Hey! Today I will give answers to your questions about mass building. Let's not delay, let's go.
Friends, thank you again for your activity. I really love answering your questions and comments.
They are still coming in.
I answered almost everyone, but when I answered, I noticed that the questions are repeated or vice versa, I come across very rare and interesting ones.
Therefore, for those who did not respond to his message, I decided to write this article, because the answers to these questions, I am sure, will be useful to many readers of my blog.
Nutrition for gaining muscle mass is a very important thing!
The fact is that if we eat incorrectly, then we can not count on muscle growth.
The bottom line is that since we want to increase our body's motor units (muscles), which consume a lot of energy, we need to eat more than we are used to.
Increased Muscle \u003d Increased Energy Consumption of Our Body
I think there is nothing complicated here.
Our body requires an increased amount of energy supplied from food, because he needs to return the body to its original state after training (state of homeostasis), as well as increase muscle cells (muscle hypertrophy) in order to overcome such a load in the future ().
All these processes require energy.
- We consume LESS calories than we expend \u003d the body lacks energy and burns up fat and muscle stores.
- We consume as much calories as we spend \u003d this equilibrium (homeostasis), in which there are enough calories, but no muscle growth.
- We consume MORE calories than we expend \u003d enough energy for recovery and for the growth of new structures (muscle and fat).
From all this we can conclude that we need an EXCESS OF DAILY CALORIE!
Those. we should consume a little more calories than we expend.
This does not mean that we should eat everything, if not into ourselves, and walk around like a swine, no.
We just need 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 time when you start to train much more, and your muscles BECOME LESS !!!
This is incredibly demotivating and annoying. 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 malfunction.
In order to avoid this, the most important thing is:
- Create a competent training program that the body is able to "digest".
- Consume the right amount of calories per day.
- Sleep 8-10 hours a day.
- Help the body with essential sports supplements.
I noted the most important, in my opinion, moments.
Create a competent training program that the body is able to "digest".
Very often, beginners who come to the gym begin to train using the schemes of professional athletes, which they took from glossy magazines.
Typically, these regimens are designed for people who use steroids. Indeed, when your regenerative abilities dramatically increase several times, almost any program works. Straight people, however, have to be very scrupulous in choosing a training program.
For beginners, I have a “System for choosing an individual training program”, which can be obtained very simply by following what is written below:
Consume the right amount of calories per day.
Nutrition is really 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 a 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 from day to day, over time, the energy, central nervous, cardiovascular, endocrine and other systems can 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 comes to EVERY ATHLETER, sooner or later.
The very moment when the previous training program stops working, the weight stays in place, the strength ones do not move. How to overcome this, let's figure it out.
- Load progression.
- Microperiodization of loads.
- A gradual increase in calorie intake.
- Macroperiodization of loads.
- Sports supplements.
- 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 for gaining muscle mass.
If the load is growing, 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 the 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 an option for LINEAR progression of loads.
And when in one workout you do 5 sets to failure in the exercise, in the range of 6-8 repetitions BEFORE FAILURE, and on the next workout you do this exercise in the range of 15-20 repetitions NOT BEFORE FAILURE, then you use a non-linear, microperiodic scheme. Rather, one of their varieties.
Microperiodization is needed for several reasons:
- Avoid overtraining.
- Break through the "weight plateau".
- Sarcoplasmic hypertrophy.
A 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 about what a person eats or how much he eats, you don’t understand at all how he was able to gain anything at all on such a meager diet.
If this is the case, then we need to gradually start increasing the calorie content of our diet, and then watch what came of it.
Macro-periodization 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 can be up to a year.
The idea 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 in the growth of muscle mass, for example, or.
Supplements are relatively inexpensive, but the effect 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 much it is powerful weapon, but only in capable hands.
The vast majority of people should NEVER take anabolic steroids, because this is the lot of athletes involved in professional bodybuilding.
I hope I managed to answer the question in sufficient detail.
There are a lot of misconceptions on this score.
There are a lot of illiterate "fitness trainers" on the Internet who advise you to load up on carbohydrates or other foods immediately after training, because GOD DON'T GOD BREAK your muscles.
