Entertaining logic. Logic and entertaining tasks (300 tasks) Math logic questions
These tasks can be given to children on the way to school, while traveling, or arrange a competition for children's holiday. It is rare that someone will be able to immediately answer the question, so you should gradually give small hints, this will make solving more fun and interesting.
We hope that you will not just put your child at the computer so that he immediately looks up all the answers. Do not forget that no car can replace parental love and attention for a son or daughter.
1. What word is always misspelled? (The task is a joke.)
Correct answer
2. How many months in a year have 28 days?
All months
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3. With what speed should the dog move (within the limits possible for it) in order not to hear the sound of a frying pan tied to its tail?
From zero. The dog needs to stand still
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4. The dog was tied to a ten-meter rope, and walked two hundred meters in a straight line. How did she do it?
Her rope was not tied to anything.
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5. How to jump off a ten-meter ladder and not hurt yourself?
Need to jump from the bottom step
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6. What can you see with your eyes closed?
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7. What does not burn in fire and does not sink in water?
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8. Who do the Australians call the sea wasp?
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9. What should you do when you see a green man?
Cross the street (this is a picture on a green traffic light)
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10. Moscow used to be called white stone. And what city was called black?
Chernihiv
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11. Residents medieval Europe sometimes wooden chocks were tied to the soles. For what purpose did they do it?
For protection against dirt, as there was no sewerage and slop was poured directly into the street
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12. In what process did water replace the sun, after 600 years sand replaced it, and after another 1100 years mechanism replaced them all?
In the process of measuring time - hours
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13. In the old days, barns were built on the outskirts, away from dwellings. For what purpose?
To prevent the fire from destroying food supplies
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14. Under Peter I, the coat of arms of the Russian Empire depicted an eagle holding maps of the four seas in its paws. List them.
White, Caspian, Azov, Baltic
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15. The name of which Germanic tribe gave the name to an entire European country?
The Germanic tribe of the Franks gave the name to France
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16. Why don't polar bears eat penguins in the wild?
Polar bears live at the North Pole, and penguins live at the South.
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17. Unwilling to admit that the Red Army could defeat them, the Germans argued that the Great Patriotic war General Frost, General Mud and General Mouse won. With regard to frost and dirt, everything is clear. And what's with the mouse?
Mice gnawed through the electrical wiring of German tanks
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18. Name five days without naming numbers (1, 2, 3, ..) and names of days (Monday, Tuesday, Wednesday ...)
The day before yesterday, yesterday, today, tomorrow, the day after tomorrow
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19. Thirty-two warriors have one commander.
Teeth and tongue
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20. Twelve brothers
They roam one after another
They don't bypass each other.
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21. How to say correctly: “I don’t see a white yolk” or “I don’t see a white yolk”?
The yolk is usually yellow
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22. Is it possible to light an ordinary match under water so that it burns out to the end?
Yes, in a submarine
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23. When is the best time for a black cat to sneak into the house?
When the door is open
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24. There were two fathers and two sons, they found three oranges. They began to divide - everyone got one. How could it be?
Correct answer
25. What dishes can not eat anything?
From empty
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26. Small, gray, like an elephant. Who is it?
Baby elephant
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27. Which hand is better to stir tea?
The one with the spoon
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28. They knock, they knock - they don’t tell you to be bored.
They go, they go, and everything is right there.
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29. Very fast two knights
They carry me through the snow - Through the meadow to the birch,
Pull two strips.
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30. When is a person in a room without a head?
When he sticks it out of the room (for example, out the window).
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31. What question cannot be answered with “yes”?
Are you sleeping?
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32. What question cannot be answered with “no”?
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33. When can the net draw water?
When the water freezes and turns to ice.
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34. Bold as ...,
insidious as ...,
cowardly like...,
cunning as...
wicked like...,
hungry like...,
hardworking like...,
faithful as...,
stubborn like...,
clueless like...,
quiet like...
free like….
Lion, snake, hare, fox, dog, wolf, ant, dog, donkey, ram, mouse, bird
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35. How do day and night end?
soft sign
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36. Magpie flies, and the dog sits on the tail. Could it be?
Yes, the dog sits on its own tail, a magpie flies nearby
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37. What should be done to keep five guys in one boot?
Each of them take off a boot
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38. How much is 2+2*2?
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39. In what month does chatty Svetochka speak the least?
February is the shortest month
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40. What belongs to you, but others use it more than you?
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41. How to find last year's snow?
Go outside right after the start of the new year.
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42. What word always sounds wrong?
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43. A man has one, a cow has two, a hawk has none. What's this?
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44. A man is sitting, but you cannot sit in his place, even if he gets up and leaves. Where is he sitting?
On your knees
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45. What stones are not in the sea?
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46. What sign should be put between 4 and 5 so that the result is more than 4 and less than 5?
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47. Can a rooster call itself a bird?
No, because he can't speak.
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48. What disease on earth has no one been ill with?
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49. Is it possible to predict the score of any match before it starts?
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50. What can be cooked but not eaten?
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51. What number will decrease by a third if it is turned over?
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52. At a square table, one corner was sawn off in a straight line. How many corners does the table have now?
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53. What knot cannot be untied?
Railway
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54. What is the cow in front and the bull behind?
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55. What is the most terrible river?
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56. What has no length, depth, width, height, but can be measured?
temperature, time
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57. What do all people on earth do at the same time?
Are getting older
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58. Two people were playing checkers. Each played five games and won five times. Is it possible?
Both people played different parts with other people.
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59. How can a thrown egg fly three meters and not break?
You need to throw an egg more than three meters, then the first three meters it will fly by.
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60. A man was driving a big truck. The headlights on the car were not on. There was also no moon. The woman began to cross the road in front of the car. How did the driver manage to see her?
It was a bright sunny day.
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61. Where is the end of the world?
Where the shadow ends
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62. Man learned from spiders to build suspension bridges, from cats he adopted the diaphragm in the camera and reflective road signs. And what invention came about thanks to snakes?
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63. What can you easily pick up from the ground, but not throw far?
Poplar fluff.
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64. What kind of comb will not comb your head?
Petushin.
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65. What do they drop when they need it and pick it up when they don't need it?
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66. What can travel around the world, staying in the same corner?
Postage Stamp.
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67. You are sitting in an airplane, a horse is in front of you, a car is behind you. Where are you?
On the carousel
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68. What notes can measure the distance?
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69. What won't fit in the biggest pot?
Her cover.
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70. Russian riddle. A wooden river, a wooden boat, and a wooden smoke streaming over the boat. What's this?
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71. A satellite makes one revolution around the Earth in 1 hour 40 minutes, and the other in 100 minutes. How can this be?
One hour and forty minutes equals one hundred minutes.
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72. Name at least three animals that Moses took into his ark?
Prophet Moses did not take animals into the ark, righteous Noah did it.
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73. In one hand the boy carried one kilogram of iron, and in the other the same amount of fluff. What was harder to carry?
Equally.
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74. In 1711, a new unit of 9 people appeared in each regiment of the Russian army. What is this division?
Regimental Band.
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Plane crashes.
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76. There is a story about a little boy who, having received new year gift, asked my mother: “Please remove the lid. I want to iron a gift." What is this gift?
Turtle
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77. What animals always sleep with their eyes open?
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78. It is known that at one time silkworm eggs were exported from China under pain of death. And what animal was taken out of Afghanistan in 1888 with the same risk?
Afghan Hound.
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79. What insects are domesticated by man?
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80. A problem invented by the learned monk and mathematician from Ireland Alcuin (735-804).
The peasant needs to be transported across the river wolf, goat and cabbage. But the boat is such that only a peasant can fit in it, and with him either one wolf, or one goat, or one cabbage. But if you leave the wolf with the goat, then the wolf will eat the goat, and if you leave the goat with the cabbage, then the goat will eat the cabbage. How did the peasant transport his cargo?
Solution 1.: It is clear that we have to start with a goat. The peasant, having transported the goat, returns and takes the wolf, which he transports to the other shore, where he leaves him, but he takes and carries the goat back to the first shore. Here he leaves her and transports the cabbage to the wolf. Then, returning, he carries a goat, and the crossing ends happily. Solution 2: First, the farmer again transports a goat. But the second one can take the cabbage, take it to the other side, leave it there and return the goat to the first bank. Then transport the wolf to the other side, return for the goat and again take it to the other side.
Correct answer
81. In the old days in Russia, married women wore a kokoshnik headdress, the name of which comes from the word "kokosh", meaning an animal. Which?
Chicken (remember what she says when she rushes?).
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82. Why can't a porcupine drown?
He has hollow needles.
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83. Name the fifth largest country after Russia, China, Canada and the USA.
Brazil.
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84. A man went to the market and bought a horse there for 50 rubles. But soon he noticed that the horses had risen in price, and sold it for 60 rubles. Then he realized that he had nothing to ride on, and bought the same horse for 70 rubles. Then he thought about how not to get a scolding from his wife for such an expensive purchase, and sold it for 80 rubles. What did he gain as a result of the manipulations?
Answer: -50+60-70+80=20
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85. The only bird that has auricles?
