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Find the minimum mass of each ball. Weigh the balls

Hello! Today I will give answers to your questions on mass gain. Let's not pull, let's go.

Friends, thank you again for your activity. I love answering your questions and comments.

They still continue to do so.

I answered almost everyone, but when I answered I noticed that the questions were repeated or vice versa, I came across very rare and interesting ones.

Therefore, for those who did not answer his message, I decided to write this article, because. The answers to these questions, I am sure, will be useful for many readers of my blog.

Nutrition for gaining muscle mass is a very important thing!

The fact is that if we eat improperly, then we can not count on muscle growth.

The bottom line is that since we want to increase the motor units of our body (muscles), which consume a large amount of energy, we need to eat more than we are used to.

Muscle growth = increase in energy consumption of our body

I think there is nothing complicated.

Our body requires an increased amount of energy from food, because. he needs to return the body to its original state after training (a state of homeostasis), as well as increase muscle cells (muscle hypertrophy) in order to overcome a similar load in the future ().

All of these processes require energy.

• We consume LESS calories than we expend= the body lacks energy and burns fat and muscle stores.
• We consume calories as much as we spend\u003d this is an equilibrium (homeostasis), in which there are enough calories, but muscles do not grow.
• We consume more calories than we expend= enough energy for recovery and for the growth of new structures (muscles and fat).

From all this we can conclude that we need an EXCESS OF DAILY CALORIES!

Those. we should consume slightly more calories than we expend.

This does not mean that we should eat everything in a row, as if not in ourselves, and walk around like a pissed-off pig, no.

We just have to create a SMALL, controlled excess of energy in our body so that the body can safely spend excess energy on hypertrophy (growth) of muscle tissue.

The question, in my opinion, is correct and very interesting.

The fact is that, indeed, quite often there comes a moment when you start to train much more, and your muscles GET SMALLER!!!

This is incredibly demotivating and annoying, because. we spend more energy and get less in return.

All this, with the wrong approach, leads us to.

We spend and destroy more than we receive and build.

As a result, even the strongest organism gives up and starts to fail.

In order to avoid this, the most important thing is:

1. Make a competent training program that the body is able to "digest".
2. Eat the right amount of calories per day.
3. Sleep 8-10 hours a day.
4. Help the body with the necessary sports supplements.

I have noted the most important, in my opinion, points.

Make a competent training program that the body is able to "digest".

Very often newcomers coming to gym, I start training using the schemes of professional athletes, which they took from glossy magazines.

As a rule, these schemes are designed for people who use steroids. Indeed, when your recovery abilities increase dramatically by several times, then almost any program works. Naturals, on the other hand, have to be very scrupulous in choosing a training program.

For beginners, I have a "Personal Training Program Selection System", which can be obtained very simply by following what is written below:

Eat the right amount of calories per day.

Nutrition is, indeed, not half, but 60-70% of the success of your workouts.

As we said above, it is necessary to create a certain excess of calories so that the body can afford to spend it on muscle growth.

Sleep 8-10 hours a day.

So far, no other way to restore the body has been invented, like healthy sleep.

The fact is that during sleep, our body produces hormones necessary for growth and recovery, such as somatotropin (growth hormone), testosterone and others.

All this creates a favorable background for muscle growth. Otherwise, when sleep is not enough day after day, over time, energy, central nervous, cardiovascular, endocrine and other systems may fail.

Help the body with essential sports supplements.

“Well, he’s talking about his pills again!” someone will say. Well, yes, just not, but about those that can really provide significant assistance to our body.

First of all, these are:

This is enough for a start.

“Weight plateau” is a thing that happens to EVERY ATHLET, sooner or later.

That very moment when the previous training program stops working, the weight stands still, the strength does not move. How to overcome this, let's see.

1. load progression.
2. Microperiodization of loads.
3. Gradual increase in calorie intake.
4. Macroperiodization of loads.
5. Sports supplements.
6. Anabolic steroid.

This is what came to my mind at the rally, in fact, there are much more points and you can increase the mass in many more ways.

Load progression- the basis of a set of muscle mass.

If the load increases, then the muscles do not make sense to increase. Many beginners make a lot of mistakes, and not only beginners, associated with an increase in load or with its absence.

Microperiodization of loads- this is a non-linear direction of the load in bodybuilding.

When you simply increase the weights from workout to workout, this is a variant of the LINEAR progression of loads.

And when in one workout you do 5 sets to failure in an exercise, in the range of 6-8 repetitions TO FAILURE, and in the next workout you do this exercise in the range of 15-20 repetitions NOT TO FAILURE, then you are using a non-linear, microperiodized scheme. Or rather, one of their varieties.

Microperiodization is needed for several reasons:

1. Avoid overtraining.
2. Break through the weight plateau.
3. hypertrophy of the sarcoplasm.

Gradual increase in calorie intake can also help break through the "weight plateau".

It often happens that training cannot cause any complaints, but when you find out what a person eats or how much he eats, you don’t understand at all how he could gain something on such a meager diet.

If this is the reason, then we need to gradually start increasing the calorie content of our diet, and then observe what comes of it.

Macroperiodization of loads. The meaning is the same as that of microperiodization, the difference is only in the VALUE of the cycle of changing the direction of the load.