In bodybuilding, the idea of \u200b\u200ba narrow CARBON WINDOW is often mentioned, which "opens" immediately after training, and at this time the body is able to absorb a particularly large amount nutrients... Carbohydrates and proteins, especially.
The idea looks reasonable, especially considering the huge number of articles on this subject in various fitness publications. Everyone recommends drinking protein or a gainer ("liquid carbohydrates" in high 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 before this period of my life I always adhered to it regularly.
Guess what happened?
I have not lost ANYTHING AT ALL !!! Quite the opposite happened. 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 carbs" right after training.
Now I always just drink water after training, calmly go home, and after 1-2 hours I calmly eat regular food. Usually these are 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 TOTAL 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 there will be more than enough information from that article.
There are a lot of ways, but the best, I think, are three:
- Weekly body weight monitoring.
- Reflection in the mirror and photography.
- Bioimpedance body analysis.
Weekly body weight control... Every week on the same day on an empty stomach we perform a control weighing.
- If our weight grows in the range of 200-500 grams per week, then, most likely, we are gaining a fairly clean muscle mass (for beginners, the mass can grow faster).
- If the weight grows by more than 1 kg per week, then we are gaining fat in addition to muscle. We need to reduce calories.
- If the weight does not change, then we are eating within our starting point, we need to slightly increase the calorie content of the diet until the weight gradually goes up.
All this is very arbitrary, 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 is true for girls.
Reflection in the mirror... The next criterion you can rely on.
Take a picture at the very beginning of your journey and take pictures of yourself, for example, every week at the same time.
From the photos you will clearly see your progress.
While you are growing smoothly, your muscles are quite prominent, the press is visible, nothing needs to be changed, we gradually increase the calorie content and progress the load.
As soon as you start to smoothly swim with 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 Body Analysis... 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 bioimpedance meter (a device designed for bioimpedance measurement) was developed for resuscitation, in order to calculate the amount of injected drugs.
Using a bioimpedance meter, a specialist will be able to estimate the volume:
- 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.
The most interesting thing for us is that we can choose for ourselves the further path of gaining muscle mass or slightly adjust the nutrition program.
- When breathing squats on initial stage legs will grow, provided that the most important rule is preserved - the progression of the load. Alternating between the classic and breathing squats is a good idea as creates the involvement of more muscle fibers in the work, which leads to a greater production of anabolic hormones (including endogenous testosterone).
- Yes of course. If you are an ectomorphic physique, then you can eat complex carbohydrates in the penultimate meal. But the point is not in what kind of food you use them, the main thing is TOTAL CALORIES CONSUMPTION!
- You can eat vegetables practically without restrictions, because they are zero calories and aid in digestion. With fruits, not everything is so simple, because they contain mostly fast-digesting high carbohydrates. The minimum amount for each is individual and depends on individual characteristics.
I have a nice blog article about. Be sure to read it.
Asterisks on their feet (telangiectasias) usually occur in people who have a genetic predisposition to their formation.
Asterisks appear under the influence of provoking factors:
- Prolonged standing still from day to day in the same position without movement.
- Workouts in the gym.
- Overweight.
- Abuse of saunas and baths.
- Pregnancy.
By themselves, spider veins on the legs are the main manifestation of reticular (reticular) varicose veins.
This diagnosis is not a sentence, but only an additional condition in your life.
Just in case, you need to consult a phlebologist to establish the severity of the disease and identify all associated factors.
What to do with training?
The main problem with varicose veins is blood stasis.
You can do ANY CARDIO that fully engages all legs.
What exercises can you do? ON TOP OF THE BODY ANYONE!
The legs are more difficult. The most important thing is to AVOID PUMPING!
Blood filling can give rise to new telangiectasias, which we do not need, so it is better to refuse high-volume training.
Hard work is possible, for example, 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 full 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 a wall) until the blood "drains".
What can be used additionally?
- Compression jersey the size of your feet. You can buy it at a pharmacy, it squeezes the legs from all sides and does not allow swelling and filling.