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86. Two approached the river at the same time. The boat on which you can cross can only support one person. And yet, without outside help, everyone crossed on this boat to the other side. How did they do it?
They sailed from different shores.
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87. In Chinese, the combination of three hieroglyphs "tree" means the word "forest". And what does the combination of two hieroglyphs "tree" mean?
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88. Residents of Kansas are very fond of Russian nuts. What is it if it is known that we can meet them in any market?
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89. The Romans made a revolutionary innovation in the design of the fork - all subsequent models became only variations of the solution found. And what was the fork before this innovation?
Single tooth.
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90. Chinese martial artists said that fighting is for fools, for smart people it is victory. And what, in their opinion, is for the wise?
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91. Name the language that is native to the largest number of people.
Chinese.
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92. In Ancient Russia they were called broken numbers. What are they currently called?
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93. A brick weighs two kilograms and half a brick. How many kilograms does a brick weigh?
Place a brick on one scale. On the other we put a 2-kilogram weight and half a brick. Now let's break the whole brick in half and remove half a brick from each scale pan. We get: on the left half a brick, on the right - a 2-kilogram weight. That is, half a brick weighs two kilograms. And two half-bricks, that is, a whole brick, weighs four kilograms.
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94. For some reason, these people, returning to their homeland, brought with them branches of exotic plants, for which they received their nickname. What are these people?
Pilgrims, they brought palm leaves.
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95. In terms of production, bananas rank first in the world, followed by citrus fruits. What fruits are on the third?
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96. In the US state of Arizona, they began to protect the desert from thieves. They steal that without which the desert is threatened by desolation and devastation. What are the thieves taking out of the desert?
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97. Name the plant that has the largest fruits.
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98. Neither fish nor meat - what was this Russian proverb originally about?
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99. In Spain they are called Portuguese, in Prussia they are called Russians. What are they called in Russia?
Cockroaches.
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100. Who do the Malays catch with a locked boombox cage with a live pig inside?
Pythons, after eating a pig, they could no longer get out of the cage.
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101. A hedgehog has 4 g, a dog has 100 g, a horse has 500 g, an elephant has 4-5 kg, and a person has 1.4 kg. What?
The mass of the brain.
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102. In 1825, the streets of Philadelphia were cleared of garbage by domestic animals. What?
Pigs.
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103. What dish was invented in the 17th century by Marco Aroni?
Pasta.
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104. What does any astronaut lose in flight?
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105. As you know, all native Russian female (full) names end either in A or in Z: Anna, Maria, Olga, etc. However, there is one female name that does not end in either A or Z. Name it.
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106. The Gallic priests found a trouble-free way to quickly mobilize soldiers in case of war. For this, they sacrificed only one person. What?
The last one to arrive.
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107. Once in the city of Nice they held a competition for the most enduring smoker. One of the participants set a record by smoking 60 cigarettes in a row. However, he did not receive the prize. Why?
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108. A person has twelve pairs of ribs. And who has more than three hundred ribs?
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109. In the mouth - a pipe, in the hand - a tambourine, under the arm - a mug. This is how buffoons were portrayed in Russia. As for the pipe and tambourine, everything is clear, but what is a mug?
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110. Everyone knows that "one cannot take dirty linen out of public." But what was supposed to be done with him if he couldn’t stand it?
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111. In what place did Russian men put on hats and mittens, regardless of the season?
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112. How is stickleback fish similar to birds?
She builds nests, laying eggs there.
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113. What is the tallest grass?
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114. Name a crop that burns 90% and 10% is thrown away.
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115. The Greeks used it to protect certain parts of their body. It was made from sandalwood bark. Name it.
Sandals.
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116. The first greenhouses appeared in France. Why do you think?
For growing oranges (orange - orange).
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117. The owner of the largest horn is the white rhinoceros (up to 158 cm). What animal has the softest horns?
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118. This is what football referees used before they used the whistle.
Bell.
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119. What is considered dirty when it is white and clean when it is green?
Blackboard.
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120. In practice, when moving along a curve, this ball makes 5,000 revolutions per minute, and when moving in a straight line, more than 20,000 revolutions per minute. Where is this ball located?
In a ballpoint pen.
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121. The great Hippocrates was asked: “Is it true that genius is a disease?” “Absolutely,” said Hippocrates, “but very rare.” What other feature of this disease was noted with regret by Hippocrates?
Non-contagious.
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122. What was the name of the city in England, where in 1873 the Indian game, popular to this day, was first demonstrated?
Badminton.
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123. Where, judging by the name, did the ancient Slavs attach a case for hunting edged weapons?
On the foot. These are scabbards.
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124. Three painters had a brother Ivan, and Ivan had no brothers. How could it be?
Ivan had three sisters.
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125. The Russian princes had various nicknames that came from the names of cities (Vladimir, Chernigov, Galitsky), from bright personal qualities (Udaloy, Wise, Kalita). What was the nickname given to Prince Vsevolod, who had twelve children?
Vsevolod the Big Nest.
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126. In 1240, the first census was conducted in Kievan Rus. Who did it and for what purpose?
Genghis Khan (to collect tribute from the population).
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127. It was the year 988 ... A large crowd of residents of ancient Kyiv for some reason moved to the Dnieper. What was the name of the road that the townspeople walked along?
988 - the year of the baptism of Russia. The street is called Khreshchatyk.
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128. Russia consisted of Great Russia (Russia proper), Little Russia (Ukraine), White Russia (Belarus). And what was the name of Manchuria, which was part of this state?
Zheltorossia.
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129. The Italian flag is red-white-green. Which cutaway berry helped the Italians choose these colors?
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130. Socrates did this "in order to sharpen the mind." So did Seneca. Horace was cured of a serious illness in this way. Suvorov was a big fan of this. A.S. Pushkin and L.N. Tolstoy also liked to do this. What were they doing?
They walked barefoot.
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131. How was a philosopher called before in Russia?
Lubomud.
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132. What flower was considered a symbol of royalty?
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133. If the Turks wanted to say "protect the village", they said "kara avyl". How are we talking now?
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134. The ancient Romans wore a tunic. And what did they wear when the cold came?
Several tunics worn one over the other.
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135. What is the Tatar word for “shoes”?
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136. We mainly use only the beginning of this proverb, and its end: "... just choked on his tail"?
Ate the dog.
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137. Say "Ole, close your eyes" in Danish.
Ole Lukoye.
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138. Barbarians were easily recognized by this piece of clothing.
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139. What literary character were 300 year old calluses?
Old man Hottabych.
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140. These three brothers can be called architects.
Three pigs.
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141. As you know, grandfather Mazay saved many hares from the flood. Name the person who saved eighteen doves and a sparrow during a fire.
Uncle Styopa.
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142. With what words does a proverb begin if its ending sounds like this: “... and cows lay eggs”?
They say that chickens are milked ...
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143. With what words does a proverb begin if its ending sounds like this: “... there will be Great Lent”?
Every day is not Sunday…
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144. How does the proverb begin: "... a large stump, but a hollow one"?
Small spool but precious.
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145. Everyone knows the expression "Keep as the apple of your eye." What is the "apple of the eye"?
Eye pupil.
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146. This word literally means "what will happen after the morning." What is this word?
Tomorrow - tomorrow.
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147. He really wanted to become a real boy and eventually became one. Who is he?
Pinocchio.
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148. What fairy tale hero from birth spoke three languages?
Dragon.
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149. In Russia, it was eaten everywhere, the Romans called it a stinking plant, and Pythagoras called it the king of spices. Name it.
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150. Before the advent of the potato, it served as the main food of the poor in Europe. And we know this better from a short work with six characters.
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151. What kind of plant is this, which embodies both a native and an adoptive relative?
Coltsfoot.
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152. Among all garden weeds, according to traditional medicine, it is very useful, especially if you cook a salad with it ...
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153. Russian riddle: "The girl is beautiful, and her heart is stone." What's this?
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154. Which peaceful ships do not have captains, but commanders?
Space.
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155. What is the most popular mode of transport for logging in hard-to-reach areas of Asia.
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156. Once upon a time, an officer named Siverst-Mering served in the Russian army, who, like Baron Munchausen, became famous for his indefatigable imagination. What phraseologism was born in connection with his name?
Lying like a gray gelding.
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157. He has four, but if they are all cut off, then he will have as many as eight. What is this about?
About the corners of a quadrilateral.
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158. Catherine II bought works of art all over the world in order to place them in a “secluded refuge”. What do we call it now?
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159. Julius Caesar ordered his soldiers to decorate their shields and weapons with jewels. What for?
To be a pity to quit.
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160. How is running different from walking? Before answering this question, remember that running can be slower than other walking, and that sometimes even running in place.
Running differs from walking not in the speed of movement. When walking, our body is always in contact with the ground at some point of the feet. When running, there are moments when our body is completely separated from the ground, not touching it at any point.
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161. All victims of accidents in the city were sent to the hospital in Kukuev. Most of all there were drivers and passengers injured in the accident. To reduce their number, the city authorities have made the use of seat belts mandatory. Drivers and passengers began to wear these belts, but the number of accidents remained unchanged, and the number of people injured in them who were admitted to the hospital even increased. Why?
