Easy logic tasks. Math Puzzles

14.02.2019

Characteristics of the hero from the comedy "Minor" D

Pyotr Andreevich Grinev, the main character of the story

The image and characteristics of svidrigailov in the novel crime and punishment dostoevsky composition

The image and characteristics of Mitrofanushka in Fonvizin's "The Minor": the description of Mitrofan Prostakov

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Fly

Two trains, located at a distance of 200 km, move towards each other at a speed of 50 km / h each. The fly takes off from one of the trains and flies towards the other at a speed of 75 km / h. Having reached another train, the fly turns around and flies back to the first. So it flies back and forth until the two trains collide and the insect dies.
How far did the fly fly?
There are two ways to solve this problem, one is simple, the other is hard.

The hard way to solve the problem: calculate each segment of the path. It is much easier to solve the problem if you simply calculate the distance that a fly can fly in 2 hours (exactly in two hours the trains will collide) at a constant speed of 75 km / h.
She will fly 150 km.

Trains

A freight train leaves Boston for New York, moving at a speed of 60 km / h. In 30 minutes, a passenger train, moving at a speed of 80 km / h, leaves New York for Boston to meet him.
Which train will be closer to New York at the time of the meeting? (Ask the students for help - they will probably cope with the task faster.)

When the trains meet, they will both be approximately the same distance from New York.
A train leaving New York will be closer to New York by about the length of one train, because the trains are moving in the opposite direction. Well, this is if by the word "meet" you mean exactly "meet" and not "intersect at the very moment when one of the trains equals all its cars with the cars of the second train."

average speed

Half of the way to the city, located at a distance of 60 km, I drove at an average speed of 30 km / h.
How fast should I drive the rest of the way so that the overall average speed of the entire trip is 60 km / h?

Wire over the equator

The circumference of the Earth is approximately 40,000 km. If you stretch a wire over the equator around the Earth so that the length of the wire is only 10 meters (0.01 km) longer than the circumference of the earth, can a flea be able to crawl under this wire? Mouse? Human?

Let's compare the original perimeter with the length of the wire. The original perimeter is 2πr (two radii times pi), while the length of the wire is 2π (new r) (two new radii times pi). The difference between them is approximately 1.6m.
A short man can easily pass under such a wire in full height, but taller people will have to bend in single file.

Diophantus

Little is known about the life of one Greek mathematician from Alexandria, who is called the ancestor of algebra. It is assumed that he lived in the 3rd century AD. According to the stories, the following epitaph was carved on his tombstone:
“Diophantus' childhood took 1/6 of his life; 1/12 of his life Diophantus grew a beard; another 1/7 of Diophantus' life passed before he got married. 5 years after the wedding, Diophantus had a son who lived only half the years that his father lived. And 4 years after the death of his son, Diophantus died. "
How many years did Diophantus live?

Ahmes Papyrus

In 1858, Scottish collector Henry Rind acquired an ancient Egyptian papyrus signed with the name "Ahmes". This scroll of papyrus, 33 cm wide and 5.25 meters long, is a copy of an even more ancient mathematical manual dating back to the time of Pharaoh Amenemhat III. Here is one problem from this oldest collection of mathematics:
One hundred measures of grain must be divided among five workers so that the second receives as much more than the first, as much as the third more than the second, and as much as the fourth more than the third, and as much as the fifth more than the fourth. How many measures of grain should each receive if the first and second workers together receive seven times less grain than the other three workers?

To solve the problem, we will compose two equalities. 5w + 10d = 100; 7 * (2w + d) = 3w + 9d, where w is the amount of grain for the first worker, d is the difference in the amount of grain between the two (next in order) workers. Answer: first worker 10/6 measures of grain, second worker 65/6 measures of grain, third worker 120/6 (20) measures of grain, fourth worker 175/6 measures of grain, fifth worker 230/6 measures of grain.

How long until midnight?

In two hours until midnight, there will be half the amount left of what it would have been in an hour.
What time is it now?

Clock hands

At noon, the hour, minute and second hands of the watch coincide at one point on the dial. A little over an hour and five minutes later, the hour and minute hands will coincide again. Find, with millisecond precision, the time when they match.
What angle will the second hand make with them at this time?

This problem can be solved in several ways, but most of all I like the following, the simplest one. This situation (when the hour and minute hands coincide) is repeated 11 times every 12 hours. It is easy to guess that the 1/11 mark of the dial circumference is at the time 1: 05: 27.273, that is, the second hand will stand at 27.273 seconds.
In this case, the angle between the hour and second hands will be 131 degrees.

Pool

The pool has four pipes, through which the rate of filling the pool can be controlled through the taps. By opening the first tap, you can fill the pool in 2 days, the second in 3 days, the third in 4 days and the fourth in 6 hours.
How long does it take to fill the pool by turning on all four taps at the same time?