- Pentoxifylline (consult your doctor first). Working drug is inexpensive.
- Lavenum gel (or heparin ointment). Smear 2 times a day. It works very slowly, the effect accumulates for months.
- Detralex.Costly, but working.
There is no question here, but I would like to say that on my blog there is a lot of information about weight loss, plus there is a powerful paid product "Extreme Fat Burning", which received a lot of positive reviews.
So the topic of losing weight is also covered 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 to put it briefly, 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 composed almost entirely of water and fast-digesting carbohydrates. This is good for replenishing energy stores and glycogen, but does not provide the required amount of protein for muscle growth.
If there are few calories and the BJU ratio is not entirely correct, then you can forget about the growth of muscle mass.
The number of repetitions IS NOT IMPORTANT AT ALL, I talked about this. Be sure to read it.
The number of sets depends on your training program and fitness level. For beginners, it is enough 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 sets, in high-volume training a little less. All this is individual, but generally speaking, the higher your fitness level, the more work sets 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 adhere to both! You can gain the required calorie content if you only eat chocolates, but is that right?
The number of calories speaks about the amount of energy received, and the BJU speaks about the ratio of the received nutrients, from which further vital activity will be built.
How to gain lean muscle mass, I told in articles.
Everything is very brief and concise here) We have already talked about nutrition in the articles, the links to which I gave above.
We analyzed the set of lean muscle mass in my last article (link to it just above). Everything is detailed there.
If you want something sweet, you can afford it, but taking into account the total daily calorie intake and, preferably, before training.
A clear relief on the legs is taken from two things:
- Hypertrophy of the leg muscles.
- Decrease in the amount of body fat.
With the first point, everything is simple, swing your legs and a relief will appear.
The second point needs to be explained. You cannot lose weight only in " the right places", Fat burning in our body is triggered by HORMONES that circulate throughout the body, triggering 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 of the second type), through which hormones interact with them.
Women have a fairly large number of alpha-2-adrenergic receptors in their thighs, so it is more difficult to lose weight in these parts of the body.
But there is nothing to do but gradually reduce the calorie content of the diet in order to cause fat burning (then there is no question of gaining mass). You can also use. This is a cool supplement that will help with weight loss and increase your sexual desire a little.
The question is not a little about weight gain, but it's okay.
The best nutritional option for fat burning for beginners, I think, is simply the so-called proper nutrition.
The question is very relevant and interesting. So much so that I will definitely write a separate article on this topic in the near future.
And so, for the very beginning, the program is suitable.
The basic principles remain the same, namely:
- Load progression.
- A gradual increase in the diet.
- The main load falls on the lower body (because there are more muscles).
- The use of microperiodization is mandatory (due to the menstrual cycle).
As for whether you gain fat or muscle, I said above. The most accurate method is bioimpedance body analysis, 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 activity, for example. The growth of muscle and adipose tissue (mainly).
Dmitry, thanks for the kind words! Very nice.
A similar food system (and more than one) will be in my new product, very soon, and even more. I'll tell you a secret. Will be painted ABSOLUTELY EVERYTHING! Completely!
And so, this is a topic for a separate article, at least.
For now, just try to figure out your starting point and start gradually increasing your calorie intake.
Michael, hello! I am very glad that progress is being made. It's hard to say, but, most likely, your muscle growth has already begun.
Your goal is quite real. I'm sure you will succeed.
I included you on the preliminary list.
The course will be very cool! I have not done anything like this, and I have not seen anywhere else.
Hi Alex!
This is real. It is necessary to focus on exercises in frames and simulators. Try the hack squat, bench press. Gradually strengthen the lower back with hyperextension.
I also had problems, but with a knee, did leg presses and grew great. You just need to probe a little what works specifically for you.
Burning fat and gaining muscle mass at the same time is almost impossible to achieve (without stimulants).
If we are talking about natural training, then first I would lose weight up to 10-12% of body fat (when the abs are clearly visible, etc.), and then I already began to gain high-quality muscle mass, through the progression of loads and a gradual increase in the 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 to supplement my new course on muscle gain. Thank you very much!