The use of seat belts has reduced the number of deaths in road accidents. Many people who would have died without a seat belt (and ended up in morgues) survived but were injured and needed treatment. Therefore, the number of people admitted to the hospital has increased.
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162. There are two guards on the road. One looks in one direction of the road, and the other in the opposite direction, but at the same time they see each other. How can this be? Options with reflections, etc. - excluded.
Although sentries look in opposite directions, they do not stand back to back, but face each other.
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163. If it is raining at 12 o'clock at night, can we expect it to be sunny in 72 hours?
No, because in 72 hours it will be midnight again.
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164. There is a round deep lake with a diameter of 200 meters and two trees, one of which grows on the shore near the water, the other - in the center of the lake on a small island. A person who cannot swim needs to cross to the island with a rope, the length of which is a little more than 200 meters. How can he do it?
Having tied the rope with one end to a tree growing on the shore, it is necessary to go around the lake with a rope stretched over the water and tie the other end of the rope to the same tree. As a result, a double rope will be stretched between the trees for crossing to the island.
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165. A person lives on the 17th floor. He takes the elevator to his floor only in rainy weather or when one of his neighbors is in the elevator with him. If the weather is good and he is alone in the elevator, then he goes to the 9th floor, and then he goes up the stairs to the 17th floor ... Why?
Correct answer
166. One person was asked:
How old are you?
“Absolutely,” he replied.
- I am older than some of my relatives almost six hundred times. How can this be?
For example, if a person is 50 years old, and his grandson or granddaughter is 1 month old.
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167. People who came to one village were often surprised by the local fool. When offered a choice between a shiny 10-ruble coin and a crumpled 100-ruble bill, he always chose the coin, even though it costs ten times less than the bill. Why did he never choose the bill?
He was not stupid at all: he understood that as long as he chose a ten-ruble coin, people would offer him money to choose from, and if he chose a hundred-ruble bill, the offers of money would stop and he would receive nothing.
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168. The day before yesterday, Petya was 17 years old. He will turn 20 next year. How can this be?
If the current day is January 1, and Petya's birthday is December 31. The day before yesterday (December 30) he was 17 years old, yesterday (December 31) he turned 18 years old, this year he will be 19 years old, and next 20.
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169. One king wanted to remove his prime minister, but did not want to offend him too much. He called the prime minister to him, put two sheets of paper in his briefcase and said: “On one sheet I wrote “Go away”, and on the second - “Stay”. The leaf you pull out will decide your fate." The Prime Minister guessed that on both sheets of paper was written "Go away." How, however, did he manage to keep his place under these conditions?
The Prime Minister pulled out a piece of paper and, without looking at it, rolled it into a ball - and swallowed it. Since on the remaining sheet was -Go away-, the king had to admit that on the swallowed sheet was -Stay-.
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170. One gentleman, showing his friend a portrait painted for him by one artist, said: "I have neither sisters nor brothers, but the father of this man was my father's son."
The portrait shows the son of this gentleman.
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171. There are 8 benches in the park. Three have been painted. How many benches are there in the park?
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172. The thermometer shows plus 15 degrees. How many degrees will two such thermometers show?
15 degrees.
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173. A long loaf was cut into three parts. How many incisions were made?
Two cuts.
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174. What is lighter than 1 kg of cotton or 1 kg of iron?
Equally.
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175. The truck was going to the village. On the way he met 4 cars. How many cars were going to the village?
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176. Twice born, once dies. Who is it?
Chick.
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177. What can't you pick up from the floor by the tail?
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178. What always increases and never decreases?
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179. The more you take from it, the more it becomes. What's this?
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180. The 9-storey building has an elevator. On the first floor there are 2 people, on the second 4 people, on the third 8 people, on the fourth 16, on the fifth 32 and so on. Which button in the elevator of this house is pressed more often than others?
First floor button
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181. What goes uphill, then downhill, but remains in place?
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182. 7 sparrows were sitting on a tree, one of them was eaten by a cat. How many sparrows are left on the tree?
Not a single one: the surviving sparrows scattered.
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183. Guests came to you, and in the refrigerator there is a bottle of lemonade, a bag of apple juice and a bottle of mineral water. What will you open first?
Fridge.
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184. What Russian city flies?
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185. What is not eaten raw, but cooked - thrown away?
Bay leaf.
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186. What two words in Russian are written with three letters "e" in a row?
Long-necked and snake-eater.
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187. When the Europeans brought her to Tahiti, the islanders, who had never seen anything like it before, christened her a pig with teeth on its head. What do we call her?
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188. In Thailand, there are schools for monkeys. What do they teach?
Collect coconuts.
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189. How, according to scientists, does a crocodile get rid of excess salts in the body?
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190. One of the Japanese airlines paints huge eyes on the nose of their planes. What for?
Scare away the birds.
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191. Why do birds choose a cold day for departure in autumn, and arrive on a warm one in spring?
Choose a tailwind.
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192. According to the writer O'Henry, she is the only animal into which nails are driven. Who is it?
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193. From the skin of this particular animal, files were first made, which were used to polish wood and even marble.
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194. What animal takes second place after a person in terms of the number of images on pedestals?
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195. The absence of what organ does not allow sharks to stop even for a moment, otherwise they will simply drown?
Swim bladder.
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196. Who has teeth in his stomach?
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197. Until the XVI century. in nature, its varieties existed only in white and yellow. However, Dutch breeders, admirers of the Duke of Orange, brought out the currently known variety of patriotic color. What are we talking about?
About carrots.
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198. Judging by the name of this country, it should consist mainly of plains and steppes. Nevertheless, most of the plains no longer belong to it, and at present about half of its territory is occupied by mountains, hills and forests. What country is it?
Poland (from the word field).
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199. The territory of Finland is 8% covered with lakes. Although it is called the country of a thousand lakes (and their number is much larger), the primacy belongs to another. Which?
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200. What metal is less common in nature than platinum or uranium, but until recently it was in almost every home?
Mercury in a thermometer.
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201. In which US state is there one woman for every 50 men?
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202. There is something so fragile that even by saying its name, you will break it. What's this?
Correct answer
203. In 1086 the sister of Vladimir Monomakh opened a school at one of the Kyiv monasteries. How did this school differ from all those that existed in Russia before that?
Correct answer
204. Where was the potato first discovered?
Correct answer
205. How to write "nineteen", and then, removing the one, get
"twenty"?
Correct answer
206. Feed him and he will come to life. Get him drunk and he'll die. What it is?
Correct answer
207. What has 5 fingers, but is not a living being.
Glove.
Correct answer
208. I am nothing, but I have a name. Sometimes I'm big, sometimes
small and cannot exist alone. Who am I?
Correct answer
209. What is most like half an orange?
For the second half.
Correct answer
210. What part of a bookcase consists of half a consonant letter?
Correct answer
211. How many ends do three sticks have? Four and a half? two and a quarter?
Three have 6, four and a half have 10, two and a quarter have 6.
Correct answer
212. How many eggs can you eat on an empty stomach?
One (the rest will no longer be on an empty stomach).
Correct answer
213. What word begins with three letters "G" and ends with three letters "I"?
Trigonometry.
Correct answer
214. What is the arithmetic mean between a bicycle and a motorcycle.
Correct answer
215. Small, gray, like an elephant?
Baby elephant.
Correct answer
216. Atthere are two dombras,harpsthere are five of them, the guitar has six. How many do the piano have?
Seven (octaves).
Correct answer
217. What baby is born with a mustache?
For example, a kitten.
Correct answer
218. When can a person race at the speed of a racing car?
When he is in it.
Correct answer
219. What do elephants have and no other animals have?
Correct answer
220. To whom do all people take off their hats?
in front of the hairdresser.
Correct answer
221. How to write a mousetrap with five letters?
Correct answer
222. Son of my father, but not my brother?
Correct answer
223. What kind of fabric cannot be used to sew a shirt?
From the railway.
Correct answer
224. What city is in compote?
Izyum (City in Ukraine, in the Kharkov region).
Correct answer
225. There were 20 light bulbs in the lamp, 5 of them burned out. How many light bulbs are left?
Twenty light bulbs (15 working and 5 burned out).
Correct answer
226. Dad on a fishing trip caught 3 fish in 10 minutes. How long will it take him to catch 10 more fish?
The problem does not have a clear answer.
Correct answer
227. There were 9 buns on a tray. 9 girls took a bun. But there was only one bun left on the tray. How did it happen?
The last girl took the bun along with the tray.
Correct answer
228. Vasya is 5 years old. Anna is 9 years old. What is the age difference between them in three years?
Four years (the difference does not change with age).
Correct answer
229. From the forest, Misha brought 2 white mushrooms, 3 aspen mushrooms, 4 fly agaric and 5 russula to his grandmother for mushroom soup. How many mushrooms will grandma need for soup?
10 mushrooms, fly agaric - inedible mushroom.
Correct answer
230. Airplane, steamer, balloon, helicopter. What word is missing here?
Steamboat (does not fly).
