This problem is a variation of the classic Microsoft interview question where applicants were asked how many times a day the hour and minute hands meet each other. Since this question has now become widely known, interviewers have begun to use a variation of it.

Let us first consider the variant of the most expected solution, the mathematical one. First, imagine a situation where the hour and minute hands overlap. Everyone knows it happens at midnight, then around 1:05, 2:10, 3:15, and so on. In other words, they overlap each other every hour, except between 11:00 and 12:00. At 11:00, the faster minute hand is at 12 and the slower hour hand is at 11:00. They will not meet each other until 12:00 noon, and therefore there will be no overlap between them around 11 o'clock.

Thus, 11 overlaps occur in each 12-hour period. They are evenly distributed in time since both hands move at a constant speed. This means that the intervals between overlays are 12/11 hours. This is equivalent to 1 hour 5 minutes 27 and 3/11 seconds. Therefore, for each 12-hour cycle, overlays occur during the periods indicated in the picture.

Let's return to the second hand. Its overlap with the minute is possible when the number of minutes coincides with the number of seconds. The exact overlap occurs at 00:00:00. In general, the minute and second hands overlap for only a fraction of a second. For example, at 12:37:37 the second hand will point to 37, lagging behind the minute hand, which at this time will be between 37 and 38 and lagging behind the hour hand. In a moment, the minute and second will overlap, but the hour will not be near them. Those. All three arrows will not overlap.

The second hand will not overlap in any of the options in the picture, with the exception of midnight and noon. This means that the final answer to the question is: twice a day.

And here is the answer welcomed by Google. The seconds hand is designed to show short time intervals rather than communicate time to the nearest second. If it is not in sync with the other two hands, that is quite normal. By "synchronizing" here we mean that at midnight and noon all three hands point exactly to 12. Most analog watches of all kinds do not allow you to precisely set the second hand. One would have to remove the battery or, in the case of a mechanical watch, wait for the spring to finish winding and then, with the second hand stopped, synchronize the minute and hour hands with each other, then wait for the time shown on the watch to return. battery or wind the watch.

To do all this, you need to be a maniac or a fan of punctuality. But if you don't do all this, the second hand won't show "real" time. It will differ from the exact seconds by some amount in a random interval up to 60 seconds. Given the random discrepancies, there is no chance that all three arrows will ever meet. This never happens.

Time cannot be seen or felt. But if you know some tricks and practical techniques, you can easily teach your child to understand time and determine it by the clock. Theory and practical tasks, games and exercises to start with - read and try.

It happens that already at a decent age people admit that they only use electronic watches. And everyone has the same reason - either their parents did not explain to them in childhood how to use a clock with hands, or they explained it erroneously. To prevent this from happening, it is important not to leave the problem unattended. Where to start teaching a child to understand time by the clock?

What does a child need to know to tell time using a clock?

Before you start learning time, check your child's understanding of the basics. Can he count? Is he oriented in key concepts related to time? Often parents encounter difficulties in learning and stubbornly do not notice the root of the problem (the child confuses “left” and “right”, does not count well enough, etc.) Therefore, it will be useful to go over the basic skills and make sure that there are no gaps that may hinder the child move forward, no.

Count to 60

Least. Or better yet, up to 100. We strengthen our counting skills with exercises:

• - name the double numbers that we see (these could be price tags in a store, house numbers, etc.);
• - train counting backwards (from 100 to 1);
• - learning to name the “neighbors” of round numbers (50 is the neighbors of 49 and 51, 90 is the neighbors of 89 and 91, etc.).

Count with numbers that are multiples of 5

Surely you have already explained to your child that such numbers always end in 5 or 0. All that remains is to learn how to list and use them without hesitation.

• - we count numbers that are multiples of 5, in direct and reverse order;
• - we simulate tasks where you need to count in fives (Vlad decided to do push-ups five times every day. How many times will he do push-ups in a week, two weeks, a month? How will these numbers change if from the second month Vlad does not 5, but 10 push-ups a day?)

Try online classes at LogicLike

• Complete the first 3 chapters of the course and gain access to different categories. Be sure to solve “Smart Counting” and “Logic Problems”.
• Try tasks of different difficulty levels: “Beginner”, “Experienced”, “Expert”.

Distinguish between "left" and "right"

For studying in general and so as not to confuse the concepts of “clockwise” and “counterclockwise” as well.