Don't stop growing and improving, friends.
Subscribe on my instagram and other social networks.
P.S. Subscribe to blog updates... Further it will only get steeper.
Regards and best wishes!
comments powered by HyperComments At first it seemed that the problem could not be solved. I got to 11 balls when dividing the original pile into smaller ones: 3-3-3-2.If the first two heaps are equal to 3 \u003d 3, then we compare any three balls from them with the third, if again equality, then the required ball in the remaining two is in 1 weighing with any ordinary ball.
If at some of the previous stages there is an inequality, then by weighing any of the unequal heaps with three ordinary balls, we find both the required heap of 3 balls and the ratio of the weights. And then it is solved 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 rest.
Accordingly, 9 +, 2; 27 +, 3.
You can try to iterate over the options. Let's number the balls as indicated in the solution: 1,2,3, ..., 12.
1. Weigh any 2 balls. There is a good option when the desired ball is one of these two balls, but there is a bad option. Next, we will consider bad options.
It turns out the problem 10-, which cannot be solved in 2 weighings in any way (in 2 moves a maximum of 9+ is solved).
2. Weigh 1.2 and 3.4. In a bad case, the problem is reduced to 8-, which also cannot be solved in 2 moves.
3. 1,2,3 and 4,5,6. In case of inequality at any stage, the problem is solved, as indicated above. In the bad case, after two equalities 1,2,3 \u003d 4,5,6 and 1,2,3 \u003d 7,8,9 we come to the problem 3-, which is not solved for 1 remaining move.
4.1,2,3,4 and 5,6,7,8. If it is equal, then in the remaining 4 balls the desired one is found quite simply by means 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 equal, then any of 11-12 with any of the usual 1-10.
If inequality, then 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 equal, then the required ball is 12.
If the 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 is solved in 1 move.
If in the first weighing there is an inequality, then, at first glance, the problem is not solved. This is discussed below.
5.1,2,3,4.5 and 6,7,8,9,10. In a bad variant, we get an inequality and the problem is not solved in the remaining 2 moves (it will take 1 move to identify the required group of 4 balls, and problem 4+ cannot be solved in one remaining move).
6. 1,2,3,4,5,6 and 7,8,9,10,11,12. In a bad case, in 2 moves we only know the group of 6 balls where the required ball is. Problem 6+ cannot be solved in the remaining turn.
In option 4, I was initially embarrassed by the fact that in the case of inequality in the first weighing, it was not possible to reduce the problem to 3+ in 1 move. The usual way: dividing any of the heaps 1-4 and 5-8 into two 2 balls each and weighing them gives a 4+ problem in the worst case. And for 1 remaining move it is not solved.
The given solution contains an indication of how to proceed and resolve this issue. You can use the proposed designations or just think logically.
It is necessary to redistribute groups 1-4, 5-8 so that no more than 3 balls remain in the logically separated subgroups. And we have 3 possible readings of the scales: \u003d,\u003e,<, которые могут указывать на искомую группу.
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 regular ones (6-8 are replaced with any three of 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 moved to another bowl or replaced. That is, if the previous attitude is observed, then the required ball is in those that remained in place, and this is 2,3,4. The task was reduced to 3+.
b) If the ratio changed to equilibrium, then this means that the desired ball has been removed from the scales. Then this is an indication of 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 balls 1 and 5. By weighing any of these balls with any ordinary (2-4 or 6-12) balls, 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 \u003d 5,6,7,8).
Gazalova Victoria and Popova Marina
This paper presents interesting methods for solving problems of transfusion and weighing. This material can be used in preparation for the Olympiads in the subject.
Download:
Preview:
- Updating
- Weighing tasks
- Transfusion tasks
- Conclusion
- Literature
The relevance of research
Mathematical problems for pouring and weighing have been known since antiquity. Now they can be found in Olympiad problems or in computer games - puzzles. The classical counterfeit coin (FM) problem has recently found application in coding and information theory - to detect errors in a code. The purpose of our work is to find and describe algorithms for solving such problems. Transfusion and weighing problems are of the combinatorial search type; their solution comes down to working with information.