Correct answer
231. Two people entered the entrance at the same time. One has an apartment on the 3rd floor, the other has an apartment on the 9th. How many times will the first one reach faster than the second?
4 times, because the 1st needs to overcome 2 gaps between floors, and the 2nd - 8.
Correct answer
232. What object, made by man before the 20th century, can move faster than sound?
The tip of the whip. We hear a characteristic click (pop) precisely because the tip overcomes the sound barrier.
Correct answer
233. Car wheel rolls to the right; its rim rotates clockwise. In which direction does the air move inside the rubber tire of the wheel - towards the rotation of the wheel or in the same direction?
The air inside the tire moves from the place of compression in both directions - forward and backward.
Correct answer
234. What is first in Russia and second in France?
Correct answer
235. A camel can withstand a load of 10 pounds for an hour. How long will he bear the burden of 1,000 poods?
None. The camel can't bear that weight.
Correct answer
236. Why are riddles dangerous for the head?
Because people break their heads over it.
Correct answer
237. What can snow and lilac bushes have in common?
Color. Lilac flowers are also white.
Correct answer
238. What does a watchman do when a sparrow sits on his head?
Correct answer
239. Where are cities without houses, rivers without water, and forests without trees?
On a geographical map
Correct answer
240. Which side of the world has one hundred and one letters in its name?
Correct answer
241. Who speaks all languages?
Correct answer
242. They go with a load, they stop without a load.
Clock with weights.
Correct answer
243. Who has a mustache longer than legs?
Cancer, cockroach.
Correct answer
244. What was "tomorrow" and will be "yesterday"?
Correct answer
245. Six legs, two heads, and one tail. What's this?
Rider on a horse.
Correct answer
246. Which clock shows the correct time only twice a day?
that have stopped.
Correct answer
247. Somehow the guys gathered at a picnic, only 6 people. They look, and instead of 6 apples they took 5. How to divide the apples equally among everyone so that no one is offended? You can't cut or break them.
You need to cook compote from apples.
Correct answer
248. If Erica lives in Washington DC and Tina lives in Buenos Aires, where does Ty live?
In Pekin. The names of people are part of the names of the country in whose capital each of them lives.
Correct answer
249. In 1849, a man went to California, where the "gold rush" was raging. He hoped to get rich by selling tents to gold miners. However, the weather was fine, and the gold diggers slept right under the open sky. Nobody bought tents. Nevertheless, the seller got rich, and his products are sold to this day. How did he do it and what was his name?
Correct answer
250. The spy sat down in the bushes and assesses the situation at the checkpoint. An officer comes up, sentry to him: "Password."
Officer: "26".
Sentry: Feedback.
Officer: "13".
Sentry: "Come in."
The second one fits: "Password!" - "22".
"Review" - "11".
"Come on."
Well, the spy thought he figured out the password system, he runs to the sentry.
Sentry: "Password".
Spy: "100".
Sentry: Feedback.
Spy: "50".
In general, they caught a spy. What would be the correct answer?
The correct answer is 3. This is the number of letters in the word one hundred.
Correct answer
251. For each of the following words, think of a word that has the same semantic meaning and begins with the letter K:
Wealth, Seal, Universe, Lattice, Hearth, Comfort, Crown, Duke, Castle, Hammer.
1. Capital. 2. Brand. 3. Space. 4. Cell. 5. Fireplace. 6. Comfort. 7. Crown. 8. Prince. 9. Fortress. 10. Sledgehammer.
Correct answer
252. The doctor prescribed three tablets to the patient and ordered them to be taken every half an hour. How long will it take to take the pills?
At first glance, it may seem that a person will drink 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. The person immediately 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.
Correct answer
253. What insect applauds the whole world?
Correct answer
254. Is she red? - No, black. Why is she white? Because green. What's this?
Black currant.
Correct answer
255. How can you put two liters of milk in a liter jar?
Cook condensed milk from it.
Correct answer
256. Comic task. A hunter is riding in a bus, he sees a hare running. He fired. Where did he get to?
To the police (Shooting in vehicles is prohibited).
Correct answer
257. Who is the master of all trades?
Glover.
Correct answer
258. How to throw a tennis ball so that after flying a short distance it stops and starts moving in the opposite direction? In this case, the ball must not hit an obstacle, it must not be hit with anything or tied to anything.
Throw it up.
Correct answer
259. The ratio of the age of one boy to the age of another boy was the same a few years ago as it is now. What is this attitude?
One to one, that is, boys of the same age.
Correct answer
260. What is the largest number that can be written with four ones?
Eleven to the eleventh power.
Correct answer
261. In the dense Murom forest, ten sources of dead water gush out of the ground, they are numbered from No. 1 to No. 10.
From the first nine sources, everyone can take dead water, but source No. 10 is located in Koshchei's cave, into which no one except Koshchei himself can get.
The taste and color of dead water is no different from ordinary water, however, if a person drinks from any source, he will die. Only one thing can save him: if he drinks poison from a source whose number is greater. For example, if he drinks from the seventh source, then he must definitely drink poison No. 8, No. 9 or No. 10. If he drinks not the seventh poison, but the ninth, only poison No. 10 can help him. And if he immediately drinks the tenth poison, then nothing will help him.
Ivan the Fool challenged Koshchei to a duel. The terms of the duel were as follows: each brings a mug of liquid with him and gives it to his opponent to drink. Koschei was delighted: “I will give poison number 10, and Ivan the Fool will not be able to escape! And I myself will drink the poison that Ivanushka the Fool will bring me, I will drink it with my tenth and be saved!
On the appointed day, both opponents met at the agreed place. They honestly exchanged mugs and drank what was in them. It turned out that Koschei died, but Ivan the Fool remained alive! How did it happen?
Ivanushka gave Kashchei plain water, and it turned out that Kashchei drank poison from the 10th spring. Before the duel, Ivanushka himself drank poison from any one source and it turned out that he washed down the poison with Kashcheev 10, and as a result, this poison was neutralized ..
Correct answer
262. Mentally divide by two the following number: one sextillion seven
Half a sixtillion three and a half
Correct answer
263. How to divide five apples among five people in such a way that one apple remains in the basket? (Joke task)
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. Highlighted in yellow, add as a note to the same task, we have it.
Correct answer
264. How can the number 66 be increased by one and a half times without performing any arithmetic operations on it?
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.
Correct answer
265. One lily leaf grows in a 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?
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 in 100 days.
Correct answer
266. Is it possible to fly to the moon by plane? (It must be taken into account that aircraft are equipped with jet engines, like space rockets, and run on the same fuel as them.)
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.
Correct answer
267. A girl dropped her ring into a cup containing instant coffee. Why is the ring dry?
The cup has not been filled with water yet.
Correct answer
268. 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?
It is necessary to turn to any of the warriors with the following question: “If I ask you, does this exit lead to freedom, then you will 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”.
Correct answer
269. If three days ago there was a day preceding Monday, what day will be the day after tomorrow?
Sunday was before Monday. If three days ago it was Sunday, then today is Wednesday. If today is Wednesday, then the day after tomorrow will be Friday.
Correct answer
270. The girl was riding in a taxi. She talked so much along the way that the driver got nervous. He told her that he was very sorry, but he could not hear a word because his hearing aid did not 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?
If the taxi driver is deaf, how did he understand where to take the girl? And one more thing: how did he then understand that she was saying anything at all?
Correct answer
271. You are in the cabin of an ocean liner at anchor. At midnight, the water was 4 m below the porthole and rose half a meter per hour. If this speed doubles every hour, how long will it take the water to reach the porthole?
The water will never reach the porthole because the liner rises with the water.
Correct answer
272. A train leaves Moscow for Vladivostok every day. Also every day a train leaves Vladivostok for Moscow. The move takes 10 days. If you left Vladivostok for Moscow, how many trains going in the opposite direction will you meet during the trip?
At first glance, it may seem that during the trip we will meet ten trains. But this is not so: we will meet not only those ten trains that left Moscow after our departure, but also those that were already on the way by the time of our departure. This means that we will meet not ten, but twenty trains.
Correct answer
273. There is an easy and cheap way to travel, which, surprisingly, no one uses. As you know, the Earth rotates around its axis, and quite quickly (in just 24 hours, each point on the earth's equator travels approximately 40,000 km - a path equal to the length of the equator). So, instead of going somewhere by train or flying by plane, or sailing on a ship, it is enough for us to rise high above the earth in a balloon or airship and stay there motionless for some time. During this time, the Earth will turn to us with another part of its surface and it will only be necessary to descend to the right place. Is this reasoning correct? If not, what is wrong with it?
This way of travel is, of course, unsuitable. The atmosphere, attracted by the Earth, rotates with it. And even if the atmosphere were motionless, then, having risen into it from the rotating Earth, we would continue the earth's movement by inertia for some time. In addition, if the atmosphere were motionless, and the Earth would continue to rotate in it (and fast enough: see the condition of the problem), then in this case the greatest hurricane would not stop raging on Earth, which would make impossible not only any travel but also human life itself.