Microcycles can be from 1-2 days to a month, on average, and macrocycles up to a year.

The meaning is the same, gradually develop several muscle structures in parallel in order to constantly increase the load.

Sports supplements. There are sports supplements that can really help with muscle growth, for example, or.

Supplements are relatively inexpensive, but the effect of them is very good (relatively, of course).

Anabolic steroid. After a while there will be a series of articles about various stimulants and steroids, but for now I will say that the growth of muscle mass on these drugs is an extremely pronounced and powerful thing.

Individual athletes can gain from 5 to 25 kg of muscle mass in a two-month course! Just imagine how powerful this weapon is, but only in capable hands.

The vast majority of people should NEVER take anabolics, because. this is the lot of athletes involved in bodybuilding professionally.

I hope I was able to answer the question in sufficient detail.

There are a lot of misconceptions about this.

There are a lot of illiterate "fitness trainers" on the Internet who advise immediately after training to load with carbohydrates or other food, because God forbid, the muscles will burn out.

A common reference in bodybuilding is the idea of ​​a narrow CARB WINDOW that "opens" immediately after a workout, at which time the body is able to absorb especially large amounts of nutrients. Carbohydrates and proteins, especially.

The idea looks reasonable, especially when you take into account the huge number of articles on this subject in various fitness publications. Everyone recommends drinking protein or a gainer (“liquid carbohydrates” in a strong concentration with a small amount of protein).

But for a very long time this idea seemed to me a little exaggerated.

In 2012-2013, I served in the army, and there I did not have the opportunity to consume carbohydrates according to the “carbohydrate window” theory, although until this period of my life I always adhered to it regularly.

Guess what happened?

I haven't lost ANYTHING AT ALL!!! It even happened the other way around. I was able to gain even more muscle mass than before. Strange, isn't it?

When I returned from the army, I was no longer loaded with "fast carbohydrates" immediately after training.

Now I always just drink water after a workout, calmly go home, and after 1-2 hours I calmly eat regular food. Usually it is eggs, or meat with vegetables.

I don't notice any negative changes. And now I even feel better, because, in my opinion, digestion is going even better than before.

It is DAILY CALORIE CONSUMPTION that plays a big role, and not one specific meal, friends.

In my opinion, there is a pronounced EXCESS of calories in the diet.

If the belly grows, then the calorie content of the diet is significantly exceeded.

I think the information from that article will be more than enough.

There are a lot of ways, but the best, in my opinion, are three:

1. Weekly body weight check.
2. Reflection in the mirror and photographs.
3. Bioimpedance analysis of the body.

Weekly body weight check. Every week on the same day on an empty stomach we make a control weighing.

• If our weight grows in the range of 200-500 grams per week, then most likely we are gaining fairly clean muscle mass (for beginners, mass can grow faster).
• If the weight grows by more than 1 kg per week, then we gain fat in addition to muscles. We need to cut calories.
• If the weight does not change, then we eat within our reference point, we need to slightly increase the caloric content of the diet until the weight smoothly goes up.

All this is very conditional, because. many factors can influence the growth of body weight: weight, age, genetics, metabolism, gender, etc.

For example, it will be much more difficult for an older athlete to gain muscle mass without fat, the same for girls.

reflection in the mirror. The next criterion that you can rely on.

Take a photo at the very beginning of your journey and take a photo of yourself, for example, every week at the same time.

The photos will clearly show your progress.

While you are growing smoothly, your muscles are quite embossed, the press is visible, you don’t need to change anything, we gradually increase the calorie content and progress the load.

As soon as you begin to smoothly swim in fat, your abs are no longer visible, then you need to reduce calories and add physical activity (you can add cardio).

So you can understand the rate of your growth of quality muscle mass.

Bioimpedance analysis of the body. A fairly accurate method that is based on diagnosing the composition of the human body by measuring the impedance (electrical resistance of body parts) in different parts organism.

Initially, a bioimpedancemeter (a device designed for bioimpedancemetry) was developed for resuscitation, in order to calculate the amount of medication administered.

With the help of a bioimpedancemeter, a specialist will be able to assess the volume of:

• Fat mass.
• Muscle mass and organs.
• Connective tissue (ligaments, tendons, etc.).
• Liquids.

Based on the results of the obtained parameters, it is possible to accurately determine the normal or impaired hydration of body tissues, fat and water-salt metabolism.

For us, the most interesting thing is that we can choose for ourselves a further path to gaining muscle mass or slightly adjust the nutrition program.

• During breathing squats at the initial stage, the legs will grow, provided that the most important rule is preserved - the progression of the load. Alternating classic and breathing squats will be a good solution, because. creates the involvement of more muscle fibers in the work, which leads to a greater production of anabolic hormones (including endogenous testosterone).
• Oh sure. If you are an ectomorph, then you can eat complex carbohydrates in the penultimate meal. But it's not about what meal you eat them, the main thing is the GENERAL CALORIE CONSUMPTION!
• You can eat vegetables almost without restrictions, because. they have zero calories and aid digestion. With fruits, not everything is so simple, because. they contain mostly fast-digesting carbohydrates with a high . The minimum amount for each individual and depends on individual characteristics.

I have a nice blog post about . Be sure to read.

Asterisks on the legs (telangiectasias) usually occur in people who have a genetic predisposition to form them.