In the course of the study, it turned out that there are a lot of different subjects for these tasks. Therefore, we looked at the most common plots for each species.
Weighing tasks.
Weighing tasks are the type of tasks in which it is required to establish one fact or another (select a fake coin among the real ones, sort a set of weights in increasing weight, etc.) by weighing on a beam scale without a dial. Coins are most often used as objects to be weighed. Less commonly, there is also a set of weights of known mass.
Very often, a problem statement is used that requires determining either the minimum number of weighings required to establish a certain fact, or 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 formulation is not very successful, since with a positive answer to the question, the problem is most often reduced to the construction of an algorithm, and the negative one almost never occurs.
The search for a solution is carried out by means of comparison operations, moreover, not only single elements, but also groups of elements with each other. Problems of this type are most often solved by the method of 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 problems using weights.
Weighing problems with weights.
Weighing tasks on scales without weights.
Task 1.1 The most classic puzzle game ever.
One of the 9 coins is counterfeit, it weighs lighter than the real one. How to determine a counterfeit coin (FM) in 2 weighings?
Decision. The key idea for solving such problems is the right one.trisection , that is, 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 FM.
We weigh coins 123 and 456, putting 789 aside.
If 123 is lighter, then FM is among them; heavier then FM among 456; are equal, then FM is 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 false (K is more than two). It is known to be lighter than the real one. What is the smallest number of weighings that FM can be found?
Decision.
ALGORITHM 1. Put on the bowls in K: 3 coins, set aside the rest (if the number of coins is not a multiple of 3, then put on the bowls the same number of coins equal to (K-1): 3 or (K + 1): 3, depending on which one is natural). Further, if one of the bowls outweighed, then FM on the other bowl, and in the case of equilibrium - FM among the deferred ones. Then we repeat this for a group of coins, among which there is FM.
FM in the condition may be heavier than the real one, in this case we also argue, only the FM coin will be on the bowl that outweighed.
Consider a problem with weights where this rule can also be applied.
Problem 1.2 There are 9 standard weights weighing 100,200, ..., 900 g. One of them was in the hands of dishonest traders and now weighs 10 grams. less. How to find it in 2 weighings?
Find two different triplets 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. Reasoning also, we find a group with a damaged weight. Then you can find the spoiled weight by adding the deliberately real ones. For example, 200 + 600 and 700 + 100.
The next problem differs in that it is not known beforehand - it is easier or heavier FM than the real one.
Problem 1.3 One of the three coins 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?
This problem has 6 possible answers (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 with markings 1d, 2d, 3d, 4d. One of them is defective - lighter or heavier. Is it possible for two weighings to find this weight and determine whether it is lighter or heavier than the real one?
There are 8 answers here. Weigh 1g + 2g and 3g, then 1g + 3g and 4g.
We get the following table of options:
The answer is yes you can.
Generalization 2. Let there be K coins and one of them is false. For what is the smallest number of weighings that FM can be determined and is it lighter or heavier?
First you need to find out the number of answer options. Their K * 2, since each coin can be lighter or heavier. Then we determine the number of weighings. One weighing determines three options:, \u003d. Two weighing determines 9 options:, \u003d,\u003e \u003d, \u003e\u003e, \u003d\u003d (there are 3 * 3, but in this problem the \u003d\u003d option is not possible). Three weighing determines 3 * 3 * 3 \u003d 27 options, etc.
ALGORITHM 2. We divide coins into three groups. If K is not divisible by 3, then either (K-1) is divisible by 3, then we put (K-1): 3 coins on the scales and (K-1) will remain: 3 coins and 1 more coin. Or (K-2) is divided by 3, then we put on the scales by (K-2): 3 coins and (K-2) will remain: 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. If the scales turned out to be in equilibrium in both cases, then FM in the deferred coins and then, according to the number of deferred coins, in one or two weighings we will find FM and it is lighter or heavier than the real one (comparing them with real coins). Further, if FM was not in the deferred coins, then we can already determine whether it is lighter or heavier than the real one. And then we act according to Algorithm 1. Having designated the groups of coins 1, 2, 3, we show the weightings 1 and 2 then 1 and 3 in this table.
