Math and games

        The game which makes Math to be recreational can take different forms like: a puzzle to be solved, a game of a competitive nature, a magic trick, a paradox, a logical error or, simply, math with curious and fun features. Are these pure math or applied math? Is hard to say about this. In a sense the recreational math is a pure math which is not contaminated by utility. In another sense we can think of the recreational math as the applied math, because it satisfies the universal human need of amusement.
       Maybe this need of fun is hiding even in the back of pure math. It's not a big difference between the pleasure which a beginner tries by solving an ingenious math puzzle and the happiness felt by a mathematician which solves a tough problem of science. Both of them are pure beauty, clear order and net defined, also mysterious and charming, which is on the base of every structure. This is why we don't need to be surprised that often is hard for us to make a distinction between the pure math and recreational math. Let's take for example the four color theorem which is an important problem of the topology which is not yet solved, but with many references on many recreational math books. No one will have any doubts that the flexagones of paper are extremely fun toys, but their structure will lead us in an advanced field of the group theory and we know that the most technical magazines of math published a lot of articles about them.
       Creator mathematicians almost never feel shame about their interest in the recreational math subjects. Topology has its origins in the analysis made by Euler over a puzzle that concerned crossing some bridges. Leibniz consecrated a lot of time for the study over a game with the jump of the horse on chess table, which was and is a true delight and a new rejuvenation of trying your intelligence. The great German mathematician David Hilbert proved one of the most important theorems in the dissection games field. Someone that visited Albert Einstein one day said that he had a shelf full of books with games and math puzzles. The interest of these bright minds for the enigmatic math game is not hard to guess, because the creative thinking which is given to these kind of simple and commonplace subjects is of the same nature with the type of thinking which leads to the mathematical and scientific discovery. At the end we can ask ourselves: what else is math, if nothing but trying to solve puzzles? And what is science if nothing but a systematic effort for getting answers for the nature's puzzles, which are better and better? 
       The pedagogical value of recreational math is at present very popular. We see that is put increasingly more emphasis on it in the published magazines for math teachers and also in the school books, especially in those written in the "modern" way.  These subject matters are very interesting for the pupils and students. We can see that there is a fascinating connection between math and games  which lead us to the conclusion that math is not just long calculations and complex theory but also  imagination and insight. Thank you

How to solve a Math problem I

      1. We need to understand the problem.

        What is the unknown? What data do we know? What is the condition?
        Can the condition be satisfied? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or a contradiction?
        Let's make a drawing. Let's introduce corresponding notations. Let's separate the different parts of the condition. Can we write them in a mathematical language?

        2. Let's make a plan.

        Let's find the relation between the data we know and the unknown.
        Did we met this problem before? Or maybe we had to do with it in the past, but somehow in a different way?
        It's not excluded to be put in the situation of considering auxiliary problems if we are not able to find a direct relation. Do we know a related problem?
        Do we know a theorem which can be used here in our problem?
        Let's investigate the unknown! And let's think of a known problem which has the same unknown or a similar one.
        Eventually, we should get a plan of the solution. Let's say we have a related problem with ours and which was solved before. Can we use it? Can we use its result? Can we use somehow its method of finding the solution to our problem? Can we not introduce an auxiliary element for making it usable? Can we reformulate the problem? Can we reformulate it in another way? Let's go back to the definitions now. If we can't solve the proposed problem, we try first to solve a related problem. Can we imagine a related problem that is more accessible? A more general problem? A more particular problem? An analogue problem? Do we know to solve a part of the problem? Let's keep a part of the problem, omitting the rest. To what extent can be determined now the unknown, how can it vary more? Can we deduce something useful from the data of the problem? Can we imagine some other data which is proper for finding the unknown? Can we change the unknown or the data, or both of them, if it's the case, so that the new unknown to be more related to our new data?       
         All data had been used? The entire condition had been used? Was it taken into account all the essential notions involved in the problem?


        3. Let's make the plan.

        When we make the plan of the solution, we verify each step. Can we realize very clear that the step is correct? Can we prove that is correct?

        4. Let's analyze the result obtained.

        Can we verify the result? Can we verify the argumentation? Can we get the result in a different way? Can we realize this just if we take a look at the problem? Can we use the result or the method in another new problem?
        