Have a general understanding of time

We explain to the child the concepts of “yesterday”, “today”, “tomorrow”; "past present Future"; “morning”, “day”, “evening”, “night”, “day”. Often children themselves associate time with a specific event: “in the morning I did exercises”, “at lunch I ate soup”, “before going to bed I brushed my teeth”, etc. Therefore, when explaining the above concepts, it is best for a parent to tie specific events to them.

Correct your child carefully if he makes mistakes somewhere. It is important that he does not develop a false understanding of time.

Have you successfully completed the preparatory stage? Now we can teach the child to understand time using a clock with arrows.

We teach a child to understand time using a clock with arrows

Oh, these adults! And why do they only allow you to watch cartoons for about 15 or 20 minutes? For children, time is an incomprehensible number. To figure out where it comes from, you will need a watch with hands. If there are no such things at home, but only electronic ones, it will be completely difficult for the child to understand what time is. Therefore, the first step for a parent is to acquire a wall or special children's clock, on which the numbers and arrows will be clearly visible.

Introducing the child to the structure of the clock

First, explain to your child the concepts of “dial”, “day”, “hours”, “minutes”, “seconds”; “exactly one hour”, “half an hour”, “quarter of an hour”, tell us about the hour, minute, second hands. Please note that all arrows have different lengths. Let the child observe which of the arrows is the fastest and which one practically stands still. And how long does it take for each to complete the whole circle?

Be sure to connect all the basic concepts into one logical chain: there are 24 hours in a day, 1 hour is 60 minutes, and 1 minute is 60 seconds. Do not ignore the concepts of “clockwise” and “counterclockwise”. Let your child understand that time always moves forward.

We teach a child to “read” the hour and minute hands at the same time

First of all, teach your child to count minutes in intervals divisible by 5. Minutes are not indicated on a regular clock, so this skill needs to be practiced. You can come up with a legend that each number on the dial has its own “shadow”. 1 is 5 minutes, 2 is 10 minutes, 3 is 15 minutes, etc. The “shadow” can only be seen when the minute hand is pointing to the number. When your child can easily navigate five-minute intervals, tell him about smaller intervals.

The hour hand also has two meanings. In the first half of the day we see the numbers as they appear on the dial, but after a hearty afternoon snack at 12:00 they begin to “get fat”: 1 turns into 12, 2 into 14, etc. A funny analogy will help your child grasp the meaning faster.

The ability to determine time using a clock with hands must be reinforced with specific examples. Draw your child's attention to the clock more often. Correct him if he says the time incorrectly.

The best gift for a child who is learning to tell time by clock is a wristwatch. With them, he will become more willing to answer the question “What time is it?” and will definitely ask you about this in order to check with his “walkers”.

Ideally, a child should have a “draft” watch that he can “exploit” as he pleases: set the time on it, add “shadows” to each of the numbers, sign the names of the hands. For training, you can use an old non-working clock (wall or table clock). You need to remove the glass in them so that the hands can be rotated. If you haven't found one like this at home, we suggest you make your own.

Making homemade watches

A homemade clock will help make time more tangible. If you have the necessary materials, their creation will take no more than 15 minutes.

How to make a watch yourself

The basis for the dial can be a disposable plate or a circle made of cardboard. We draw the circle in half, then in half again and apply the first numbers. Next, carefully divide each quarter into three parts and add the remaining numbers. The dial is ready, which means it's time to attach the hands. We cut them out of cardboard of different colors and attach them to the circle using a button. We place the resulting model of the clock next to the real clock.

When creating your own watch, it will be useful to go over the concepts you have already learned. We drew the circle into four parts - we remembered about the “quarter hour”, attached an hour hand - we remembered its function, etc.

Homemade watches may look unusual. For example, like this:

Games and tasks with a clock

Games and tasks will help you strengthen your ability to tell time using a clock.

"What time is it now"

Show your child how the arrows move. Change their position and call the time. Then have the child do the same exercise. Change the time clockwise and counterclockwise.

Let's complicate the game. We show the time on the clock and associate it with events (“it’s 7:00”, at this time we wake up”, “it’s 18:00”, at this time we have dinner”, etc.). Now we invite the child to pretend to live the whole day.