Correct answer
274. Is it possible to boil water on an open flame in a paper box?
The question of the problem, at first glance, seems very strange, because if you hold paper over a fire, it will definitely catch fire. But the fact is that the boiling point of water is much lower than the ignition temperature of paper. Since the heat from the flame is taken away by the boiling water, the paper cannot reach the required temperature and therefore does not ignite. It is only necessary that the paper is thick enough, otherwise the water will simply tear it and pour out onto the flame. A cardboard box is quite suitable for boiling water. The same explanation underlies such a phenomenon as a fireproof piece of paper tightly wound around a metal rod (or steel nail) and brought into the flame of a candle. The rod will take the heat of the fire, preventing the paper from heating up to the desired temperature and catching fire.
Correct answer
275. 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 in the class wrote an essay on a free topic, which had to end with the phrase: "Everything written here is true" or "Everything written here is a lie." There were 17 truth-tellers and 18 liars in the class. How many essays turned out with a statement about the veracity of what was written?
All the truth-seekers rightly claimed that everything they wrote was true, but all the liars falsely claimed that everything they wrote was true. Thus, all 35 essays contained a statement about the veracity of what was written.
Correct answer
276. How many great-great-grandparents and great-great-grandmothers did you have in total?
Each person has 2 parents, 2 grandmothers and 2 grandfathers, 4 great-grandparents and 4 great-grandparents, 8 great-great-grandparents and 8 great-great-grandparents.
Correct answer
277. Dialogue in a household goods store:
How much does one cost?
- 20 rubles, - the seller answered.
How much is 12?
- 40 rubles.
Okay give me 120.
- Please, 60 rubles from you.
What did the visitor buy?
Room for an apartment.
Correct answer
278. 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?
At first glance, it may seem that a bottle costs 1 ruble, and a cork 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 r. 05 k., and the cork costs 5 k.
Correct answer
279. 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?
At first glance, it may seem that Olya walks 30 steps - half as many as Katya, since she lives two times lower than her. 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.
Correct answer
280. How to pour out exactly half of a mug, ladle, pan and any other dish of regular cylindrical shape, filled to the brim with water, without using any measuring instruments?
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. It is necessary to drain water 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 dishes:
Correct answer
281. Three hens lay three eggs in three days. How many eggs will 12 hens lay in 12 days?
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.
Correct answer
282. Name two numbers in which the number of digits is equal to the number of letters that make up the name of each of these numbers.
One hundred (100) and one million (1000000)
Correct answer
283. "I guarantee," said the salesman in 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 mute as a fish. However, the seller did not lie. How is this possible? (The task is a joke.)
The parrot can indeed repeat every word it hears, but it is deaf and does not hear a single word.
Correct answer
284. There is a candle and a kerosene lamp in the room. What will you light first when you enter this room in the evening?
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, then - from it - everything else).
Correct answer
285. Half of half of a number is equal to half. What is this number?
Correct answer
286. Over time, man 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?
The reasoning is wrong. It is not necessary that Sasha Ivanov eventually visit Mars. The external correctness of this reasoning is created due to the use of one word ("man") in it in two different senses: in the broad (abstract representative of humanity) and in the narrow (concrete, given, this particular person).
Correct answer
287. 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.)
Of course, a composer, as well as an artist, writer or scientist, must be born, because if a person is not born, then he will not be able to compose music, draw pictures, write novels or make scientific discoveries. This joke problem is based on the ambiguity of the question: "Do you really have to be born?" This question can be understood literally: is it necessary to be born in order to engage in any type of activity; and also this question can be understood in a figurative sense: is the talent of a composer (artist, writer, scientist) innate, given by nature, or is it acquired during life by hard work.
Correct answer
288. 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?
The reasoning is, of course, wrong. Its external correctness is based on the almost imperceptible exclusion of one more option, which in this reasoning also had to be considered. This is an option when not a single eye sees. It was he who was omitted: “Without the right eye we see, without the left too, which means that the eyes are not necessary for vision.” The correct statement should be: “Without the right eye, we see, without the left we also see, but without two together we cannot see, which means that we see either with one eye, or the other, or two together, but we cannot see without eyes, which, thus necessary for vision."
Correct answer
289. 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?
At first glance, it may seem that a parrot can be asked up to 99 questions. In fact, you can get by with a much smaller number of questions. Let's ask him like this: "Are you over 50 years old?" If he answers "yes", then his age is from 51 to 99 years; if he answers “no”, then he is from 1 year to 50 years old. The number of options for his age after the first question is halved. The next similar question: “Are you more (you can ask - less) 25 years old?”, “Are you more (less than) 75 years old?” (depending on the answer to the first question) reduces the number of options by four times, etc. As a result, the parrot needs to be asked only 7 questions.
Correct answer
290. 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?
The narrator divided the rope not across, as it most likely might seem, but along it, making two ropes of the same length out of it. When he tied the two pieces together, the rope became twice as long as it was at first.
Correct answer
291. Make a question out of five consecutive letters of the Russian alphabet. Hint: it might not be just one word.
Correct answer
292. Before you is an electronic clock. How many times a day will they show the time so that all cells on the dial (hours, minutes, seconds) will be filled with the same digit?
Three times: 00.00.00; 11/11/11; 22.22.22
Correct answer
293. A man tossed and turned in bed for a long time at night and could not fall asleep in any way ...
Then he picked up the phone, dialed someone's number, after listening to a few long beeps, hung up and fell asleep peacefully. Q: Why couldn't he sleep before?
The truck ran out of fuel getting to the center of the bridge.
Correct answer
298. I was invited to a party. There I saw a man with a very rare watch. How do I know this watch was stolen?
Because this watch was mine.
Correct answer
299. 8 + 7 = 13 or 7 + 8 = 13?
8 + 7 = 15 not 13
Correct answer
300. Frau and Herr Meyers have 4 daughters. Each daughter has one brother. How many children do the Myers have in total?
5. Four daughters and one son.
Correct answer
The words of Sherlock Holmes: “How many times have I told you, drop everything impossible, then what remains will be the answer, no matter how incredible it may seem,” could serve as an epigraph to this chapter.
If solving a puzzle requires only the ability to think logically and does not need to perform arithmetic calculations at all, then such a puzzle is usually called a logical problem. Logic problems, of course, are among the mathematical ones, since logic can be considered as very general, fundamental mathematics. Nevertheless, it is convenient to single out and study logical puzzles separately from their more numerous arithmetic sisters. In this chapter, we will outline three common types of logical problems and try to figure out how to approach them.
The most common type of problem that puzzle lovers sometimes call the “Smith-Jones-Robinson problem” (by analogy with the old puzzle invented by G. Dudeni).
It consists of a series of parcels, usually reporting certain information about the characters; On the basis of these assumptions, certain conclusions must be drawn. For example, here is what the latest American version of the Dudeney problem looks like:
1. Smith, Jones and Robinson work in the same train crew as a driver, conductor and fireman. Their professions are not necessarily named in the same order as their surnames. There are three passengers with the same surnames on the train served by the brigade.
In the future, we will respectfully call each passenger "Mr" (Mr).
2. Mr. Robinson lives in Los Angeles.
3. The conductor lives in Omaha.
4. Mr. Jones has long forgotten all the algebra he was taught in college.
5. Passenger - conductor's namesake lives in Chicago.
6. The conductor and one of the passengers, a well-known specialist in mathematical physics, go to the same church.
7. Smith always beats the stoker when they happen to meet for a game of billiards.
What is the name of the driver?
These problems could be translated into the language of mathematical logic, using its standard notation, and a solution could be sought using appropriate methods, but such an approach would be too cumbersome. On the other hand, without abbreviations of one kind or another, it is difficult to understand the logical structure of the problem. It is best to use a table empty cells which we will inscribe all possible combinations of elements of the sets under consideration. In our case, there are two such sets, so we need two tables (Fig. 139).
Rice. 139 Two tables for the problem of Smith, Jones and Robinson.
In each cell we enter 1 if the corresponding combination is admissible, or 0 if the combination contradicts the conditions of the problem. Let's see how it's done. Condition 7 obviously excludes the possibility that Smith is a stoker, so we enter 0 in the box in the upper right corner of the left table. Condition 2 tells us that Robinson lives in Los Angeles, so in the lower left corner of the table we enter 1, and 0 to all other cells in the bottom row and left column to show that Mr. Robinson does not live in Omaha or Chicago, and Mr. Smith and Mr. Jones do not live in Los Angeles.
Now we have to think a little. From conditions 3 and 6 we know that the mathematical physicist lives in Omaha, but we do not know his last name. He cannot be either Mr. Robinson or Mr. Jones (after all, he has forgotten even elementary algebra).
Therefore, it must be Mr. Smith. We note this circumstance by putting 1 in the middle cell of the upper row of the right table and 0 in the remaining cells of the same row and empty cells in the middle column. The third unit can now be entered in only one cell: this proves that Mr. Jones lives in Chicago. From condition 5, we learn that the conductor also has the last name Jones, and we enter 1 in the central cell of the left table and 0 in all other cells of the middle row and middle column. After that, our tables take the form shown in Fig. 140.
Rice. 140 Table eggs shown in fig. 139, after prefilling.