Asterisks appear under the influence of provoking factors:

1. Prolonged standing still day after day in the same position without moving.
2. Workouts in the gym.
3. Overweight.
4. Abuse of saunas and baths.
5. Pregnancy.

By themselves, spider veins on the legs are the main manifestation of reticular (net) varicose veins.

This diagnosis is not a sentence, but only an additional condition in your life.

Just in case, it is necessary to consult a phlebologist to determine the severity of the disease and identify all concomitant factors.

What to do with workouts?

The main problem with varicose veins is blood stasis.

You can do ANY CARDIO that fully engages all of your legs.

What exercises can be done? ON THE TOP OF THE BODY ANY!

The legs are more difficult. The most important thing is to AVOID PUMPING!

Blood filling can give rise to new telangiecstasias, which we do not need, so it is better to refuse high-volume training.

Hard work is possible, for example, a warm-up, then 1-2 sets of heavy squats, then 15-20 minutes of cardio.

After training, you should have fatigue in the muscles of the legs, but not fullness of blood.

If there is still a feeling of pumping, then I advise you to lie on the floor and raise your legs up (for example, lean against the wall) until the blood “drains”.

What can be used additionally?

• Compression stockings according to your foot size. You can buy it at a pharmacy, it squeezes your legs from all sides and does not allow you to swell and fill up.
• Pentoxifylline(check with your doctor first). Working drug, inexpensive.
• Lavenum gel(or heparin ointment). Apply 2 times a day. It works very slowly, the effect accumulates for months.
• Detralex. It's expensive, but it works.

There is no question here, but I would like to say that there is a lot of information on my blog about losing weight, plus there is a powerful paid product "Extreme Fat Burning", which received a lot of positive feedback.

So the topic of losing weight is also discussed very closely on my blog. It's just not the season

There will be a separate detailed article on this topic on my blog.

But in short, soy protein, despite the fact that it is as close as possible in amino acid composition to animal protein, still does not have a complete set of amino acids.

Fruits are also made up almost entirely of water and fast-digesting carbohydrates. This is good for restoring energy reserves and glycogen, but does not provide the necessary amount of protein for muscle growth.

If there are few calories and the BJU ratio is not quite correct, then you can forget about the growth of muscle mass.

The number of repetitions DOES NOT MATTER AT ALL, I talked about this. Be sure to read.

The number of approaches depends on your training program and fitness. It is enough for beginners to do 2-3 working approaches, and only then, with an increase in fitness, increase the number of working approaches.

Let's say in low-catabolic training we do more approaches, in high-volume training a little less. All this is individual, but generally speaking, the higher your fitness, the more working approaches you should perform. And most importantly, not a huge number of approaches, but their quality.

Over time, based on the results of experiments, you will learn to understand how many approaches you should do.

You need to stick to both! You can gain the required calories if you eat only chocolates, but is that right?

The number of calories indicates the amount of energy received, and BJU indicates the ratio of the nutrients received, from which further life activity will be built.

I also talked about how to gain lean muscle mass in articles.

Everything is very short and concise here) We have already talked about nutrition in the articles, links to which I gave above.

We discussed the set of lean muscle mass with you in my last article (the link to it is just above). Everything is detailed there.

If you want sweets, then you can afford it, but taking into account the daily calorie content of the diet and, preferably, before training.

A clear relief on the legs comes from two things:

1. Hypertrophy of the muscles of the legs.
2. Reducing the amount of fat in the body.

With the first point, everything is simple, swing your legs and the relief will appear.

The second point needs to be clarified. You can’t lose weight only in the “right places”, fat burning in our body is triggered by HORMONES that circulate throughout the body, starting fat burning IN ALL CELLS!

Another thing is that in different tissues of our body there is a different ratio of ALPHA and BETA receptors (especially the second type), through which hormones interact with them.

In the hips of women, a sufficiently large number of alpha-2-adrenergic receptors, so it is more difficult to lose weight in these parts of the body.

But there is nothing left but to gradually reduce the calorie content of the diet in order to cause fat burning (there is no question of mass gain then). You can also use . This is a cool supplement that will help with weight loss and increase sexual desire a little.

The basic principles remain the same, namely:

1. load progression.
2. Gradual increase in diet.
3. The main load falls on the bottom of the body (because there are more muscles).
4. The use of microperiodization is mandatory (because of the menstrual cycle).

About whether you gain fat or muscle, I said above. The most accurate way is bioimpedance analysis of the body, at least once a month. This will be enough to understand the growth dynamics of certain body tissues.

In centimeters, volumes increase due to the growth of body tissues under the influence of physical exertion, for example. Growth of muscle and adipose tissue (mostly).

Dmitry, thanks for the kind words! Very nice.

A similar power system (and not just one) will be in my new product, very soon, and even more. I'll tell you a secret. ABSOLUTELY EVERYTHING will be painted! Fully!

And so, this is the topic of a separate article, at least.

For now, just try to figure out your baseline and start gradually increasing your calorie intake.

Michael, hello! I'm glad that progress is being made. It's hard to say, but most likely, your muscle growth has already begun.

Your goal is very real. I'm sure you will succeed.

Included in the preliminary list.

The course is going to be awesome! I have never done anything like this, and I don’t see it anywhere.