        If we apply these steps to every problem we need to solve we will not fail for sure. I know this from my experience and we need to think also if we can find more than one solution to a proposed problem. Thank you 

About the theory of the probability II

        Let's go back now to the game of our distinguished French men, Antoine Gombauld. The question was if the number 24 was a number of throws which you can rely on in this game of throwing two dices and get at least a double of six. An old rule of this game said that the bank could afford to bet with equal chances that any player should get at least a double of six from 24 throws of a pair of dices. However, de Mere was not convinced by that and he believed that this was not the right number. It seems that he realized from his own experience that 24 was not a sure number to bet on. However this statement is not likely to be true because the different tolerances of the chances were so small that would have had someone to play a long time to get a satisfactory experimental demonstration. It's almost sure now that his conclusion was based on theory and not on practice.     
        We don't know for sure how de Mere found the correct solution to this problem. Today we can solve this very easy and we will see that very soon. The idea is that de Mere consulted back then if he was right or not, a young French, Blaise Pascal, which he met shortly before at a reception offered by another important French men. Pascal was a philosopher, theologian and mathematician and he solved the problem proposed by de Mere and showed that the chances were slightly unfavorable to the bank if the bet was for 24 throws and was slightly favorable if the bet was for 25 throws.
        Pascal solved also another problem which was much more interesting and harder and also proposed by de Mere. This second problem, already famous, has been discussed so many times but never solved correctly. It was called "the problem of the parts" or "the problem of the division". In general the problem was of this kind: in what way we could share the money between the players if from one reason or another, the games had to be interrupted before it was finished, the players having just partial results? Pascal introduced the important idea that the size of the price earned by a player in a partial game must depend on the probability of that player to win the game, in the case of ending it. Pascal showed with details how we can find the probability of one to win by knowing the nature of the game and the partial result of very player.

        De Mere questions and the solutions proposed by Pascal proved to be very stimulative and important. Pascal wrote about these problems to another famous French, Pierre de Fermat, which had a great reputation back then as a mathematician and also he was a distinguished jurist in Toulouse. The exchange of letters between these two mathematicians conducted to a deeper understanding of the mathematical aspect of these fortune games and become very popular those days. This way, in the elegant, but less recommended atmosphere of the game rooms, evoked in the correspondence of some eminent Frenchmen of the 17th century, was born "Mrs. Chance". Thank you.

About the theory of the probability I

People manifested interest and curiosity for probability and hazard since ancient times. The practice of the simple gambling games, the drawing, the fear and worry - which were inevitable in a world dominated by the whims of the gods - all of these make us believe that people back then, were forced to think over the random probability and fate and on something that happens by accident, even that their reasoning was very simple and wrong, if we judge by our moderns standards today.
        Is very strange that the mathematical theory of the probability has a well defined beginning that is easy to recognize. It was born in France more than three centuries ago in 1654. Antoine Gombauld, a distinguished French men, which was Knight of Mere, Sire of Baussay, was admired back then for his advices full of wisdom about the subtle things of life and also for his charming personality. Without no doubt he spent some of his time in the gambling houses and he had a strong curiosity for the gambling games.
        Back then was a famous gambling game which was played by a century and it's still played today. Let's describe the game now: "the bank" which is the professional player or the gambling house management, offers to bet on equal chances that a player will get at least once the face with number six from four dice throws. It's somewhat easy to find that this game was slightly favorable for the bank and not for the player. We can check in fact that the bank will win on average in 671 times and loses in 625 times. However, the Knight of Mere was not interested in this game, but in a more similar complicated one. Let's say that instead of throwing just one dice we have a pair of dices. Now we need to ask ourselves as Antoine did back then: why is not favorable for the bank to bet that the player will get at least a double of six from 24 throws? Why was he interested in finding this? We probably because he discovered in this game example a contradiction between a result he got from applying some probability theory and a result which was somehow different and was given by an old rule of the players.
        This rule determines in a dice game the critical number of samples or throws that is the number of throws from which the chances are changing from bad (for a smaller number of throws than this critical number) to favorable (for every number bigger than this critical number). The rule applied in this case says that the critical number of throws for a game with a single dice is four and for a game with a pair of dices is six times four which is twenty four. The multiple of six appears because by the rule if you throw two dices, then you have six times more ways of getting a result in comparison with the game with one single dice. So what was the finding of Antoine Gombauld, how did he do it to get a favorable result for this game and was 24 a number of throws which you can rely on in this game? You can find more by reading the next article. Thank you

