does Olya need to climb to get to the second floor?
48. A mathematician wrote a two-digit number on a piece of paper. When he turned the paper upside down, the number decreased by 75. What number was written?
49. A rectangular sheet of paper is folded in half 6 times. On a folded sheet, not on the folds, 2 holes were made. How many holes will there be on the sheet if it is unfolded?
50. Two fathers and two sons caught three hares: one each.
How is this possible?
51. The interlocutor invites you to think of any three-digit number. Then he asks to duplicate it to get a six-digit number. For example, you thought of the number 389, duplicating it, you get a six-digit number - 389,389; or 546 - 546 546, etc.
Further, the interlocutor offers you to divide this six-digit number by 13. “Suddenly it will turn out without a trace,” he says. You divide with a calculator (you can do it without it) and indeed your number is divisible by 13 without a remainder. Then he offers you to divide the result by 11. You divide, and again it turns out without a remainder. And finally, the interlocutor asks you to divide the resulting result by 7. The division not only goes without a remainder, but also results in the same three-digit number that you arbitrarily chose first. How does this happen?
52. Divide the figure, consisting of three identical squares, into four equal parts (Fig. 50):
53. One hundred schoolchildren simultaneously studied English and German. At the end of the course, they took an exam, which showed that 10 students did not master either one or the other language. Of the remaining German students, 75 passed, and 83 passed the English exam. How many test takers speak both languages?
54. How to pour exactly half from a mug, ladle, pan and any other dishes of the correct cylindrical shape, filled to the brim with water, without using any measuring instruments?
55. Hour and minute hands sometimes coincide, for example at 12 o'clock or at 24 o'clock. How many times will they coincide between 6 o'clock in the morning of one day and 10 o'clock in the evening of another day?
56. The ship sails from Nizhny Novgorod to Astrakhan in 5 days, it makes the return journey at the same speed in 7 days. How many days does it take a raft to sail from Nizhny Novgorod to Astrakhan?
57. Three hens lay three eggs in three days. How many eggs will 12 hens lay in 12 days?
58. How to write the number 100 using five units and action signs?
59. Let's calculate how many days a year we work and how many we rest. There are 365 days in a year. Everyone sleeps eight hours a day, that's 122 days a year. Subtract, 243 days remain. Eight hours a day is spent resting after work, which is also 122 days a year. Subtract, there are 121 days left. On weekends, which are 52 a year, no one works. Subtract, there are 69 days left. Further, a four-week vacation is 28 days. Subtract, there are 41 days left. Approximately 11 days a year are occupied by various holidays. Subtract, there are 30 days left. Thus, we work only one month a year.
60. In one row are three glasses filled with water and three empty (Fig. 51). How to make it so that filled and empty glasses alternate if you can only take one glass in your hands?
61. If 1 worker can build a house in 12 days, then 12 workers will build it in 1 day. Therefore, 288 workers will build a house in 1 hour, 17,280 workers will build it in 1 minute, and 1,036,800 workers will be able to build a house in 1 second. Is this reasoning correct? If not, what is the error?
62. What word is always spelled wrong? (The task is a joke.)
63. "I vouch," said the salesman at the pet store, "that this parrot will repeat every word it hears." A delighted buyer bought a miracle bird, but when he came home, he found that the parrot was as dumb as a fish. However, the seller did not lie. How is this possible? (The task is a joke.)
64. There is a candle and a kerosene lamp in the room. What will you light first when you enter this room in the evening?
65. Peter was very tired and went to bed at 7 o'clock in the evening, setting a mechanical alarm clock for 9 o'clock in the morning. How many hours will he get to sleep?
66. The negation of a true sentence is a false sentence, and the negation of a false one is true. However, the following example says that this is not always the case. The sentence "This sentence contains six words" is false because it has five words instead of six. But the negation: “This sentence does not contain six words,” is also false, since it has just six words. How to resolve this misunderstanding?
67. How many eight-digit numbers are there whose sum of digits is two?
68. The perimeter of a figure made up of squares is six (Fig. 52). What is its area?
69. What is the difference between the cube of the sum of the squares of the numbers 2 and 3 and the square of the sum of their cubes?
70. Half of half a number is equal to half. What is this number?
71. Over time, a person will definitely visit Mars. Sasha Ivanov is a man. Consequently, Sasha Ivanov will eventually visit Mars. Is this reasoning correct? If not, what is wrong with it?
72. To get orange paint, mix 6 parts yellow paint with 2 parts red. There are 3 g of yellow paint and 3 g of red.
How many grams of orange paint can be obtained in this case?
73. 4 squares are made out of 12 matches (Fig. 53). How should 2 matches be removed so that 2 squares remain?