“Drawing pizza”

The good thing about a homemade dial is that you can make your own notes on it. Ask your child to draw lines from the center of the dial to the numbers and shade each sector with a different color. You will get a “colored pie” or “colored pizza” (this will make it easier to understand 5-minute intervals). Label the second values ​​of each of the numbers (2 - 10, 3 - 15) and minutes (from 1 to 60).

"Daily regime"

Take a piece of paper, write down the daily routine, and together with your child, illustrate it with images of a clock that indicates a period of time (8:00 - time for school, 15:00 - time to do homework, etc.). Hang it above your child's bed or desk. This way the child will learn not only to do everything on time, but also to navigate in time.

Pay your child's attention to how much time he spends on this or that action. This way you can teach him to be punctual from an early age.

“Two options for telling the time”

Tell your child that time can be called in different ways (for example, 1 hour 18 minutes is eighteen minutes past two, etc.). Write down the second, more complex option on a piece of paper, and indicate the hint numbers to make it easier for the child to cope (example: “five minutes to eight”, the hint numbers are 9, 5, 5, 1). Gradually remove the prompts.

"Cubes"

To play you will need 4 dice and our homemade clock. We throw the dice in pairs. The first pair of cubes will determine the hours, the second pair - the minutes. The time that has fallen must be set on a toy watch.

There are also interactive games with clocks on the LogicLike platform. We have more than 3,500 exciting tasks for children of preschool and primary school age that help develop logic, thinking, and memory.

Introducing the child to electronic, sun, and hourglasses

When your child has learned to tell time using a clock with hands, it’s time to introduce him to other clocks. You have room to move forward! Getting to know electronic, sun, and hourglasses will help your child deepen his understanding of time. Moreover, it will be no less interesting to deal with them.

Digital Watch are more conventional than a clock with hands; they cannot be used to visually track the passage of time. But if the child understands how hours and minutes are counted, then there should be no problems. Get an electronic watch and instruct your child to keep track of the time on it as well. The same TV program always shows the time in electronic format, so the first thing you can do is remember what time cartoons and children's programs start.

Sundial They look more like a clock with hands, so they will be easier to understand. All that remains is to wait for a sunny day, draw a circle in the sand, place a wooden stick in the center, check the time with a mechanical watch and finish drawing the dial. And you can watch in fascination as the shadow of the wand gradually creeps clockwise.

Hourglass It will also be most convenient to compare with arrows. They measure very short periods of time. Invite your child to simultaneously watch the second hand on a mechanical watch and the passage of time in an hourglass. By the way, with them it is much more fun to complete tasks for a while: make the bed, put all the toys in a box, etc., until the sand stops falling.

Teaching a child to understand time is not as difficult as it seems. By solving this problem in childhood, you will help your child become a punctual person, for whom the sense of time will not be a weak point.

At 5-7 years old, most children have a peak of cognitive activity. And this is in many ways the best time to develop together in an interesting and varied way. Until the child was drawn into school everyday life.

To help parents - entertaining logic tasks, exercises for the development of thinking, attention, memory and speech.

Try to decide for yourself!
If something doesn’t work out, don’t despair, the answer and solution are located below.

1. How many times a day do clock readings have the property that by swapping the minute and hour hands we will arrive at a meaningful clock reading?

2. How many times a day do the hour and minute hands form a right angle?

3. How many minutes later will the (normal) clock hands overlap again after alignment?

4. How many times is the number showing how many times the speed of the second hand is greater than the speed of the minute hand, greater than the number showing how many times the speed of the minute hand is greater than the speed of the hour hand?

5. How many times will the hour hands be on top of each other in 12 hours?

6. Some work was started at the fifth hour and completed at the eighth hour, and the clock readings at the beginning and end of the work are converted into each other if the hour and minute hands are swapped. Determine the duration of the work and show that at the beginning and at the end of the work the arrows were equally deviated from the vertical direction.

7. How many times a day does the minute hand overtake the hour hand? What about a second?

8. The clock struck midnight. How many times and at what points in time before the next midnight will the hour and minute hands be aligned?

9. Between what numbers is the second hand located when the hour hand first aligns with the minute hand in the afternoon?

10. Why do the clock hands move from left to right (clockwise), and not vice versa?

11. On a watch with three hands - hour, minute and second - at 12 o'clock all three hands coincide. Are there other times when all three arrows coincide?

12. Problem proposed Lewis Carroll : Which clocks tell time more accurately: those that are behind by a minute per day, or those that do not go at all?