Hi Alex!

This is real. You need to focus on exercises in frames and simulators. Try the hack squat, leg press. Gradually strengthen the lower back with hyperextension.

I also had problems, but with a knee, did a leg press and grew well. You just need to feel a little about what works specifically for you.

Simultaneous fat burning and muscle gain is almost impossible to implement (without stimulants).

If we are talking about natural training, then at first I would lose weight to 10-12% of body fat (when the press is clearly visible, etc.), and then I began to gain high-quality muscle mass, through the progression of loads and a gradual increase in calorie intake.

Let's summarize a little

Thank you again for your questions. It was interesting for me to talk to you again.

Now I have an almost clear understanding of how I should supplement my new course about gaining muscle mass. Thank you very much!

Keep growing and improving, friends.

Subscribe to my instagram and other social networks.

P.S. Subscribe to blog updates. It will only get worse from there.

With respect and best wishes, !

At first it seemed that the problem could not be solved. Reached 11 balls when dividing the original pile into smaller ones: 3-3-3-2.
If the first two piles are equal to 3=3, then we compare any three balls from them with the third one, if again equality, then the desired ball in the remaining two, is 1 weighing with any ordinary ball.
If there is an inequality at some of the previous stages, then by weighing any of the unequal piles with three ordinary balls, both the desired pile of 3 balls and the ratio of weights are found. And then it is decided for 1 weighing.

You can enter the notation:
3+,1 - this means that the problem of finding a ball in a pile of three balls is solved in one weighing, if it is known whether the ball is lighter or heavier than the others.
Accordingly, 9+,2; 27+,3.

You can try iterating over the options. We number the balls as indicated in the solution: 1,2,3,...,12.
1. Weigh any 2 balls. There is a good option when the required ball is one of these two balls, and there is a bad option. Next, we will consider bad options.
It turns out the problem 10-, which is not solved in 2 weighings in any way (in 2 moves a maximum of 9+ is solved).
2. Weigh 1.2 and 3.4. In the worst case, the problem is reduced to 8-, which is also not solved in 2 moves.
3. 1,2,3 and 4,5,6. In case of inequality at any stage, the problem is solved as mentioned above. In the worst case, after two equalities 1,2,3=4,5,6 and 1,2,3=7,8,9 we come to problem 3-, which is not solved for 1 remaining move.
4. 1,2,3,4 and 5,6,7,8. If equality, then in the remaining 4 balls the required one is found quite simply with the help of two weighings and the possibility of using ordinary balls. It is this point that is not covered correctly in the proposed solution.
a) You can weigh 9 and 10, if equality, then any of 11-12 with any of the usual 1-10.
If inequality, then we weigh any of 9-10 with any of the usual 1-8 or 11-12.
b) You can weigh any three of 1-8 and 9,10,11, if equality, then the desired ball is 12.
If inequality, then the ball is at 9,10,11 and we know whether it is heavier or lighter. The problem is reduced to 3+ and solved in 1 move.

If there is inequality in the first weighing, then, at first glance, the problem is not solved. We will discuss this below.
5. 1,2,3,4,5 and 6,7,8,9,10. In a bad version, we get an inequality and the problem is not solved in the remaining 2 moves (1 move will be spent on identifying the desired group of 4 balls, and problem 4+ is not solved in one remaining move).
6. 1,2,3,4,5,6 and 7,8,9,10,11,12. In the worst case, in 2 moves we will only know the group of 6 balls where the desired ball is. Problem 6+ is not solved for the remaining move.

In option 4, at first I was confused by the fact that in the case of inequality in the first weighing, it was not possible to further reduce the problem to 3+ in 1 move. The usual way: dividing any of the heaps 1-4 and 5-8 into two by 2 balls and weighing them gives a 4+ problem in the worst case. And for 1 remaining move, it is not solved.
In the above solution there is an indication of how you can proceed and resolve this issue. You can use the proposed notation or simply reason logically.
It is necessary to redistribute groups 1-4, 5-8 so that no more than 3 balls remain in logically selected subgroups. And we have 3 possible readings of the scales: =, >,<, которые могут указывать на искомую группу.
We remove one ball from the first group, say, 1, and transfer it to the second group. And from the second we transfer one ball, say, 5, to the first. From the second group, we replace the three remaining balls with ordinary ones (we replace 6-8 with any three from 9-12).
We weigh (5,2,3,4 and 1,9,10,11).
a) The ratio between the masses on the bowls will change if the desired ball was transferred to another bowl or replaced. That is, if the previous ratio is observed, then the desired ball is in those that have remained in their place, and these are 2,3,4. The task was reduced to 3+.
b) If the ratio has changed to equilibrium, then this means that the desired ball has been removed from the balance. Then this is an indication of the balls 6,7,8. The task was reduced to 3+.
c) If the ratio has changed to the opposite, then this means that the desired ball has been moved from one bowl to another. Those. this is an indication of the balls 1 and 5. By weighing any of these balls with any ordinary (2-4 or 6-12) the required ball is found.

The solution presented in the answer is correct, except for the confusion in the first part (after the equality in the first weighing 1,2,3,4 = 5,6,7,8).