Probability and the future

   

        The theory of probability and mathematical statistics became today an important and even indispensable part in our lives. The economy and insurances use largely probabilistic laws. Every science that is based on measurements is inevitably interested in this branch of the theory of probabilities known as "the theory of errors". In the last analysis, physical world was found to be of probabilistic nature. Some of the fundamental aspects of biology are also probabilistic. Any call on experience, using selection processes, depends, for its interpretation, on the statistical theory. Many of the judgments and decisions that we all have to take daily are based on a weighing, conscious or intuitive - and for sure more intuitive - of probabilities. Even the courage, like Socrate said, is the knowing the reason of fear or hope and therefore depend(s) of the probabilistic estimations of the risks implied in our fears and hopes.        
        However, despite its importance, those who fall within the responsibility of education issues have not yet recognized as it should the universal significance of the theory of probability and mathematical statistic. Without a doubt, the mathematicians will get in the coming years deeper, stronger and more general results in every of these domains. It seems very likely that all areas of social sciences, as they will be more solidly grounded, will use increasingly more the theory of the probabilities. The question of legal judgments have been examined by the probabilistic people since the time of publication of the anonymous article, "A calculation of confidence in the testimony of people" which appeared in England in 1699 in a number of the magazine "Philosophical transactions". With little more than a century later, the Frenchman Condorcet wrote an essay called "The essay on probabilistic analysis applications to taking multiple decisions".
There was an article that illustrates how interesting and important are the applications of the theory of the probabilities in the functioning of a democratic system.       
        If you don't feel weird to end this at somewhat humorous note, let's consider the following problem: "If A, B, C and D tell each the truth (independently) just in one case from three and if A say that B deny that C declares that D is a liar , then what is the probability that D to tell the truth?". This problem, which is called "the truth of the four liars" is not too complicated. Sir Arthur Eddington used it in a book that he wrote in 1935, but he gave a solution back then which subsequently confessed that it was not correct. For this problem to have rational sense, we have to give some additional information. The solution of Eddington, of 25/71, is valid only if we add a condition that is a little ridiculous. With the additions that seem most reasonable, the solution is 13/41. You can make a probabilistic thinking exercise for this problem.
        We do hope that this scientific progress will accompany the progress in education that elements of the probability theory will be included in the high school programs and the extremely interesting results of this one will replace outmoded various mathematical methods, of purely historical interest that are still in use. We do must hope also that secondary schools will have better care for those wishing to study physical sciences, biological, medical or social, to know the foundations of the theory of probability and mathematical statistics - disciplines that have proved their worth and have matured fully. Thank you.

Geometric probabilities

        Let's think about of a segment of line, which has 10 cm length. And let's put a random point on this line - i.e. so that this point to have equal chances to fall in one place like in any other. What is the probability that this point to fall in the middle of this segment line? The number of points on this segment of line is infinite - or, more clearly, as the segment is divided into segments increasingly smaller, the number of these segments increases indefinitely. Here, we don't deal with a finite number of cases. It has little sense to seek the likelihood that the point will fall exactly in the center. And this is because we cannot decide if this event ever occurred or not. However, is there a logical sense to ask ourselves what is the probability that the point to fall in a segment of line of 0.01 cm length, placed in the center? It is clear that a reasonable answer would be 0.01/10.      
        A beautiful problem of geometric probabilities is the next one we propose. Let's consider a segment of line AB and a point placed somewhere at the right of the middle of our segment - that divides the segments in two parts of lengths "a" and "b", with "b" smaller than "a". Now let's choose a point C at random on the longest segment, and another point D, also at random, on the shortest segment. These two points, C and D, divide our initial segment of line in three parts - AC, CD, DB. What is the probability that with any of these three segments to build a triangle? This problem is not so simple, but the answer "p=b/(2a)"is, instead, very simple and elegant.
        Another classical problem of geometric probabilities is the problem of Bufon. We suppose that on a plane surface parallel lines are drawn at the distance "a" one of another (as cracks on a floor planks). We throw on this plane, at random, a bar (a needle or match needle) of a length "c" that is smaller than "a". What is the probability that the object thrown to intersect one of the lines? The answer is "2c/(πa)". Therefore, throwing the bar a large number of times N and noting the number of crossing with "m", can be calculated experimentally the value of π, with the formula "π=2cN/(am)". At first sight it seems almost magic that the answer involve the number π. However, do not forget that π is always related with the circles and the angles. If the center of the needle or the needle match falls in a random point, its extremities can fall anywhere on a circle which surrounds that point because from the conditions of our problems, it's equal probable to have an orientation as well as any other. This way the circle comes in the problem - and with him also π.
        With more than 100 years ago a certain teacher named Wolf from Frankfurt he had the patience to throw a needle 5,000 times; the needle had 36 mm length and the plane was crossed by parallel lines at the distance of 45 mm one from each other. He observed that the needle had crossed the lines for 2,532 times; from here we get an approximation of the value of
π (with only 0.6% bigger than the exact value of 3.1416). At the end of the last century it was said that a certain captain Fox has made over 1,120 throws which gives a value for π of 3.1419, and later a mathematician named Lazzarini made 3,408 throws getting a value of 3.1415929, which differs from the real value π=3.14159265 at the seventh decimal. It's easy to guess that it was something strange in the experiment of Lazzarini. Thank you