13. How many degrees does the minute hand rotate per minute? Hour hand?

14. Determine the angle between the hour and minute hands of a clock indicating 1 hour 10 minutes, provided that both hands move at constant speeds.

15.

16. But you probably noticed that this is not the only moment when the hands of the clocks meet: they overtake each other several times during the day. Can you point out all the times this happens?

17. When will the next meeting take place?

18. At 6 o'clock, on the contrary, both hands are directed in opposite directions. But does this only happen at 6 o’clock or are there other moments when the hands are positioned like this?

19. I looked at the clock and noticed that both hands were the same distance from the number 6, on both sides of it. What time was this?

20. At what time is the minute hand ahead of the hour hand by exactly the same amount as the hour hand is ahead of the number 12 on the dial? Or maybe there are several such moments a day or none at all?

21. What angle does the clock hand make at 12:20 o'clock?

22. Find the angle between the hour and minute hands a) at 9 o'clock 15 minutes; b) at 14:12?

23. When the angle between the hour and minute hands of a clock is greater than a) at 13:45 or 22:15; b) at 13:43 or 22:17; c) t minutes after noon or t minutes before midnight?

24. The clock hands have just aligned. After how many minutes will they “look” in opposite directions?

25. How can we explain that in a working watch the minute hand has passed 6 minutes in one second?

26. Using a precision chronometer, it was established that the hour and minute hands of a clock running evenly (but at the wrong speed!) coincide every 66 minutes. How many minutes per hour is this clock fast or slow?

27. In Italy they produce watches in which the hour hand makes one revolution per day, and the minute hand - 24 revolutions, and, as usual, the minute hand is longer than the hour hand (in a regular watch, the hour hand makes two revolutions per day, and the minute hand - 24). Let's consider all the positions of the two hands and the zero division, which are found on both Italian watches and ordinary ones. How many such provisions are there? (The zero mark marks 24 hours in Italian watches and 12 hours in regular watches).

28. Vasya measured with a protractor and wrote down in a notebook the angles between the hour and minute hands, first at 8:20, and then at 9:25. After that, Petya took his protractor. Help Vasya find the angles between the arrows at 10:30 and 11:35.

29. How many times do the minute and hour hands of a clock coincide from 12:00 to 23:59?

30. It's noon. When will the hour and minute hands coincide next time?

31. Indicate at least one point in time other than 6:00 and 18:00 when the hour and minute hands of a correctly running clock point in opposite directions.

32. When Petya began to solve this problem, he noticed that the hour and minute hands of his watch formed a right angle. While he was solving it, the angle was always obtuse, and the moment Petya finished solving it, the angle became right again. How long did Petya spend solving this problem?

33. Petya woke up at eight o'clock in the morning and noticed that the hour hand of his alarm clock bisected the angle between the minute hand and the bell hand pointing to the number 8. After what time should the alarm clock ring?

34. Kolya went for mushrooms between eight and nine o'clock in the morning at the moment when the hour and minute hands of his watch were aligned. He returned home between two and three o'clock in the afternoon, while the hands of his watch were directed in opposite directions. How long did Kolya’s walk last?

35. The student started solving the problem between 9 and 10 o'clock and finished between 12 and 13 o'clock. How long did it take him to solve the problem if during this time the hour and minute hands of the clock swapped places?

36. How many times during the day do the hour and minute hands of a properly running clock form an angle of 30 degrees?

37. There is a clock in front of you. How many hand positions are there that cannot tell the time unless you know which hand is the hour hand and which hand is the minute hand? (It is believed that the position of each of the arrows can be determined accurately, but it is impossible to monitor how the arrows move.)

38. In the world of the Antipodes, the minute hand of a clock moves at normal speed, but in the opposite direction. How many times per day the hands of the antipodean clocks a) coincide; b) opposite?

39. How many times a day can antipodean clocks be indistinguishable from normal ones (if you don’t know what time it really is)?

40. At noon, a fly sat on the second hand of the clock and drove off, adhering to the following rules: if it overtakes some hand or is overtaken by some hand (in addition to the second hand, the clock has hour and minute hands), then the fly crawls onto this hand. How many circles will a fly travel in an hour?

Pattern of time

Find out the pattern in the time change on the clock and determine what the clock at number five should show.