Since there are 24 hours in a day, the first tap will fill 1/48 of the pool in an hour, the second tap will fill 1/72, the third tap will fill 1/96, and the fourth will fill the pool 1/6. From here we get: (6 + 4 + 3 + 48) / 288 = 61/288. The pool will fill up in 288/61 hours, i.e. after 4 hours, 43 minutes and about 17 seconds.

Crossing the desert

A military vehicle with an important message must cross the desert. However, a full gas tank is only enough for half the journey. The military base has several such vehicles at its disposal, and gasoline can be pumped from one tank to another. They cannot use any canisters and cables.
How to deliver a message without leaving a single vehicle in the desert? (Try to simulate the situation with toy cars for clarity.)

Magic belt

The magic belt that fulfills the wishes of the owner is reduced by half in length and 3 times in width after each fulfilled desire. After the fulfillment of three wishes, the area of the front side became 4 cm2.
What was the original length of the belt if its original width was 9 cm?

Baldwill

All residents of Boldville different amount hair on the head. There is not a single inhabitant who has exactly 518 hairs on his head. The population of the city exceeds the number of hairs on the head of any of the inhabitants of Boldville.
What is the maximum possible population for the city of Boldville?

Unfaithful wives

An anthropologist studying a tribe in a remote corner of the Amazon jungle discovered strange custom... When the husband found out that his wife was cheating, he had to publicly execute her at midnight that same day. All the inhabitants of the tribe, except for her husband, always knew about any woman who cheats on her husband. But no one ever told her husband about his wife's betrayal, because it was contrary to the code of honor. The same code of honor did not allow wives to notify a wife whose husband was unfaithful to her. Otherwise, she would have shot her husband that very evening. On the day of his departure, the anthropologist called all the representatives of the tribe and announced: "I know that there are unfaithful wives in this tribe." And on the ninth day, all unfaithful husbands were executed.
How many unfaithful husbands were there?

If we take the number of unfaithful husbands for the number "n", then the number of unfaithful husbands known to every wife of an unfaithful husband is "n-1" (because everyone knows everything for sure - only one has to guess about the loyalty of one's own husband). Now let's build the next logical chain.
Suppose the number of unfaithful husbands is one. Then all but one of the wives know that there is one unfaithful husband among the inhabitants, while the wife of this unfaithful husband is sure that all husbands are loyal to their wives. As soon as she hears that there is at least one unfaithful husband among the residents, she will immediately understand that there can only be her husband, so that same evening she will shoot him without hesitation.
Now imagine that there are two unfaithful husbands among the residents. Each wife of such unfaithful husbands is sure that there is only one unfaithful husband among the inhabitants, so she is waiting for one of the wives to shoot her husband. But that evening, no one shot anyone, and this can only mean one thing: her OWN husband she is ALSO unfaithful and is the SECOND unfaithful husband in the tribe. The first wife of the first unfaithful husband comes to exactly the same conclusions (she also expected that one of the wives would shoot her husband). Thus, both offended wives on the very first evening understand that their husbands are cheating on them, and the next evening (the second day) they shoot both husbands.
Following this logic, it is easy to guess that the number of unfaithful husbands "n" will be shot on the "n" evening.

1 = 2

Find the error in the math:

X = 2
x (x-1) = 2 (x-1)
x2 -x = 2x-2
x2 -2x = x-2
x (x-2) = x-2
x = 1

Connect the 9 dots with four straight lines without lifting your arms or outlining the lines.

Motto

In my youth, I found that thumb legs sooner or later make a hole in the sock. So I stopped wearing socks.
Albert Einstein

In connection with the beginning school year we decided to test how smart and resourceful our subscribers are. Can you solve all the problems presented by us?

"COUNT-KA"
Let's see if you can count?

Solve this example without the help of a calculator: To 1000 you need to add 40, then another 1000. Then add 30. Got it? Now 1000 again. Add 20. 1000 again. And finally 10.
How much did it turn out?
Now check everything again with your phone. Did it match?

"WHAT COOLES IN THE MORNING?"
And now a logic puzzle.
The woman dropped her ring into a glass full of coffee. How could he stay dry?
What do you think is the secret?

"MATCHES FOR CHILDREN ARE NOT A TOY"
How many matches are in the picture?

"GREEN MAN"
This is the riddle that you will solve with the help of childish naivety. We are sure you can guess it the first time! Answer the question: what should you do when you see a green man?

"CIRCLES"
The teacher draws several circles on a piece of paper and asks one student: "How many circles are there?" "Seven" - the student answers. "Right. So how many circles are there? " the teacher of another student asks again. "Five" - he answers. “That's right,” the teacher says again. So how many circles did he draw on a piece of paper?

Do you think everything is so easy? Now try to solve the most difficult problems in the world!

"SUPER SUDOKU"
The first thing we invite you to puzzle over is the hardest Sudoku in the world.

Sudoku is a Japanese number puzzle. Its principle is not complicated at all. But the one that we offered you, definitely not everyone can solve!