Knowing whether it is heavier or lighter than the real FM, we can use the algorithm1 described in generalization 1. As you can see, here the division into three as equal parts as possible.
Let's check the algorithm with more coins.
Problem 1.5 There are 80 coins, one of which is counterfeit. What is the smallest number of weighings on a scale without weights that a counterfeit coin can be found?
Decision. We carry out the first weighing: put on the bowls at (80-2): 3 \u003d 26 coins. In the case of equilibrium - FM among the remaining 28;weighing real 26 coins with 26 "suspicious" ones, we will determine whether FM is lighter or heavier than real(in case of equilibrium, it is in the remaining two and then 2 more weighings are needed). If during the first weighing the scales were not in equilibrium, then a fake one - 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 counterfeit one by 9, 9, 8, weigh, then weigh 3 coins, and then one at a time.
Answer: for 5 weighings.
Algorithm 1. 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 by 3.3 and 4 9 to 3.3 and 3 | 3 by 1.1 and 1 4 by 1,1 and 2 | 2 to 1 and 1 |
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10 over 3.3 and 4 9 to 3.3 and 3 | 3 by 1.1 and 1 4 by 1,1 and 2 | 2 to 1 and 1 |
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K is a multiple of 3 | K: 3 K: 3 K: 3 | divide similarly |
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K: 3 with stop. one | (K-1): 3 (K-1): 3 (K-1): 3 + 1 |
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K: 3 with stop. 2 | (K + 1): 3 (K + 1): 3 (K + 1): 3-1 |
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- If there are 2 or 3 coins, then 1 weighing is required to find a false coin among them.
- If the coins are from 4 to 9 inclusive, then the smallest number of weighings to find a counterfeit coin is 2.
- If coins are from 10 to 27 inclusive, then it is equal to 3.
- If the coins are from 28 to 81 inclusive (due to the fact that 81 \u003d 3 * 27), then the smallest number of weighings is 4.
Regularity ... The numbers 9, 27, 81 are consecutive powers of a triple, and the numbers 4, 10, 28 are, respectively, the previous powers of a triple, increased by 1: 4 \u003d 3 + 1, 10 \u003d 32
+1, 28 = 3
3
+1.
Algorithm 2. In the 2nd weighing, 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 |
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9 to 3.3 and 3 | 3 by 1.1 and 1 | ||||
9 +1 | 10 by 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 to 1 and 1 |
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9 +2 | 10 over 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 to 1 and 1 |
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K is a multiple of 3 | K: 3 K: 3 K: 3 | K: 3 K: 3 K: 3 | If in the first or in the second case the scales were not in equilibrium, then it is possible to determine the group of coins containing FM, and also to conclude whether it is lighter or heavier than the real coin. Next, we act according to Algorithm 1. (otherwise *) | In general, let the number of coins k satisfy the inequality |
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K: 3 with stop. one | (K-1): 3 (K-1): 3 (K-1): 3 + 1 | (K-1): 3 (K-1): 3 (K-1): 3 +1 |
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K: 3 with stop. 2 | (K-2): 3 (K-2): 3 (K-2): 3 + 2 | (K-2): 3 (K-2): 3 (K-2): 3 +2 |
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* In the second weighing, we find the group of coins containing FM. If in 1 and 2 weighings the scales were in equilibrium, then FM is among the remaining one or two. If there is 1 coin left, then it is FM and weighing it with the real one, we find out whether it is lighter or heavier than the real coin. If there are 2 left, then weighing them among themselves, and then one of them with the present, we answer the question of the problem. If in the first or in the second case the scales were not in equilibrium, then it is possible to determine the group of coins containing FM, and also to conclude whether it is lighter or heavier than the 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 false coin among them.
- If the coins are from 4 to 9 inclusive, then the smallest number of weighings to find a counterfeit coin is 3.
- If the coins are from 10 to 27 inclusive, then it is equal to 4.
- If the coins are from 28 to 81 inclusive (due to the fact that 81 \u003d 3 * 27), then the smallest 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 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).