We take four balls on each scale and weigh them. Let's call those balls about which we EXACTLY know that they are not what we are looking for, reference. We will identify them by analyzing the results of weighing
I) If the scales came into equilibrium, then the desired ball remained in those four balls that did not participate in the weighing. Suspicious in this case, we will have those balls that did not participate in the first weighing, and the reference ones that lay on the scales.
A) We put two "suspicious" balls on one scale pan, one "suspicious" ball on the second one and supplement this pan with one of the reference balls.
a) If the scales have come into equilibrium, then the desired ball is the one that remains. We weigh it with any of the standard balls and find it lighter or heavier.
b) If the scales are out of balance, then we remember the position of the scales (This is important if we want not only to identify, but also to accurately determine the desired ball easier or heavier than the others). Let's agree to call the FIRST bowl the one on which TWO "suspicious balls" lay The SECOND bowl is the one on which ONE suspicious ball and one reference ball lay.

B) We remove one of the "suspicious" balls from those two that were on the same bowl (On the second bowl, as we remember, there was one "suspicious" and one reference), transfer the "suspicious" ball from the second bowl to the first bowl and complete the second the bowl of the scales with another reference ball. Thus, it turns out that on the first bowl we again have TWO "suspicious" balls, and on the second - two reference ones. WE WEIGH. We analyze taking into account the previous weighing.
1) the scales came into balance: the ball that we removed from the first pan of the scales is to blame. If the first bowl of the scales in the previous weighing was higher, then it is lighter than the rest, if lower, it is heavier.
2) If the scales have not changed their state, then the “blame” is the ball from the first pan of the scales, which we did not touch. If in the previous weighing the first bowl of the scales was higher than the second, then it is lighter than the others, if it is lower, it is heavier.
3) If the scales came to a state opposite to that which was in the previous weighing, then the “suspicious” from the second bowl, which we transferred to the first bowl, is “guilty”. If the First bowl in the previous weigh-in was
higher than the second, then the ball is heavier than the rest, if lower, it is heavier.

II) The scales are out of balance. remove one ball from each scale (any of them are ALL "suspicious", Reference, in this case, those that did not participate in the first weighing)
We transfer TWO "suspicious" balls from one bowl of scales to another, and from the second bowl we transfer ONE suspicious ball. So we divide the balls into threes. WE WEIGH.

Gazalova Victoria and Popova Marina

This paper presents interesting methods for solving transfusion and weighing problems. This material can be used in preparation for the Olympiads in the subject.

Download:

Preview:

1. Update
2. Weighing tasks
3. Tasks for transfusion
4. Conclusion
5. Literature

The relevance of research

Mathematical tasks for transfusion and weighing have been known since antiquity. Now they can be found in Olympiad problems or in computer games - puzzles. The classic counterfeit coin problem (FM) has recently found application in coding and information theory - to detect errors in the code. The purpose of our work is to find and describe algorithms for solving such problems. Transfusion and weighing problems belong to the type of combinatorial search problems; their solution comes down to working with information.

In the course of the study, it turned out that there are a lot of different plots of these tasks. Therefore, we examined the most common plots for each type.

Weighing tasks.

Weighing tasks are a type of tasks in which it is required to establish one or another fact (select a counterfeit coin among the real ones, sort a set of weights in ascending order of weight, etc.) by weighing on a balance scale without a dial. Coins are most often used as weighted objects. Less commonly, there is also a set of weights of known mass.

Very often, a problem statement is used, requiring either to determine the minimum number of weighings required to establish a certain fact, or to give an algorithm for determining this fact for a certain number of weighings. Less common is a statement that requires an answer to the question of whether it is possible to establish a certain fact for a certain number of weighings. Often such a statement is not very successful, since with a positive answer to a question, the problem most often comes down to constructing an algorithm, and a negative answer is almost never found.

The search for a solution is carried out by comparison operations, and not only single elements, but also groups of elements among themselves. Problems of this type are most often solved by reasoning.

Having studied the literature on this topic, we came to the conclusion that all weighing tasks can be divided into the following types:

Comparison tasks using weights.

Tasks for weighing on scales with weights.

Problems for weighing on scales without weights.

Task 1.1 The most classic puzzle.

One of the 9 coins is fake, it weighs lighter than the real one. How to determine a counterfeit coin (FM) for 2 weighings?

Solution. The key idea for solving such problems is the correct trisection , i.e. sequential division of the set of options into three equal parts. After the first trisection, no more than three suspicious coins should remain, after the second - no more than one PM, which is the PM.

We weigh coins 123 and 456, setting aside 789.

If 123 is lighter, then among them is FM; heavier then FM among 456; are equal, then FM among 789.

Hypothesis . There are algorithms for determining the FM in the least number of weighings if it is known that the FM is heavier or lighter than the real one (algorithm 1) and if it is not known (algorithm 2).

Generalization 1. Let there be K coins and one of them is counterfeit (K is greater than two). It is known that it is lighter than the real one. What is the least number of weighings to find the FM?

Solution.

ALGORITHM 1. Put K:3 coins on the bowls, set aside the rest (if the number of coins is not a multiple of 3, then put the same number of coins on the bowls, equal to (K-1):3 or (K+1):3, depending on whether which one is natural). Further, if one of the bowls outweighed, then the FM is on the other bowl, and in the case of balance, the FM is among the pending ones. Then we repeat this for a group of coins, among which is the FM.