"GODS OF LOGICAL PROBLEMS"
There are three gods, A, B, and C, one of which is the god of truth, the other is the god of lies and the third is the god of chance, and it is not clear which of them is which. The god of truth always speaks the truth, the god of lies deceives, and the god of chance can say both in no particular order. It is necessary to determine who each of the gods is by asking three questions that can be answered "yes" or "no", with each question being asked only to one god. The gods understand the questions, but they answer in their own language, which contains the words "da" and "ja", but it is not known which word means "yes" and which "no".

This logical problem, authored by the American philosopher and logician George Boulos, was first published in the Italian newspaper "la Repubblica" in 1992. There are also comments from the creators in the puzzle:
- You can ask one god more than one question (therefore, other gods may not be asked a single question at all).
- What the next question will be and to whom it will be asked may depend on the answer to the previous question.
- The God of chance answers in a random way, depending on the toss of a coin hidden in his head: if the obverse falls out, then he answers truthfully, if the reverse, then he is lying.
- The God of chance answers "da" or "ja" to any question that can be answered "yes" or "no."

Answers to all tasks can be viewed by

Intelligence is the most important thing that distinguishes people from other representatives of the animal world. Man used his mind to reach unprecedented heights in science and technology, but sometimes mind games were not only purely practical and utilitarian in nature: this is how many different puzzles were born, for the solution of which you have to thoroughly "brainwash". You will find ten of them in this collection.

1. World's Hardest Sudoku

One of the most popular crossword puzzles in the world is Sudoku, a Japanese number puzzle. Its principle is simple, so many amateurs try to create their own versions. In 2012, Finnish mathematician Arto Inkala announced that he had developed "the world's hardest Sudoku."

According to the British newspaper "The Telegraph", if the simplest of the common variants of Sudoku on the scale of difficulty is designated as "1", and the most difficult of the popular ones are rated at "5", then the variant proposed by the mathematician is "11".

2. The hardest logic puzzle

There are three gods, A, B, and C, one of which is the god of truth, the other is the god of lies and the third is the god of chance, and it is not clear which of them is which. The god of truth always speaks the truth, the god of lies deceives, and the god of chance can say both in no particular order. It is necessary to determine who each of the gods is by asking three questions that can be answered "yes" or "no", with each question being asked only to one god. The gods understand the questions, but they answer in their own language, which contains the words "da" and "ja", but it is not known which word means "yes" and which "no".

This logical problem, authored by the American philosopher and logician George Boulos, was first published in the Italian newspaper "la Repubblica" in 1992. In the comments to the riddle, Bulos does important note: each god can be asked more than one question, but more than three cannot be asked.

3. The most difficult sum-do-ku in the world

One of the popular varieties of Sudoku is sum-do-ku, it is also called "killer sudoku". The only difference is that additional numbers are set in the sum-do-ku - the sums of values in groups of cells, while the numbers contained in the group should not be repeated. In the popular puzzle service Calcudoku.org, you can track the difficulty rating of the published problems, one of which is the sum-do-ku, which is shown here.

4. The most difficult "Recognition problem" of Bongard

This type of puzzle was invented by the outstanding Russian cyberneticist, the founder of the theory of pattern recognition, Mikhail Moiseevich Bongard: in 1967 he first published one of them in his book The Problem of Recognition. The "Bongard problems" gained wide popularity when the famous American physicist and computer scientist Douglas Hofstadter mentioned them in his work "Gödel, Escher, Bach: This Endless Garland."

The two most complex examples such problems are taken from Foundalis.com, to solve them you have to find a rule that matches six images on the left page, but six images on the right do not match.

5. The most difficult tracing paper puzzle

This type of Sudoku is similar to sum-do-ku, but, firstly, any arithmetic operations are used to calculate the value of the cells, and not only addition, and secondly, the field can be a square of any size (the number of cells is not limited), and in - thirdly, unlike Sudoku, hints from 1 to 9 in each 3 × 3 square do not have to be present here. Such problems were developed by the Japanese mathematics teacher Tetsuya Miyamoto.

You can try to figure out the most difficult tracing document, which was published on Calcudoku.org on April 2, 2013. Only 9.6% of regular visitors to the resource managed to solve it.

6. The hardest challenge from IBM

It is necessary to develop an information storage system that would encode 24 bits of information on eight disks of four bits each, provided that:

Eight 4-bit disks are united by one 32-bit system, in which any function from 24 to 32 bits can be calculated by no more than five mathematical operations from the set (+, -, *, /,%, &, |, ~).
After failure of any two disks out of eight, you can recover these 24 bits of information.

On the IBM website there is a regular column “Think about it!”, In which interesting logical problems have been published since 1998. The given task is one of the most difficult.

7. Hardest Kakuro puzzle

Kakuro puzzles combine elements of Sudoku, logic, crosswords and basic math. The goal is to fill the cells with numbers from one to nine, and the sum of the numbers in each horizontal and vertical block must converge with the specified number, and the numbers inside one block must not be repeated. For horizontal blocks, the required amount is written directly to the left, and for vertical blocks, at the top.