Now it is not difficult to continue the reasoning leading to the final answer. In the column labeled "Stoker", a unit can only be placed in the bottom cell. It immediately follows from this that 0 should be in the lower left corner. Only the cell in the upper left corner of the table remains empty, where only 1 can be put. So, the name of the driver is Smith.
Lewis Carroll liked to invent extremely complex and ingenious problems of this kind. The dean of mathematics at Dortmouth College, John J. Kemeny, programmed one of the monstrous (with 13 variables and 12 conditions, from which it follows that "no judge sniffs tobacco") Carroll problems for the IBM-704 computer. The machine completed the solution in about 4 minutes, although printing out the complete "truth table" of the problem (a table showing whether the possible combinations of truth values of the problem's variables are true or false) would have taken 13 hours!
For readers who want to try their luck with a more difficult problem than the Smith-Jones-Robinson problem, we offer a new puzzle. Its author is R. Smullyan of Princeton University.
1. In 1918, the first World War. On the day of the signing of the peace treaty, three married couples gathered to celebrate this event at the festive table.
2. Each husband was the brother of one of the wives, and each wife was the sister of one of the husbands, that is, among those present, three related pairs of “brother and sister” could be indicated.
3. Helen is exactly 26 weeks older than her husband, who was born in August.
4. Mr. White's sister is married to Ellen's brother-in-law and married him on her birthday, in January.
5. Margaret White is shorter than William Blake.
6. Arthur's sister is prettier than Beatrice.
7. John is 50 years old.
What is Mrs Brown's name?
No less common is another variety of logical problems, which, by analogy with the following well-known example, can be called problems of the “colored caps problem” type. Three people (let's call them A, B and FROM) blindfold and say that each of them was put on either a red or a green cap. Then their eyes are untied and they are asked to raise their hand if they see a red cap, and to leave the room if they are sure that they know what color the cap is on their head. All three hats turned out to be red, so all three raised their hands. Several minutes passed and FROM, which is more intelligent than BUT and AT, left the room. How FROM was able to determine what color the hat is on it?
[The problem of the wise men in green caps is formulated in the text in such a way that it cannot have a solution. This is especially evident when the number of wise men is large. How long will it take the first wise man to guess the true situation?
At the end of the forties this problem was intensively discussed in Moscow in school mathematical circles, and a new version of it was invented, in which discrete time was introduced. The task looked like this.
In ancient times, wise men lived in one city. Each of them had a wife. In the mornings they came to the market and found out all the gossip of the city there. They were gossipers themselves. It gave them great pleasure to learn about the infidelity of any of the wives - they found out about it immediately. However, one unspoken rule was strictly observed: nothing was ever reported to the husband about his wife, since each of them, having learned about his own shame, would have driven his wife out of the house. So they lived, enjoying intimate conversations and remaining completely ignorant of their own affairs.
But one day a real gossip came to town. He came to the bazaar and publicly declared: “But not all wise men have faithful wives!” It would seem that the gossip did not say anything new - and so everyone knew it, every sage knew it (only with malice he thought not about himself, but about the other), so none of the residents paid any attention to the gossip's words. But the wise men thought - that's why they are wise men - and n-th day after the arrival of the gossip n wise men were expelled n unfaithful wives (if there were n).
It is not difficult to restore the reasoning of the sages. It is more difficult to answer the question: what information did the gossiper add to that which was known to the sages even without him?
This problem has been repeatedly encountered in the literature].
C asks himself if his cap can be green. If that were the case, then BUT would immediately recognize that he was wearing a red cap, because only a red cap on his head could make AT raise a hand. But then BUT would leave the room. AT would have begun to reason in exactly the same way and would also have left the room. Since neither one nor the other came out, FROM concluded that his own cap should be red.
This problem can be generalized to the case when there are any number of people and all of them are wearing red caps. Assume that a fourth actor has appeared in the problem D, even more insightful than C.D could reason like this: “If my cap were green, then A, B and FROM would find themselves in exactly the same situation that has just been described, and in a few minutes the most perceptive of the trio would certainly leave the room.
But five minutes have already passed, and none of them comes out, therefore, my cap is red.
If there were a fifth member who was even smarter than D, he could have come to the conclusion that he was wearing a red cap after waiting ten minutes. Of course, our reasoning loses its persuasiveness due to assumptions about different degrees of ingenuity. A, B, C... and rather vague considerations as to how long the most perceptive person should wait before he can confidently name the color of his cap.
Some other "color cap" problems contain less uncertainty. Such, for example, is the following problem, also invented by Smullyan. Each of the three A, B and FROM- is fluent in logic, that is, he knows how to instantly extract all the consequences from a given set of premises and knows that the rest also have this ability.
We take four red and four green stamps, blindfold our “logicians” and stick two stamps on each of their foreheads. Then we remove the bandages from their eyes and, in turn, ask A, B and FROM the same question: "Do you know what color the stamps are on your forehead?" Each of them answers in the negative. We then ask again BUT and again we get a negative answer. But when we ask the same question a second time AT, he answers in the affirmative.
What color is the mark on the forehead AT?
The third type of popular logic puzzles are problems about liars and those who always tell the truth. AT classic version problem we are talking about a traveler who ended up in a country inhabited by two tribes. Members of one tribe always lie, members of another always tell the truth. The traveler meets two natives. "Do you always tell the truth?" he asks the tall native. He replies: "Tarabar". "He said yes," explains the smaller native who knows English, "but he's a terrible liar." To which tribe does each of the natives belong?
A systematic approach to solving would be to write out all four possibilities: AI, IL, LI, LL (I means "true", L - "false") - and exclude those that contradict the data of the problem. An answer can be obtained much more quickly if one observes that the tall native must answer in the affirmative whether he is lying or telling the truth. Since the smaller native told the truth, he must belong to the tribe of the truthful, and his tall friend - to the tribe of liars.
The most famous problem of this type, complicated by the introduction of probability weights and a not very clear formulation, can be found quite unexpectedly in the middle of the sixth chapter of the book New Pathways in Science by the English astronomer A. Eddington. "If a A, B, C and D tell the truth one time out of three (independently) and BUT States that AT denies that FROM says as if D liar, what is the probability that D told the truth?"
Eddington's answer, 25/71, was met with a hail of protest from readers and gave rise to a ridiculous and confused dispute that was never finally resolved. The English astronomer G. Dingle, the author of a review of Eddington's book published in the journal Nature (March 1935), believed that the problem does not deserve attention at all as meaningless and only indicates that Eddington had not sufficiently thought through the basic ideas of probability theory. The American physicist T. Stern (Nature, June 1935) objected to this, stating that, in his opinion, the problem is by no means meaningless, but there is not enough data to solve it.
In response, Dingle remarked (Nature, September 1935) that if one takes Stern's point of view, then there is enough data for a decision and the answer will be 1/3. Here Eddington entered the fray, publishing (Mathemetical gazette, October 1935) an article explaining in detail how he got his answer. The dispute ended with two more articles that appeared in the same journal, the author of one of them defended Eddington, and the other put forward a point of view different from all the previous ones.
The difficulty lies mainly in understanding Eddington's formulation. If a AT, expressing his denial, speaks the truth, then can we reasonably assume that FROM said that D speak the truth? Eddington believed that there were not enough grounds for such an assumption. Likewise, if BUT lies, can we be sure that AT and FROM did they say anything at all? Fortunately, we can get around all these linguistic difficulties by making the following assumptions (Eddington did not make them):
1. None of the four remained silent.
2. Statements A, B and FROM(each of them separately) either confirm or deny the following statement.
3. A false assertion coincides with its negation, and a false negation coincides with an assertion.
All four lie independently of each other with a probability of 1/3, that is, on average, any two of their three statements are false. If a true statement is denoted by the letter And, and false - letter L, then for A, B, C and D we get a table consisting of eighty-one different combinations. From this number, one should exclude those combinations that are impossible due to the conditions of the problem.
Number of valid combinations ending with a letter And(i.e. truthful - true - statement D), should be divided by the total number of all valid combinations, which will give the answer.
The formulation of the problem about a traveler and two natives should be clarified. The traveler realized that the word "gibberish" in the language of the natives means either "yes" or "no", but he could not guess what exactly. This would have alerted several emails, one of which I reproduce below.
The tall native, apparently, did not understand a word of what was said to him (in English language) traveler, and could not answer "yes" or "no" in English. Therefore, his "gibberish" means something like: "I do not understand" or "Welcome to Bongo-Bongo." Consequently, the little native lied when he said that his friend answered "yes", and since the little one was a liar, he also lied when he called the tall native a liar. Therefore, a tall native should be considered truthful.
So female logic dealt a blow to my male vanity. Doesn't it hurt your author's pride a little?
Answers
The first logical problem is best solved using three tables: one for combinations of first and last names of wives, the second for first and last names of husbands, and the third for family ties.
Since Mrs. White's name is Margaret (condition 5), we are left with only two possibilities for the names of the other two wives: a) Helen Blake and Beatrice Brown, or b) Helen Brown and Beatrice Blake.