OGE tasks

Unified State Exam assignments

1. The clock with hands shows 8 hours 00 minutes. In how many minutes will the minute hand line up with the hour hand for the fourth time?

This task is no more difficult than the task of moving in a circle. Our hour and minute hands move in a circle. The minute hand travels a full circle in an hour, that is, 360°. Means, its speed is 360° per hour. The hour hand moves through an angle of 30° per hour (this is the angle between two adjacent numbers on the dial). Means, its speed is 30° per hour.

At 8:00 a.m. the distance between the hands is 240°:

Let the minute hand meet the hour hand for the first time after t hours. During this time, the minute hand will travel 360°t, and the hour hand 30°t, and the minute hand will travel 240° more than the hour hand. We get the equation:

360°t-30°t=240°

t=240°/330°=8/11

That is, after 8/11 hours the hands will meet for the first time.

Now, before the next meeting, the minute hand will travel 360° more than the hour hand. Let this happen in x hours.

We get the equation:

360°x-30°x=360°. Hence x=12/11. And so on two more times.

We get that the minute hand will align with the hour hand for the fourth time in 8/11+12/11+12/11+12/11= 4 hours= 240 minutes.

Answer: 240 min.

2. The clock with hands shows 1 hour 35 minutes. In how many minutes will the minute hand line up with the hour hand for the tenth time?

In this problem, we will express the speed of movement of the arrows in degrees/minute.

The speed of the minute hand is 360˚/60=6˚ per minute.

The speed of the hour hand is 30˚/60=0.5˚ per minute.

At 0 o'clock the position of the hour and minute hands coincided. 1 hour 35 minutes is 95 minutes. During this time, the minute hand moved 95x6=570˚=360˚+210˚, and the hour hand moved 95x0.5˚=47.5˚. And we have this picture:

The hands will meet for the first time after a time when the hour hand turns by , and the minute hand turns by 150˚+47.5˚ more. We get the equation for:

The next time the hands meet is when the minute hand passes one circle longer than the hour hand:

And so 9 times.

The minute hand will align with the hour hand for the tenth time in minutes

1. in 12 hours 132, in 24 hours 264 moments plus 22 overlays, total 286

2. The hour hand makes 2 revolutions per day, and the minute hand makes 24 revolutions. From here, the minute hand overtakes the hour hand 22 times and each time two right angles are formed with the hour hand, i.e. answer - 44 .

3. It is not difficult to figure out that this will happen after 1 hour 5 5/11 minutes, that is, at 2 hours 10 10/11 minutes. The next one is after another 1 hour 5 5/11 minutes, that is, at 3 hours 16 4/11 minutes, etc. All meetings, as you can easily see, will be 11; The 11th will occur 1 1/11 -12 hours after the first, that is, at 12 o'clock; in other words, it coincides with the first meeting, and further meetings will be repeated again at the same moments.

Here are all the moments of the meetings:

1st meeting - at 1 hour 5 5/11 minutes

2nd " - "2 hours 10 10/11 "

3rd " - "3 hours 16 4/11 "

4th " - "4 hours 21 9/11 "

5th " - "5 o'clock 27 3/11 "

6th " - "6 o'clock 32 8/11 "

2 hours 46, 153 min.

7. The hour hand makes 2 revolutions per day, and the minute hand makes 24 revolutions. From here the minute hand overtakes the hour hand 22 times.

9 . 4 and 5

10. This is exactly how the shadow moves in the very first hours - the sun. And then mechanical watches copied the direction of movement of the hands. By the way, in the Southern Hemisphere the opposite is true - the shadow in a sundial moves counterclockwise. In an hour, the minute hand makes a full revolution. This means that in a minute it rotates through 1/60th of an angle of 360°, that is, 6°. The hour hand travels 1/12 of the circle in an hour, that is, it moves 12 times slower than the minute hand. In a minute it rotates 0.5°.

14 . At 1:00 the minute hand was 30° behind the hour hand. In the 10 minutes that have passed since this moment, the hour hand will “travel” 5°, and the minute hand will “travel” 60°, so the angle between them is 60° – 30° – 5° = 25°.

15 . Let x be the period of time in minutes that must pass before the arrows are placed on the same straight line and directed in different directions. During this time, the minute hand will have time to travel x minute divisions of the dial, and the hour hand will have time to travel x/12 minute divisions. When the hands are placed on the same straight line and directed in different directions, they will be separated by 30 minute divisions of the dial. This means that at this time x – x/12 = 30, hence x = 32 (8/11). After 32 (8/11) minutes the arrows will “look” in opposite directions.