Transfusion tasks.
Description: having several vessels of different volumes, one of which is filled with liquid, it is required to divide it in some ratio or to pour out any part of it using other vessels for the least number of transfusions.
In tasks for transfusion, 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 believed that
All vessels without division,
Do not pour liquids "by eye"
It is impossible to add liquids from nowhere and drain anywhere.
We can tell exactly how much liquid is in the vessel only in the following cases:
- we know that the vessel is empty,
- we know that the vessel is full, and the problem gives its capacity,
- the problem gives how much liquid is in the vessel, and no transfusions were made using this vessel,
- two vessels participated in the transfusion, in each of which it is known how much liquid was, and after the transfusion, all the liquid was placed in one of them,
- the transfusion involved two vessels, each of which knows 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 was left in another vessel.
Most often, a verbal solution is used (i.e., a description of the sequence of actions) and a solution method using tables, where the volumes of these vessels are indicated in the first column (or row), and in each next column - the result of the next transfusion. Thus, the number of columns (other than the first one) indicates the number of transfusions required. The same methods (verbal and tabular) were used in solving weighing problems. However, we found another interesting waythat can solve such problems. This is the mathematical billiards method. ME AND. Perelman, in his book "Entertaining Geometry", proposed to solve 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 two sides of which are numerically equal to the volume of two smaller vessels. Further, from an acute angle of this table along one of the sides, you need to "launch" a ball, which, according to the law "angle of incidence equal to the angle reflections "will collide with the sides of the table, thus showing the sequence of transfusions. On the sides of the table there is a scale, the graduation value of which corresponds to the selected unit of volume. As a result of the movement, the ball either hits the side 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 forming a grid on a parallelogram. After hitting the side of the parallelogram, the ball is reflected and continues to move along the side emerging from the point where the collision occurred, fully characterizes how much water is in each of the vessels.
An old entertaining task.
The eight-bucket keg is filled to the brim with kvass. Two must share the kvass equally. But they only have two empty barrels, one of which holds 5 buckets, and the other holds 3 buckets of kvass. The question is, how can they divide the kvass using only these three barrels?
In the considered to the problem, the sides of the parallelogram must have sides of 3 units and 5 units. Horizontally we will put the amount of kvass in buckets in a 5-bucket keg, and vertically - in a 3-bucket keg. |
Let the ball be at point O and after impact hits point A. This means that the 5-bucket keg is full to the brim, and the 3-bucket is empty. Reflecting elastically from the starboard side, the ball will roll up and to the left and hit the top side at the point with coordinates 2 horizontally and 3 vertically. This means that there are only 2 buckets of kvass left in the 5-bucket keg, and the buckets from it were poured into a smaller keg. Reflecting elastically from the upper side, the ball will roll down and to the left and hit the lower side at the point with coordinates 2 horizontally and 0 vertically. This means that there are 2 buckets of kvass left in the 5-bucket keg, and the kvass was poured from the 3-bucket vessel into an 8-bucket keg. Reflecting elastically from the lower side, the ball will roll up and to the left and hit the left side at the point with coordinates 0 horizontally and 2 vertically. This means, from a 5-bucket keg, 2 buckets of kvass were poured into a 3-bucket keg. Reflecting elastically from the left side, the ball will roll to the right and hit the starboard side at the 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. Reflecting elastically from the starboard side, the ball will roll up and to the left and hit the top side at the point with coordinates 4 horizontally and 3 vertically. This means, from a 5-bucket keg, 1 bucket of kvass was poured into a 3-bucket keg, where there were 3 buckets, and 4 buckets remained in a 5-bucket. Reflecting elastically from the upper side, the ball will roll down and to the left and hit the lower side at the point with coordinates 4 horizontally and 0 vertically. This means that there are 2 buckets of kvass left in a 5-bucket keg, and kvass was poured from a 3-bucket keg into an 8-bucket keg. The problem was solved with 7 transfusions. At the same time, we fill in the table:
Transfusion No. | ||||||||
8 l | ||||||||
5 l | ||||||||
3 l |
Let's see how our billiard ball will behave if we first fill a 3-bucket keg with kvass.