Let us assume that the second of the possibilities takes place. White's sister must be either Helen or Beatrice. But Beatrice cannot be Wyne's sister, because then Blake would be Helen's brother, and Blake's two brothers-in-law would be White (his wife's brother) and Brown (his sister's husband); Beatrice Blake is not married to either of them, which contradicts condition 4. Therefore, White's sister must be Helen. From this, in turn, we conclude that Brown's sister is called Beatrice, and Blake's sister is Margaret.
It follows from condition 6 that Mr. White's name is Arthur (Brown cannot be Arthur, since such a combination would mean that Beatrice is more beautiful than herself, and Blake cannot be Arthur, since from condition 5 we know his name: William). So, Mr. Brown can only be John. Unfortunately, from condition 7 we see that John was born in 1868 (50 years before the signing of the peace treaty). But 1868 is a leap year, so Helen must be older than her husband by one day more than the 26 weeks stated in condition 3. (From condition 4 we know that she was born in January, and from condition 3 that her husband was born in August. She could be exactly 26 weeks older than her husband if her birthday was on January 31st and his on August 1st, and if there was no February 29th between these dates!) So, the second of the possibilities, with which we started should be discarded, which allows us to name the wives: Margaret White, Helen Blake and Beatrice Brown. There is no contradiction here, since we do not know the year of Blake's birth. From the conditions of the problem, it can be concluded that Margaret is Brown's sister, Beatrice is Blake's sister, and Helen is White's sister, but the question of the names of White and Brown remains unresolved.
In the problem with stamps AT there are three possibilities. His stamps can be: 1) both red; 2) both green; 3) one is green and the other is red. Let's assume that both stamps are red.
After all three have answered once, BUT can reason like this: “The marks on my forehead cannot be both red (because then FROM would have seen four red stamps and would have recognized at once that he had two green stamps on his forehead, and if FROM both stamps were green, then AT, seeing four green stamps, would have realized that he had two red stamps on his forehead). That is why I have one green and one red mark on my forehead.”
But when BUT asked a second time, he didn't know what color his brand was. It allowed AT discard the possibility that both of his own stamps are red. Arguing in exactly the same way as A, B ruled out the case when both of his stamps are green. Therefore, he was left with only one possibility: one stamp is green, the other is red.
Several readers quickly noticed that the problem can be solved very quickly without having to analyze the questions and answers. Here is what one of the readers wrote about this: “The conditions of the problem are completely symmetrical with respect to the red and green marks.
Therefore, by distributing stamps between A, B and FROM if all the conditions of the problem are met and replacing the red marks with green and, conversely, green with red, we will arrive at a different distribution, for which all conditions will also be satisfied. It follows that if the solution is unique, then it must be invariant (should not change) when replacing green labels with red ones, and red ones with green ones. Such a solution can only be such a distribution of stamps, in which B will have one green and one red stamp.
As W. Manheimer, Dean of the Department of Mathematics at Brooklyn College, put it, this elegant solution comes from the fact that not A, B and FROM(as stated in the condition of the problem), and Raymond Smullyan!
In the Eddington problem, the probability that D tells the truth, is 13/41. All combinations of true and false that contain an odd number of times false (or true) should be discarded as contradicting the conditions of the problem. As a result, the number of possible combinations is reduced from 81 to 41, of which only 13 end in a true statement. D. Because the A, B and FROM tell the truth in cases that correspond to exactly the same number of valid combinations, the probability of telling the truth is the same for all four.
Using the Equivalence Symbol
which means that the propositions connected by it are either both true or both false (then the false proposition is true, otherwise it is false), and the negation symbol ~, Eddington's problem in the propositional calculus can be written as follows:
or after some simplifications like this:
The truth table of this expression confirms the answer already received.
Notes:
That's frustrating- upset, make something futile, hopeless, doom to failure (English).
See the chapter on Raymond Smullyan in the book M. Gardner"Time Travel" (M.: Mir, 1990).
Eddington A. New Pathways in Science. - Cambridge: 1935; Michigan: 1959.
Logic tasks, as well as mathematics, is called "mind gymnastics". But unlike mathematics, logic tasks- this is an entertaining gymnastics, which in a fun way allows you to test and train thought processes, sometimes in an unexpected perspective. To solve them, you need quick wit, sometimes intuition, but not special knowledge. Solving logic problems is to thoroughly analyze the condition of the problem, to unravel the tangle of contradictory connections between characters or objects. Logic tasks for children- these are, as a rule, whole stories with popular characters, in which you just need to get used to, feel the situation, visualize it and catch the connections.
Even the most challenging tasks on logic do not contain numbers, vectors, functions. But the mathematical way of thinking is necessary here: the main thing is to comprehend and understand the condition logical task. The most obvious decision on the surface is not always the right one. But more often than not, solving a logic problem turns out to be much simpler than it seems at first glance, despite the confusing condition.
Interesting logic tasks for children in a variety of subjects - mathematics, physics, biology - arouse their increased interest in these academic disciplines and help in their meaningful study. Logic tasks for weighing, transfusion, tasks for non-standard logical thinking will help to solve everyday problems in a non-standard way in everyday life.
In the process of decision logic tasks you will get acquainted with mathematical logic - a separate science, otherwise called "mathematics without formulas." Logic as a science was created by Aristotle, who was not a mathematician, but a philosopher. And logic was originally part of philosophy, one of the methods of reasoning. In the work "Analytics" Aristotle created 20 schemes of reasoning, which he called syllogisms. One of his most famous syllogisms is: “Socrates is a man; all men are mortal; So Socrates is mortal. Logic (from other Greek. Λογική - speech, reasoning, thought) is the science of correct thinking, or, in other words, "the art of reasoning".
There are certain methods solving logic problems:
way of reasoning, with the help of which the simplest logical problems are solved. This method is considered the most trivial. In the course of the solution, reasoning is used that consistently takes into account all the conditions of the problem, which gradually lead to a conclusion and the correct answer.
tables way, used in solving text logic problems. As the name implies, solving logical problems consists in building tables that allow you to visualize the condition of the problem, control the process of reasoning, and help draw the right logical conclusions.
graphs way consists in sorting out possible scenarios for the development of events and the final choice of the only correct solution.
flowchart method- a method widely used in programming and solving logical transfusion problems. It consists in the fact that, first, operations (commands) are allocated in the form of blocks, then the sequence of execution of these commands is established. This is the block diagram, which is essentially a program, the execution of which leads to the solution of the task.
billiard way follows from the theory of trajectories (one of the sections of probability theory). To solve the problem, it is necessary to draw a billiard table and interpret the actions of the movements of the billiard ball along different trajectories. In this case, it is necessary to keep records of possible results in a separate table.
Each of these methods is applicable to solving logical problems from different areas. These seemingly complex and scientific techniques can be used in solving logic problems for grades 1, 2, 3, 4, 5, 6, 7, 8, 9.
We present you with a variety of logical tasks for grades 1, 2, 3, 4, 5, 6, 7, 8, 9. We have selected for you the most interesting logic puzzles with answers which will be interesting not only for children, but also for parents.
- choose for the child logic tasks according to his age and development
- do not rush to open the answer, let the child find it himself logical solution tasks. Let him come to the right decision himself and you will see what pleasure and a sense of delight he will have when his answer coincides with the given one.
- in the process solving logic problems Leading questions and indirect clues indicating the direction of reflection are acceptable.
With our selection logical tasks with answers you will really learn how to solve logical problems, expand your horizons and significantly develop logical thinking. Dare!!!
Solving logic problems - the first step in child development.
E.Davydova
Logic is the art of coming to an unpredictable conclusion.
Samuel Johnson
Without logic, it is almost impossible to bring into our world ingenious finds of intuition.
Kirill Fandeev
A person who thinks logically stands out nicely against the background of the real world.
American saying
Logic is the morality of thought and speech.
Jan Lukasiewicz
1. Explanatory note
1.1 Relevance
1.2 Purpose of the program
1.3 Program objectives
1.4 Terms of the program implementation, age of children, forms of conducting classes
1.5 Stages of program implementation
1.6 Program content
1.7 Expected results
2. Methodological support
2.1 Perspective-thematic plan of the circle "Entertaining logic"
3. Diagnostic program logical thinking older preschool children.
5. Information resources
1. Explanatory note.
Why logic for a little preschooler?
According to L.A. Wenger, “for five-year-old children, the external properties of things alone are clearly not enough. They are quite ready to gradually get acquainted not only with external, but also with internal, hidden properties and relationships that underlie scientific knowledge about the world ... All this will benefit the mental development of the child only if the training is aimed at developing mental abilities, those abilities in the field of perception, figurative thinking, imagination, which are based on the assimilation of samples of the external properties of things and their varieties ... "
The skills acquired by the child in the preschool period will serve as the foundation for gaining knowledge and developing abilities at an older age - at school. And the most important among these skills is the skill of logical thinking, the ability to "act in the mind." It will be more difficult for a child who has not mastered the methods of logical thinking to solve problems; performing exercises will require a lot of time and effort. As a result, the child's health may suffer, interest in learning may weaken or even fade away.
