THE PROBLEM OF EIGHT CASES
According to an ancient legend, the famous Caliph Al Motacém Billah, king of the Arabs, called one morning the crafty Sabag, his vizier, treasurer, and said gravely, as if dictating a life sentence:- Within a few hours, my dear vizier, will receive a visit from young Beremisz Samir, nicknamed "The Man Who Counted." Do not ignore, of course, that this talented Beremiz has dazzled our glorious Baghdad with clear statements of his incomparable wit and his very acute intelligence. The most intricate puzzles, calculations are more difficult, by an eminent mathematician, explained and resolved in quick time. It is my desire to present the illustrious Beremiz large sum. I would, however, also try the much praised in calculating shrewdness to offer him during our interview a problem that is related in some way, with the prize that will give you gold coins. One problem that has left our visitor enchanted, true, but also perplexed and confused.
The vizier Sabag was not the man who let himself be entibiando to the whims and fancies of the powerful Emir. After listening, downcast and thoughtful, the king's words, raised his tanned face, gazed serenely glorious caliph, and thus said:
- I hear and obey, O Prince of Believers! From the tone of your words, I guess perfectly the course followed by the caravan of your intentions. Is your desire to reward a valuable amount wise geometer. It appears, in this intention, the unparalleled generosity of heart. Would you, however, that this award is a problem with exornada original and unpublished, can surprise the most ingenious mathematicians and delight the most delicate of philosophers. This memory brings out the elegance of your attitudes, because the visitor to be argued before the court, could once again demonstrate the strength of his ingenuity and power of their culture.
Uttered these words, the vizier retired to his room to work. After some time, returned to the king, preceded by two Nubian slaves who drove heavy silver tray. Rested on the tray eight wooden boxes, all the same size, numbered from one to eight.
There was little astonishment of the caliph of Baghdad to see that particular apparatus. What would be the reason behind those boxes numbered one through eight? What a mystery in the field of accounts and calculations, they could get involved? Cheiques and nobles, who stood beside the king, looked at each other aghast.
It was up to honored Sabag, minister of the court, explain why this strange preparation. Let us hear, therefore, the report made by the worth vizier:
- Each one of these boxes contains a number of currencies. The total contained in the boxes is the prize that will be offered to the calculator. The boxes, as you can see, are numbered from one to eight, and arranged according to the number of coins that each contains. For this arrangement of the boxes, I adopted the ascending order. Thus, the box designated by number 1 contains the smallest number of coins, comes after that is indicated by the number 2, then the number 3 appears, and so on until the last, which has the largest number of coins. To avoid any doubt, let me say straight away that you can not find two boxes with the same number of coins.
The caliph, seriously intrigued, challenged the vizier:
- I do not understand, O eloquent Sabag, that problem would be formulated with these dinars allocated for eight boxes. By Allah! I do not understand!
The vizier Sabag, when young, was a schoolteacher and had learned, before the classes, teaching the illiterate, to clarify the doubts of less Attila and resolve the issues suggested by the smartest. Firmly resolved to elucidate the glorious king, the old schoolmaster spoke thus:
- I should say, O King of Time, that the dinars were not distributed at random to the eight boxes. Each box contains a number of currencies. Altogether there are therefore eight dinars in amounts. With the amounts distributed to eight boxes, can make any payment, provided a dinar to the total number contained in eight boxes, without having to open any box or touching any currency. Just separate from the collection that is over the tray, one, two, three, four or more boxes, which will retrieve the desired total.
- Iallah! It is curious! - The emir said in wonder. - According to your explanation I can infer, the arrangement of dinars, divided by eight boxes, allows you can withdraw from the total amount they want without violating any of the boxes, without removing any money?
- That's right! - Eagerly confirmed the vizier. - Say it was your desire to withdraw, for example, the total amount of 212 dinars. Nothing simpler. In the group of eight cases there are some whose portions therein comprise the sum of 212. Consist of the difficulty of the problem, in each case to determine which boxes should be separated in order to obtain a certain amount, for what was done to 212 will be able to make 200, 49, 157, or any integer up to the total number of coins.