Games and Probability I

        Someone said with 100 years before Christ: "The king should ban gambling and betting in his kingdom because these vices can destroy kingdoms." And he was soooo right. We cannot deny that Mrs. chance was born in the halls of game of the 17th century. The early beginning of the mathematical theory of probability consisted to a large extent into solving problems proposed by the fortune games.
        The fortune games represent much more than what the probabilities which are implied into these, can say. These games can be fun and interesting, they can make gains or result in disaster; they may give rise to a dinner with champagne or bring a pearl necklace, but can also bring suicide; they can be a very casual fun in a relaxing moment, in reasonable spending limits for fun, but sometimes they can become an addiction leading to redness of money which are perhaps necessary for more serious purposes.
        Although the players learned without no doubt so much from the theory of probability, it is clear that they have never learned or accepted the main lesson which the theory gives regarding the fortune games. And this lesson says briefly: "if you continue to play you will inevitably lose." Let's talk about game systems. The book of Levinson "The science of chance" contains a very interesting and detailed analysis of multiple game systems, including the game called "Martingale" in which the bet doubles every time the player loses and comes back to the original normal bet after every win. Suppose you are using this system of doubling the bet in a game with a coin, the success being considered the tail; let's say the normal bet for every play will be one dollar.  if the tail comes at the first throw you win one dollar, else if the first throw is a head then the second throw is a tail you will lose one dollar at the first play and win two dollars at the second play - so you will have a net gain of 1 dollar. If you get head the first 2 times and the third one you get the tail, then you will have three dollars loss in the first two throws, but win four dollars in the third one so again you will have a net gain of one dollar. No matter how many times the head will come up in the first throws when you get a tail the net gain will be of one dollar. The total gain will be given by the product of one dollar and the number of tails you get during the game.
        All seems beautiful until here. However keep in mind two aspects of the law of great numbers and these are: the first one is that the relative frequency of getting a tail or head comes closer to 1/2 and the second aspect is that the difference between the absolute frequencies of getting the tail and the head tends to increase. So the excess of tails as of the heads can be big, if you are lucky, when the game stops and you will win a number of dollars which is slightly higher than half of the number of throws. However also it is equally probable that the number of heads to exceed so much the number of tails. What is the translation of this to the player? Except the case when you have very large resources and the courage to engage them against the bad luck that is after you during the game, you will be clearly ruined. Doubling the bet every time you'll come up with enormous amounts of money. Thank you

Invention and discovery in Math

       