FM in the condition can be heavier than the real one, in this case we also argue, only the FM coin will be on the bowl that outweighed.

Consider the problem with weights, where this rule can also be applied.

Task 1.2 There are 9 standard weights weighing 100.200, ..., 900 gr. One of them has been in the hands of dishonest traders and now weighs 10 grams. less. How to find it in 2 weighings?

Let's find two different triples of weights that are the same in weight. For example, let's weigh 100+500+900 and

200+600+700 and 300+400+800 will remain. Arguing also, we find a group with a damaged kettlebell. Then you can find a damaged weight by adding obviously real ones. For example, 200+600 and 700+100.

The next task differs in that it is not known in advance whether the FM is lighter or heavier than the real one.

Problem 1.3 Of the three coins, one is counterfeit, and it is not known whether it is lighter or heavier than the real one. How to find it in two weighings and determine whether it is lighter or heavier than the real one?

There are 6 possible answers in this problem (each of the three coins can be either lighter or heavier than the real one).

Answer: yes, you can, while the least number of weighings is 2.

Task 1.4 There are 4 weights marked 1g, 2g, 3g, 4g. One of them is defective - lighter or heavier. Is it possible to find this weight in two weighings and determine whether it is lighter or heavier than the real one?

There are 8 possible answers here. Weigh 1g + 2g and 3g, then 1g + 3g and 4g weights.

We get the following table of options:

Answer: yes, you can.

Generalization 2. Let there be K coins and one of them is counterfeit. What is the least number of weighings to determine the FM and is it lighter or heavier?

First you need to find out the number of possible answers. Their K * 2, since each coin can be lighter or heavier. Then we determine the number of weighings. One weighing determines three options: ,=. Two weightings determines 9 options: , =, >=, >>, ==(there are 3*3 of them, but in this problem the option == is impossible). Three weightings determines 3*3*3= 27 options, etc.

ALGORITHM 2. Divide the coins into three groups. If K is not divisible by 3, then either (K-1) is divisible by 3, then we put on the scales each (K-1): 3 coins and there will be (K-1): 3 coins and 1 more coin. Or (K-2) is divided by 3, then we put on the scales each (K-2): 3 coins and there will be (K-2): 3 coins and 2 more coins. Weighing the first and second groups, and then the second and third, we conclude in which group the FM is located. If the scales were in equilibrium in both cases, then the FM is in the coins set aside, and then, according to the number of coins set aside, in one or two weighings we will find the FM and it is lighter or heavier than the real one (comparing them with real coins). Further, if the FM was not in the set aside coins, then we can already determine whether it is lighter or heavier than the real one. And then we act according to algorithm 1. Denoting the groups of coins 1, 2, 3, we will show the weightings 1 and 2 then 1 and 3 in this table.

Knowing whether the FM is heavier or lighter than the real one, we can use the algorithm1 described in Generalization 1. As you can see, here the division into three parts is as equal as possible.

Let's test the algorithm with more coins.

Problem 1.5 There are 80 coins, one of which is counterfeit. What is the least number of weighings on a scale without weights that can find a counterfeit coin?

Solution. We carry out the first weighing: we put on bowls on (80-2): 3 = 26 coins. In the case of equilibrium, FM among the remaining 28;by weighing the real 26 coins with 26 "suspicious" ones, we will determine whether the FM is lighter or heavier than the real one(in case of balance, it is in the remaining two and then 2 more weighings are needed). If at the first weighing the scales were not in balance, then the false one is in one of the bowls on the scales. We compare the first group of coins with the real ones from the third and draw a conclusion. Then we divide the group of coins where there is a fake one by 9, 9, 8, weigh it, then weigh it by 3 coins, and then one by one.

Answer: for 5 weighings.

Algorithm 1. We weigh the first two groups of coins (highlighted in color).

Qty coins	1 division	2 division	3 division	4 division
		9 to 3,3 and 3	3 by 1,1 and 1
		10 to 3,3 and 4 9 to 3,3 and 3	3 by 1,1 and 1 4 by 1,1 and 2	2 by 1 and 1
		10 to 3,3 and 4 9 to 3,3 and 3	3 by 1,1 and 1 4 by 1,1 and 2	2 by 1 and 1
K is a multiple of 3	K:3 K:3 K:3	divide similarly	and among them there is one false, which is known to be lighter or heavier than the real ones. Then the smallest number of weighings on a pan balance without weights required to find a counterfeit coin is n.
K:3 from stop. one	(K-1):3 (K-1):3 (K-1):3+1
K:3 from stop. 2	(K+1):3 (K+1):3 (K+1):3-1

• If there are 2 or 3 coins, then 1 weighing is required to find a counterfeit coin among them.
• If there are 4 to 9 coins inclusive, then the least number of weighings to find a fake coin is 2.
• If coins are from 10 to 27 inclusive, then it is equal to 3.
• If coins are from 28 to 81 inclusive (due to the fact that 81 = 3*27), then the least number of weighings is 4.

regularity . The numbers 9, 27, 81 are successive powers of the triple, and the numbers 4, 10, 28 are respectively the previous powers of the triple, increased by 1: 4 = 3+1, 10 = 3 2 +1, 28 = 3 3 +1.