16 . Let's start watching the movement of the hands at 12 o'clock. At this moment, both arrows cover each other. Since the hour hand moves 12 times slower than the minute hand (it describes a full circle at 12 o’clock, and the minute hand at 1 hour), then, of course, the hands cannot meet during the next hour. But an hour passed; the hour hand is at number 1, having made 1/12 of a full revolution; The minute clock has made a full revolution and stands again at 12 - 1/12 of a circle behind the hour clock. Now the conditions of the competition are different than before: the hour hand moves slower than the minute hand, but it is ahead, and the minute hand must catch up with it. If the competition lasted a whole hour, then during this time the minute hand would go a full circle, and the hour hand would make 1/12 of a circle, that is, the minute hand would make 11/12 of a circle more. But in order to catch up with the hour hand, the minute hand needs to travel more than the hour hand, only by that 1/12th of a circle that separates them. This will take time not a whole hour, but less by the same amount of time as 1/12 is less than 11/12, that is, 11 times. This means that the hands will meet in 1/11 of an hour, that is, in 60/11 = 5 5/11 minutes. So, the hands will meet 5 5/11 minutes after 1 hour has passed, that is, at 5 5/11 minutes past two.

21. Answer: It is not difficult to figure out that this will happen after 1 hour 5 5/11 minutes, that is, at 2 hours 10 10/11 minutes. The next one is after another 1 hour 5 5/11 minutes, that is, at 3 hours 16 4/11 minutes, etc. All meetings, as you can easily see, will be 11; The 11th will occur 1 1/11 -12 hours after the first, that is, at 12 o'clock; in other words, it coincides with the first meeting, and further meetings will be repeated again at the same moments. Here are all the moments of the meetings:

24. Let both hands stand at 12, and then the hour hand moves away from 12 by a certain part of a full revolution, which we will denote by the letter x. During the same time, the minute hand managed to turn 12x. If no more than one hour has passed, then to satisfy the requirement of our task it is necessary that the minute hand is at a distance from the end of the whole circle by the same amount as the hour hand has time to move away from the beginning; in other words: 1 - 12 x = x Hence 1 = 13 x. Therefore, x = 1/13 of a whole turn. The hour hand completes this fraction of a revolution at 12/13 o'clock, that is, it shows 55 5/13 minutes past midnight. The minute hand at the same time has traveled 12 times more, that is, 12/13 of a full revolution; both arrows, as you can see, are equally spaced from 12, and therefore equally spaced from 6 on opposite sides. We found one position of the arrows - exactly the one that occurs during the first hour. During the second hour, a similar situation will occur again; we will find it, reasoning according to the previous one, from the equality 1 - (12x - 1) = x, or 2 - 12x = x, whence 2 = 13x, and, therefore, x = 2/13 of a full revolution. In this position, the hands will be at 1 11/13 o'clock, that is, at 50 10/13 minutes past. The third time the hands will take the required position, when the hour hand moves away from 12 to 3/13 of a full circle, that is, 2 10/13 hours, etc. There are 11 positions, and after 6 o’clock the hands change places: the hour hand takes those places where the minute hand was previously, and the minute hand takes the place of the hour hand. If you carefully watch the clock, then perhaps you have seen the exact opposite arrangement of the hands as described now: the hour hand is ahead of the minute hand by the same amount, by how much has the minute moved forward from the number 12. When does this happen? Answer: For the first time, the required arrangement of the hands will be at that moment, which is determined by the equality: 12x - 1 = x/2, whence 1 = 11 ½ x, or x = 2/23 of a whole revolution, that is, 1 1/23 hours after 12. This means that at 1 hour 21 4/23 minutes the hands will be positioned as required. Indeed, the minute hand should be in the middle between 12 and 1 1/23 o'clock, that is, at 12/23 o'clock, which is exactly 1/23 of a full revolution (the hour hand will travel 2/23 of a whole revolution). The second time the arrows will be positioned in the required manner at the moment, which is determined from the equality: 12x - 2 = x/2, from which 2 = 11 1/2 x and x = 4/23; the required moment is 2 hours 5 5/23 minutes. The third desired moment is 3 hours 7 19/23 minutes, etc.