It is clearly seen that this task was solved as a result of 8 transfusions.
Let's solve the famouspoisson's 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 hesitated in his choice life path, a friend showed him the texts of several tasks that he could not cope with himself. Poisson solved them all in less than an hour. But especially to him |
i liked the problem about two vessels. “This task determined my destiny,” he said later. - I decided that I would certainly be a mathematician
A task. Someone has 12 pints of wine and wants to give half of it. But it doesn't have a 6 pint vessel. He has 2 vessels. One at 8, the other at 5 pints. The question is, how do you put 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. Horizontally we will set aside the amount of wine in the vessel at 8 pints, and vertically at 5 pints. We reason in a similar way.
12 l
5 l
8 l
It turns out 7 transfusions. However, if poured first into a 5 pint vessel, 18 transfusions are required.
Do problems of this type always have solutions?
The billiard ball method can be applied to the problem of pouring liquid using no more than three vessels. If the volumes of the two smaller vessels do not have a common divisor (that is, they are mutually prime), and the volume of the third vessel is greater than or equal to the sum of the volumes of the two smaller vessels, then using these three vessels you can measure any whole number of liters, starting with 1 liter and ending with the volume middle vessel. With, 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 divider. 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 of the rhombic table must be cut off. Then the ball can hit any point from 1 to 9, except for point 6. Despite the fact that 7 and 9 are mutually simple, 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 in 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 mathematical billiard method to the case of four vessels is reduced to the motion of a ball in a spatial domain (parallelepiped). But the difficulties arising in the image of 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 research work:
1. Collected theoretical and practical material on the research problem.
2. Based on the results of this work, we have systematized the tasks for transfusion and weighing.
3. Compiled algorithms for the solution.
4. A presentation was made to familiarize classmates with these tasks and to help them prepare for the Olympiad.
Thus, we can conclude that the work we have done turned out to be fruitful, the students got acquainted with the ways and methods of solving problems for weighing and transfusion. We learned how to correctly apply the best ways to solve them. According to the students' feedback, the work done allowed them to master the methods of solving transfusion problems and broadened their horizons. The students noted the possibility and practicality of using the billiard method when 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
1. Galperin GA, Mathematical billiards - M .: Nauka, - 1990.- 290s.
2. Galperin GA, Periodic motion of a billiard ball / Kvant. 1989. No. 3.
3.F.F.Nagibin, E.S. Kanin Matematicheskaya casket M .: Education, 1988
4. Ya.I. Perelman Entertaining geometry Moscow: GIFML, 1959
5.V.N.Rusanov Mathematical Olympiads junior schoolchildren M., Education, 1990
6. EPKolyada Development of logical and algorithmic thinking of students // Informatics and education. 1996. N1.
7. IF Sharygin Mathematical vinaigrette M., AGENCY "ORION", 1991
8.http: //www.i-u.ru/biblio/archive/makovelskiy_logic_history/4.aspx (website of the Russian Humanities Internet University, article history of logic)
9.http: //ru.wikipedia.org/wiki/ (WIKIPEDIA-modern encyclopedia)
10.http: //wiki.syktsu.ru/index.php/ Methods for solving logical problems.
11. Bayif Zh-K. Logical tasks. Moscow: Mir, 1983.171 p.
12. Bulk M.B., Bulk G.D. Mathematics after school. M .: Education, 1971.
13. Drums A.I., Chernyavsky I.Ya. Problems and exercises in mathematics. Saratov: Saratov University, 1965.234 p.
14. Barr S. Placers of puzzles. Moscow: Mir, 1978.414 p.
15. Berrondo M. Entertaining tasks. Moscow: Mir, 1983.229 p.
16. Ball W., Coxeter G. Mathematical essays and entertainment. M .: Mir, 1986. 472 s.
17. Perelman Ya.I. Entertaining arithmetic.
18. Perelman Ya.I. Entertaining algebra.
19. Perelman Ya.I. Interesting geometry.
20. Perelman Ya.I. Live mathematics.