        A great discovery solves a great problem, but you can find a grain of discovery in the solution of every problem. You can have a modest problem in front of you; but if it will arouse your curiosity and put into play your faculties of inventiveness, and if you solve that problem by your own without any external help, then you can feel the tension before discovery and you can enjoy the triumph of its creation. Such experiences, at an age where you have maximum receptivity, can create the taste for the intellectual work putting its imprint on the mind and the character for life.
        In these conditions the Math teacher has great possibilities. If he fills his time by reprimanding his students with routine operations, he kills their interest, he hampers their intellectual development and he uses really bad his great possibilities. But if he arouse their curiosity, proposing problems which are proportional with their knowledge and if he helps them to solve the problems by stimulating questions, he can inculcate them a taste for independent thinking and suitably develop their aptitudes.
        The student whose program also includes a math course, has big possibilities. Of course, these possibilities loose themselves, if he considers Math just an object in which he just needs to give a certain number of exams and and that he will forget as soon as possible after the graduation exam. These possibilities also can be lost in the case when the student has
certain natural skills for mathematics, because, just as any other one, he must discover his talents and his tastes; he doesn't know if he likes the pudding with raspberries in case he never tasted before. But he can get to find that a math problem can be just as captivating as solving a crossword puzzle or that a strained intellectual work can represent an exercise so appropriate as a tennis match. After he has tasted the pleasure of mathematics, he will never forget it easily and exists then big chances that mathematics may come to mean something to him: an amateur concern, an instrument in his profession, maybe even his profession or a big ambition.
        When I was a student I was eager to understand a little bit of Math and Physics. I was listening to lectures, reading books, I was searching to assimilate solutions and facts that were presented to me, but there was a question which always troubled me: " Yes, the solution seems to work, makes the impression that is correct; but how can be invented such a solution? Yes, this experiment seems to achieve his goal, it shows like a fact; but how can people discover such facts? And how can I invent or discover myself this kind of things? ". I imagine or hope that some of the students with more thirst for knowledge put themselves similar questions and, for this reason, I am trying to satisfy their curiosity. Searching to understand the solution of such and such problems, also the motivation for the solution found and the processes by which it was obtained, I can explain to others also my findings. The interest to discover or invent seems more widely spread than at first glance. The space that the newspapers and the magazines for the general public give to crossword puzzles and other games seems to show that people spend some time to solve problems without practical value. At the base of this wish to solve different problems which do not give material advantages can stay a more profound curiosity, a desire to understand the ways and meanings, themotivations and solution procedures. Thank you

Teaching math in the classroom

          

       1. Let's help the student. One of the most important tasks for a teacher is to help his students in the right way. This task is not so easy as it seems; it requires time, experience, dedication and healthy principles. The student must gain a larger independently work experience. If he is let alone with his problem, hot helped or helped partially and insufficient, he might not progress at all. If the teacher helps too much, the student has nothing left to do. The teacher must help, but not too much and not too less, so that the student to have a rational part of work for himself also. Mo matter if the concerned student is not able to do a big deal, the teacher must give him the illusion that he works independently. To realize this, the teacher must help his student in a discrete, not annoying way. The best is that the student to be helped in a natural way. The teacher must place himself in the student position, to see his learning difficulties, to try to understand what happens in his mind and to put him a question or to show him one step that he could also had in mind.
        2. Questions, recommendations, intellectual operations. Seeking to help the student in an efficient way, but in a natural and discrete way, the teacher is headed to put always the same question and to show the same steps. This way, in countless problems, we need to ask: "Which is the unknown in our problem?". We can change the words and ask the same thing in a different way than before: What is required? What do we want to find? What do we need to search? The goal of these questions is to concentrate the attention of the student over the unknown. Sometimes we get the same effect in a more natural way by a recommendation like: "Let's check the unknown!" The question and the recommendation follow the same result; they tend to trigger the same mental operation. It is useful to gather and group the questions and recommendations typical worthy for the student when a problem is discussed with him. The list we need to study contains questions and recommendations of this kind, chosen and ordered carefully; they are just as useful for someone who solve problems working individually. The list enumerate, not in a direct way, intellectual operations typical helpful in solving problems. These operations are passed in the list according in the order they occur most frequently.
        3. The generality it's an important characteristic of the questions and recommendations which are in our list. Let's take the following questions: "Which is the unknown in our problem we need to find? What are the data we know in our problem that we can use? Which is the condition in our problem we need to respect? " . These questions have a general application, we can formulate them with good results in any kind of problems. Their benefit is not limited by the subject of the problem. We can have an algebra or geometry problem, mathematical or not, theoretical or practical, a serious problem or just simply a game; no matter its state, these questions keep their sense and can help us solve the problem. It's true that there is a restriction or limitation, but it has nothing to do with the subject of the problem. Some questions and recommendations from the list are applicable only for the problems in which we need to find something and not for the problems in which we need to prove something. If we have a problem of the last kind, we must take in consideration other questions. We will continue. Thank you