Algorithm 2. In the 2nd weighing, we put the second and third groups of coins on the scales. In the rest, we weigh 1 and 2 groups of coins.

Qty coins	1 division 2 weighings		2 division	3 division	4 division
			9 to 3,3 and 3	3 by 1,1 and 1
		9 +1	10 to 3,3 and 4 9 to 3,3 and 3 1 and 1	3 by 1,1 and 1 4 by 1,1 and 2	2 by 1 and 1
		9 +2	10 to 3,3 and 4 9 to 3,3 and 3 1 and 1	4 by 1,1 and 2 1 and 1 3 by 1,1 and 1	2 by 1 and 1
K is a multiple of 3	K:3 K:3 K:3	K:3 K:3 K:3	If in the first or second cases the scales were not in balance, then it is possible to determine the group of coins containing FM, and also to conclude whether it is lighter or heavier than a real coin. Next, we proceed according to algorithm 1. (otherwise *)	In general, let the number of coins k satisfy the inequality When provinggiven and among them there is one false, about which it is not known whether it is lighter or heavier than the real ones. Then the smallest number of weighings on a pan balance without weights required to find a counterfeit coin is n.
K:3 from stop. one	(K-1):3 (K-1):3 (K-1):3+1	(K-1):3 (K-1):3 (K-1):3 +1
K:3 from stop. 2	(K-2):3 (K-2):3 (K-2):3+2	(K-2):3 (K-2):3 (K-2):3 +2

*In the second weighing, we find a group of coins containing FM. If in the 1st and 2nd weighings the scales were in balance, then FM was among the remaining one or two. If there is 1 coin left, then it is FM and weighing it with a real one, we find out if it is lighter or heavier than a real coin. If there are 2 left, then weighing them together, and then one of them with the real one, we answer the question of the problem. If in the first or second cases the scales were not in balance, then it is possible to determine the group of coins containing FM, and also to conclude whether it is lighter or heavier than a real coin.

• If there are 2 coins, then problem 2 has no solution.
• If there are 3 coins, then 2 weighings are required to find a fake coin among them.
• If there are 4 to 9 coins inclusive, then the least number of weighings to find a fake coin is 3.
• If coins are from 10 to 27 inclusive, then it is equal to 4.
• If coins are from 28 to 81 inclusive (due to the fact that 81 = 3*27), then the least number of weighings is 5.

Let's summarize the tasks.

The hypothesis was confirmed. We have described algorithms for determining the FM in the least number of weighings in case the FM is known to be heavier or lighter than the real one (algorithm 1) and in case it is not known (algorithm 2).

Transfusion tasks.

Description: having several vessels of different volumes, one of which is filled with liquid, it is required to separate it in some respect or cast some part of it with the help of other vessels in the least number of transfusions.

In transfusion tasks, it is required to indicate the sequence of actions in which the required transfusion is carried out and all the conditions of the task are met. Unless otherwise stated, it is assumed that

All vessels without divisions,

Do not pour liquids "on the eye"

It is impossible to add liquids from anywhere and drain anywhere.

We can tell exactly how much liquid is in a vessel only in the following cases:

1. we know that the vessel is empty,
2. we know that the vessel is full, and in the problem its capacity is given,
3. in the task it is given how much liquid is in the vessel, and transfusions using this vessel were not performed,
4. two vessels participated in the transfusion, in each of which it is known how much liquid was, and after the transfusion all the liquid fit into one of them,
5. two vessels participated in the transfusion, in each of which it is known how much liquid was, the capacity of the vessel into which it was poured is known, and it is known that all the liquid did not fit into it: we can find how much of it remained in another vessel.

Most often, a verbal solution method (i.e. a description of the sequence of actions) and a solution method using tables are used, where the volumes of these vessels are indicated in the first column (or row), and the result of the next transfusion is indicated in each next column. Thus, the number of columns (except the first one) shows the number of transfusions required. The same methods (verbal and tabular) were also used in solving weighing problems. However, we have discovered another interesting way in which such problems can be solved. This is the method of mathematical billiards. ME AND. Perelman, in his book "Entertaining Geometry", proposed solving transfusion problems using a "smart" ball. For each case, it was proposed to build a billiard table of a special design from equilateral triangles, the lengths of the two sides of which are numerically equal to the volume of two smaller vessels. Further, from the acute angle of this table along one of the sides, you need to “launch” a ball, which, according to the law “the angle of incidence is equal to the angle of reflection”, will collide with the sides of the table, thereby showing a sequence of transfusions. On the sides of the table there is a scale, the division value of which corresponds to the selected unit of volume. As a result of the movement, the ball either hits the edge at the desired point (then the problem has a solution) or does not hit (then it is considered that the problem has no solution). A billiard ball can only move along straight lines that form a grid on a parallelogram. After hitting the sides of the parallelogram, the ball is reflected and continues to move along the edge emerging from the point where the collision occurred, completely characterizing how much water is in each of the vessels.

An old-fashioned puzzle.

An eight-bucket keg is filled to the top with kvass. The two must share the kvass equally. But they have only two empty barrels, one of which contains 5 buckets, and the other - 3 buckets of kvass. The question is, how can they divide kvass using only these three casks?

In the In the problem, the sides of a parallelogram must have sides of 3 units and 5 units. We will plot the amount of kvass in buckets in a 5-bucket keg horizontally, and in a 3-bucket keg vertically.