74. What sign should be placed between the numbers 5 and 6 so that the resulting number is greater than 5 but less than 6?
75. There are 11 players on a football team. Their average age is 22 years. During the match, one of the players was eliminated. At the same time, the average age of the team became equal to 21 years. How old is the eliminated player?
76. – How old is your father? the boy is asked.
“As much as I do,” he replies calmly.
- How is this possible?
- Very simple: my father became my father only when I was born, because before my birth he was not my father, so my father is the same age as me.
Is this reasoning correct? If not, what is wrong with it?
77. There are 24 kg of nails in a bag. How can you measure 9 kg of nails on a pan balance without weights?
78. Peter lied from Monday to Wednesday and told the truth on other days, and Ivan lied from Thursday to Saturday and told the truth on other days. One day they said in the same way: "Yesterday was one of the days when I lie." What day was yesterday?
79. A three-digit number was written in numbers, and then in words. It turned out that all the numbers in this number are different and increase from left to right, and all words begin with the same letter. What is this number?
80. An error was made in the equality made up of matches: How should one match be shifted in order for the equality to become true?
81. How many times will a three-digit number increase if the same number is added to it?
82. If there were no time, there would not be a single day. If there were no day, it would always be night. But if it were always night, there would be time. Therefore, if there were no time, there would be. What is the reason for this misunderstanding?
83. Each of two baskets contains 12 apples. Nastya took a few apples from the first basket, and Masha took from the second as many as were left in the first. How many apples are left in the two baskets together?
84. One farmer has 8 pigs: 3 pink, 4 brown and 1 black.
How many pigs can say that in this small herd there is at least one more pig of the same color as her own? (The task is a joke.)
85. The only son of a shoemaker's father is a carpenter. Who is the cobbler to the carpenter?
86. If 1 worker can build a house in 5 days, then 5 workers will build it in 1 day. Therefore, if 1 ship crosses the Atlantic Ocean in 5 days, then 5 ships will cross it in 1 day. Is this statement correct? If not, what is the error in it?
87. Returning from school, Petya and Sasha went to the store, where they saw a large scale.
“Let's weigh our portfolios,” suggested Petya.
The scales showed that Petya's portfolio weighed 2 kg, while Sasha's portfolio weighed 3 kg. When the boys weighed the two briefcases together, the scales showed 6 kg.
- How so? Petya was surprised. Because 2 plus 3 does not equal 6.
- Can't you see? Sasha answered him. - The arrow has shifted on the scales.
What is the real weight of portfolios?
88. How to place 6 circles on the plane in such a way that you get 3 rows of 3 circles in each row?
89. After seven washes, the length, width and height of a bar of soap has halved. How many washes will the remaining piece last?
90. How to cut off 1/2 m from a piece of matter 2/3 m without the help of any measuring instruments?
91. It is often said that one must be born a composer, or an artist, or a writer, or a scientist. Is this true? Is it really necessary to be born as a composer (artist, writer, scientist)?
(The task is a joke.)
92. In order to see, it is not at all necessary to have eyes.
We see without the right eye. We also see without the left. And since we have no other eyes besides the left and right eyes, it turns out that neither eye is necessary for vision. Is this statement correct? If not, what is wrong with it?
93. The parrot lived less than 100 years and can only answer yes and no questions. How many questions does he need to ask to find out his age?
94. Say how many cubes are shown in Figure 54:
95. Three calves - how many legs? (The task is a joke.)
96. One man who fell into captivity tells the following: “My dungeon was in the upper part of the castle. After many days of effort, I managed to break one of the bars in the narrow window. It was possible to crawl through the resulting hole, but the distance to the ground was too great to simply jump down. In the corner of the dungeon, I found a rope forgotten by someone. However, it turned out to be too short to be able to go down it. Then I remembered how one wise man lengthened a blanket that was too short for him, cutting off part of it from below and sewing it on top. So I hurried to split the rope in half and re-tie the two resulting parts. Then it became long enough, and I safely went down it. How did the narrator manage to do this?
97. The interlocutor asks you to think of any three-digit number, and then offers to write down its numbers in reverse order to get another three-digit number. For example, 528 - 825, 439 - 934, etc. Then he asks to subtract the smaller number from the larger number and tell him the last digit of the difference. After that, he names the difference. How he does it?
98. Seven walked - they found seven rubles. If not for seven, but for three, would you find a lot? (The task is a joke.)
99. Divide the drawing, consisting of seven circles, with three straight lines into seven parts so that each part contains one circle:
100. The globe was pulled together by a hoop along the equator. Then the length of the hoop was increased by 10 m. At the same time, a small gap formed between the surface of the globe and the hoop. Can a person get through this gap? The length of the earth's equator is approximately 40,000 km.