Having mastered logical operations, the child will be more attentive, learn to think clearly and clearly, and be able to concentrate on the essence of the problem at the right time. Learning will become easier, which means that both the learning process and school life itself will bring joy and satisfaction.
This program shows how through special games and exercises it is possible to form the ability of children to independently establish logical relationships in the surrounding reality.
Working with preschoolers on the development of cognitive processes, you come to the conclusion that one of the necessary conditions for their successful development and learning is consistency, i.e. system special games and exercises with consistently developing and becoming more complex content, with didactic tasks, game actions and rules. Separately taken games and exercises can be very interesting, but using them outside the system, one cannot achieve the desired learning and developmental result.
1.1 Relevance
For the successful development of the school curriculum, the child needs not only to know a lot, but also to think consistently and conclusively, to guess, to show mental tension, to think logically.
Teaching the development of logical thinking is of no small importance for the future student and is very relevant today.
Mastering any method of memorization, the child learns to single out a goal and carry out certain work with the material to achieve it. He begins to understand the need to repeat, compare, generalize, group material for the purpose of memorization.
Teaching children about classification contributes to the successful mastery of a more complex way of remembering - the semantic grouping that children encounter at school.
Using the opportunities for the development of logical thinking and memory of preschoolers, it is possible to more successfully prepare children for solving the problems that school education sets before us.
The development of logical thinking includes the use of didactic games, ingenuity, puzzles, solving various logic games and labyrinths and is of great interest to children. In this activity, important personality traits are formed in children: independence, resourcefulness, ingenuity, perseverance is developed, and constructive skills are developed. Children learn to plan their actions, think about them, guess in search of a result, while showing creativity.
When dealing with children, you can notice that many children cannot cope with simple at first glance logical tasks. For example, most children of older preschool age cannot correctly answer the question of what is more: fruits or apples, even if they have a picture in their hands on which fruits are drawn - many apples and several pears. Children will answer that there are more pears. In such cases, he bases his answers on what he sees with his own eyes. They are “let down” by imaginative thinking, and by the age of 5 children do not yet have logical reasoning. In senior preschool age they begin to show elements of logical thinking, characteristic of schoolchildren and adults, which must be developed in identifying the most optimal methods for the development of logical thinking.
Logic games help educate children cognitive interest, contribute to the research and creative search, the desire and ability to learn. Didactic games as one of the most natural activities of children and contributes to the formation and development of intellectual and creative manifestations, self-expression and independence. The development of logical thinking in children through didactic games is important for the success of subsequent schooling, for the correct formation of the personality of the student and in further education will help to successfully master the basics of mathematics and computer science.
1.2 Purpose of the program: creating conditions for maximum development logical thinking of preschoolers in preparation for successful schooling.
1.3 Program objectives:
- teach children basic logical operations: analysis, synthesis, comparison, negation, classification, systematization, limitation, generalization, inference
- teach children to navigate in space
- develop in children higher mental functions, the ability to reason, prove
- to cultivate the desire to overcome difficulties, self-confidence, the desire to help a peer
1.4 Terms of the program implementation, age of children, forms of conducting classes
Program implementation terms – 1-2 years
The program is designed for children 5-7 years old.
The program provides for conducting circle classes in various forms:
- Individual independent work children.
- Work in pairs.
- Group forms of work.
- Differentiated.
- Frontal check and control.
- Self-assessment of the work done.
- Didactic game.
- Competition.
- Contests.
1.5 Stages of program implementation
The technology of activity is built in stages:
- Diagnosis of the initial level of development of cognitive processes and control over their development.
- Planning the means by which one or another quality can be developed (attention, memory, imagination, thinking), taking into account the individuality of each child and the available knowledge
- Building an interdisciplinary (integral) basis for training in a developing course.
- Gradual complication of the material, a gradual increase in the amount of work, increasing the level of independence of children.
- Acquaintance with the elements of theory, teaching methods of reasoning, self-argumentation of choice.
- Integration of knowledge and methods cognitive activity, mastering its generalized techniques.
- Evaluation of the results of the developmental course according to the developed criteria, which should include the child (self-esteem, self-control, mutual control).
1.
6
Program content
Short description sections and topics of classes (sections correspond to a certain logical operation that children will learn in class):
1. Analysis - synthesis.
The goal is to teach children to divide the whole into parts, to establish a connection between them; learn to mentally combine parts of an object into a single whole.
Games and exercises: finding a logical pair (cat - kitten, dog - ? (puppy)). Complementing the picture (pick up a patch, draw a pocket to the dress). Search for opposites (light - heavy, cold - hot). Work with puzzles of varying complexity. Laying out pictures from counting sticks and geometric shapes.
2. Comparison.
The goal is to teach to mentally establish the similarities and differences of objects according to essential features; develop attention, perception of children. Improve orientation in space.
Games and exercises: consolidation of concepts: big - small, long - short, low - high, narrow - wide, higher - lower, further - closer, etc. Operating with the concepts "same", "most". Search for similarities and differences in 2 similar pictures.
3. Restriction.
The goal is to teach to single out one or more objects from a group according to certain characteristics. Develop children's observation skills.
Games and exercises: “circle only red flags with one line”, “find all non-circular objects”, etc. Exclusion of the fourth superfluous.
4. Generalization.
The goal is to teach to mentally combine objects into a group according to their properties. Contribute to the enrichment of vocabulary, expand everyday knowledge of children.
Games and exercises for operating with generalizing concepts: furniture, dishes, transport, vegetables, fruits, etc.
5. Systematization.
The goal is to teach to identify patterns; expand the vocabulary of children; learn to tell from a picture, retell.
Games and exercises: magic squares (pick up the missing part, picture). Drawing up a story based on a series of pictures, arranging the pictures in a logical sequence.
6. Classification.
The goal is to teach to distribute objects into groups according to their essential characteristics. Consolidation of generalizing concepts, free operation with them.
7. Inference.
The goal is to teach with the help of judgments to make a conclusion. Contribute to the expansion of household knowledge of children. Develop imagination.
Games and exercises: search for positive and negative in phenomena (for example, when it rains, it nourishes the plants - this is good, but the bad thing is that in the rain a person can get wet, catch a cold and get sick). Evaluation of the correctness of certain judgments (“the wind blows because the trees sway.” Right?). Solving logical problems.
1.7 Expected results
Planned results:
Children should know:
- principles of constructing patterns, properties of numbers, objects, phenomena, words;
- the principles of the structure of puzzles, crosswords, chainwords, labyrinths;
- antonyms and synonyms;
- names of geometric shapes and their properties;
- the principle of programming and drawing up an algorithm of actions.
Children should be able to:
- determine patterns and perform a task according to this pattern, classify and group objects, compare, find common and particular properties, generalize and abstract, analyze and evaluate their activities;
- through reasoning, solve logical, non-standard problems, perform creative search, verbal-didactic, numerical tasks, find the answer to mathematical riddles;
- respond quickly and correctly during the warm-up to the questions posed;
- perform tasks to train attention, perception, memory
- perform graphic dictations, be able to navigate in a schematic representation of graphic tasks;
- be able to set a goal, plan the stages of work, achieve results with your own efforts.
Way to check the results of work : generalizing classes after each section and 2 diagnostics (initial (September) and final (May)) of the level of mastering the operations of logical thinking.
Some of the readers, who are familiar with the nature of the former teaching of logic at school, may question the expediency of entertaining logic. However, the reader will probably agree that everyone should be able to think consistently, judge conclusively, and refute wrong conclusions: a physicist and a poet, a tractor driver and a chemist. Especially in our time, which constantly brings a lot of unusual and amazing discoveries and inventions in various fields: in geography, politics, in public life.
Automatic fuel sorter.
The warehouse, which has two rooms for storing large quantities of two types of fuel - coal and coke, each separately, receives trucks, each time with one of these types of fuel. The mechanism that opens the shafts is required to open the shaft to the coal room if a truck with this fuel arrives, and the shaft to the coke room if a truck of coke arrives. To ensure good sorting of fuel, an additional requirement was made: only one truck is allowed into the warehouse at any time and only one shaft is opened.
The question is whether this mechanism also has the following property: if a coal truck does not enter the warehouse, the coal mine will not open, and if a coke truck does not enter, the coke mine will not open.
Note. This problem can be solved without the means of propositional logic, by simple reasoning. A more difficult, and perhaps speculatively impracticable, solution will be in the case when the number of fuels exceeds two and when several trucks can enter the warehouse at the same time. Let the reader try to solve this problem also for three types of fuel.
Free download e-book in a convenient format, watch and read:
Download the book Entertaining Logic, Kolman E., Zikh O., 1966 - fileskachat.com, fast and free download.
- Mathematics and Design, Grade 1, Textbook for educational organizations, Volkova S.I., 2016
- Mathematics, Oral exercises, Grade 1, Textbook for educational institutions, Volkova S.I., 2016
- A course of lectures on the theory and technology of teaching mathematics in elementary grades, Part 2, Ruchkina V.P., 2019
The following tutorials and books:
- Mathematics, Algebra and the beginnings of mathematical analysis, Grade 11, Vilenkin N.Ya., Ivashev-Musatov O.S., Shvartsburd S.I., 2014