Let the ball be at point O and after hitting hit point A. This means that the 5-pail keg is filled to the brim, and the 3-pail is empty. Reflected elastically from the starboard side, the ball will roll up and to the left and hit the upper side at a point with coordinates 2 horizontally and 3 vertically. This means that only 2 buckets of kvass remained in the 5-bucket keg, and the buckets from it were poured into a smaller keg. Having reflected elastically from the upper side, the ball will roll down and to the left and hit the lower side at a point with coordinates 2 horizontally and 0 vertically. This means that 2 buckets of kvass remained in a 5-bucket keg, and kvass was poured from a 3-bucket vessel into an 8-bucket keg. Reflected elastically from the lower side, the ball will roll up and to the left and hit the port side at a point with coordinates 0 horizontally and 2 vertically. This means that 2 buckets of kvass were poured from a 5-bucket keg into a 3-bucket keg. Reflected elastically from the port side, the ball will roll to the right and hit the starboard side at a point with coordinates 5 horizontally and 2 vertically. This means that 5 buckets of kvass were poured into a 5-bucket keg, and 2 buckets remained in a 3-bucket keg. Reflected elastically from the starboard side, the ball will roll up and to the left and hit the upper side at a point with coordinates 4 horizontally and 3 vertically. This means that 1 bucket of kvass was poured from a 5-bucket barrel into a 3-bucket barrel, where there were 3 buckets, and 4 buckets remained in the 5-bucket barrel. Reflected elastically from the upper side, the ball will roll down and to the left and hit the lower side at a point with coordinates 4 horizontally and 0 vertically. This means that 2 buckets of kvass remained in a 5-bucket keg, and kvass was poured from a 3-bucket keg into an 8-bucket keg. The problem was solved with the help of 7 transfusions. At the same time, we fill in the table:

number of transfusions
8 l
5 l
3 l

Let's see how our billiard ball will behave if we first fill a 3-bucket barrel with kvass.

It is clearly seen that this problem was solved as a result of 8 transfusions.

We solve the famous billiard method the Poisson problem.

This problem is associated with the name of the famous French mathematician, mechanic and physicist Simenon Denny Poisson (1781 - 1840). When Poisson was still very young and hesitating about choosing a path in life, a friend showed him the texts of several problems that he could not cope with on his own. Poisson solved them all in less than an hour. But especially to him

I liked the problem about two vessels. “This task determined my fate,” he later said. - I decided that I would definitely be a mathematician

A task. Someone has 12 pints of wine and wants to donate half of it. But he doesn't have a 6-pint jar. He has 2 vessels. One at 8, the other at 5 pints. The question is how to pour 6 pints into an 8 pint vessel?

Let's build a billiard table in the form of a parallelogram. We take the sides equal to 5 units and 8 units. We will plot the amount of wine in the vessel horizontally at 8 pints, and vertically at 5 pints. We argue similarly.

12 l

5 l

8 l

It turns out 7 transfusions. However, if poured first into a 5-pint vessel, 18 pourings would be required.

Do problems of this type always have solutions?

The billiard ball method can be applied to the problem of pouring a liquid with no more than three vessels. If the volumes of two smaller vessels do not have a common divisor (i.e., they are coprime), and the volume of the third vessel is greater than or equal to the sum of the volumes of the two smaller ones, then using these three vessels, any integer number of liters can be measured, starting from 1 liter and ending with middle vessel. Having, for example, vessels with a capacity of 15, 16 and 31 liters, you can measure any amount of water from 1 to 16 liters. This procedure is not possible if the volumes of the two smaller vessels have a common divisor. When the volume of the larger vessel is less than the sum of the volumes of the other two, new restrictions arise. If, for example, the volumes of the vessels are 7, 9 and 12 liters, then the lower right corner should be cut off at the rhombic table. Then the ball can hit any point from 1 to 9, with the exception of point 6. Despite the fact that 7 and 9 are coprime, it is impossible to measure 6 liters of water due to the fact that the largest vessel has too small a volume. It is easy to see that the points with the number 6 form a regular triangle on the diagram, and we cannot in any way get to this triangle from any other point lying outside it. We also note that the generalization of the method of mathematical billiards to the case of four vessels is reduced to the motion of a ball in a spatial region (parallelepiped). However, the resulting difficulties in depicting trajectories make the method inconvenient.

The advantage of this elegant method of mathematical billiards lies, first of all, in its clarity and attractiveness.

Conclusion

Summing up, we can say that in the course of the research work:

1. Collected theoretical and practical material on the research problem.

2. Based on the results of this work, we systematized the tasks for transfusions and weighing.

3. Algorithms for solving are compiled.

4. A presentation was made to familiarize classmates with these tasks and help them prepare for the Olympiad.

Thus, we can conclude that the work performed by us turned out to be fruitful, the students got acquainted with the methods and methods for solving problems of weighing and transfusion. Learned how to properly apply the best ways to solve them. According to the students, the work done allowed them to master the methods of solving transfusion problems, expanded their horizons. The students noted the possibility and practicality of using the billiard method in solving this type of problem. Continuing this study in the future, you can still try to find a formula for calculating the least number of weighings (transfusions).

List of sources used

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