Made brief pause to allow the king could establish ideas and reflect on the case, the clever vizier wrapped up:
- Behold, O Commander of Believers, in short, the problem that could be offered before the court, in calculating genius, "Knowing that these boxes, numbered from one to eight contain dinars in numbers that are not repeated, knowing that it is possible to make any payment until the total number of coins, without opening any box, ask yourself:
1 - How many coins contain, respectively, each of the boxes?
2 - How to determine, through reasoning, mathematically certain, the amount contained in each one?
3 - What is the total number of coins?
4 - Is it possible to solve the same problem by distributing the coins by a smaller number of boxes? "
The divan of the caliphate, that is, the royal hall of audience, stood and full of noble guests while at deaf and solemn sound of the gong, announced the visit Beremiz Samir, "the man who calculated. At the heart of the magnificent hall, about luxurious carpet, the tray was placed with eight boxes that would provide the basis for the problem.
Al-Motacém Billah, Prince of Believers, who was on his throne of gold and purple, surrounded by his viziers and cadis sent the mathematical friendly greeting:
- Be thou welcome, O Beremiz! Be welcome under the inspiration of Allah! That your presence on this couch is cause for rejoicing for all of our friends, and that of your words can reap the delicious date of wisdom that elevates the soul and purifies the hearts.
It took a moment of impressive silence. It was up to the visitor that honorable greeting thank. Leaning Beremiz before the king spoke thus:
- Allah badique, would Sidi! - God leads you, O Chief! Admire, appreciate and extol those who govern with justice, kindness and wisdom. That is your case, O Emir of the Arabs, and all your subjects proclaim this truth. Your justice ensures the power of the state and its precious dedications create your kindness, your wisdom and perpetuates and strengthens the confidence of the people. Woe to those whose rulers are wise but govern life by the injustice of the actions that they do! Woe to those whose heads and leaders are just unaware but goodness! And Allah, the Merciful, have mercy on those who are under the yoke of men ignorant, wicked and treacherous.
- Your words, the calculation - the king answered quietly - for me are like rubies and gold earrings. They serve me with encouragement and fill me with pride. I will, once again, to abuse your kindness. It will be a delight, not only for me, as for all the nobles, and viziers cheiques think that here, listen to your word, your most learned opinion, always original and brilliant, about an arithmetic problem that seems to defy the ingenuity of the most distinguished mathematicians. This problem, formulated by the vizier Sabag, could be stated as follows: "On that tray are eight boxes. Each box contains a number of currencies, and there are two boxes with the same number of coins. Vizier Sabag says that the distribution of coins by the eight boxes was made so that it can allow the total, highlighting any amount from one dinar, without opening any box, ie without touching the coins. It remains now to determine how many coins each box contains and what the total number of coins. For ease of exposition, the boxes are numbered from one to eight, according to the ascending order of amounts that close. "
And the Caliph wrapped up, after a brief pause:
- How the East, by calculating the ingenious solution of this problem?
Beremiz Samir, "The Man Who Counted", as a good subject, did not let them beg. Slowly crossed his arms, lowered his face and began to meditate. After coordinating ideas, began lecturing about the case, as follows:
- In the name of Allah, Most Gracious and Merciful! This problem is really one of the most interesting I have heard, and its solution, it is simple and smooth, highlights the beauty and simplicity unrivaled of Mathematics. Let's see. The distribution of dinars by the eight boxes was made to allow us to separate any one amount from one dinar, highlighting a collection of two, three or more boxes. It remains to determine the contents of each box. Clearly, the first box should contain one dinar, because otherwise we could not highlight the unity of the total. That is the conclusion handcuffed by the evidence: the box contains a number assigned by a dinar.
The second box should contain, necessarily, two dinars, because the amount of one dinar can not be repeated, and if the second box had three, four or more dinars would not be possible to separate two dinars of the total. Conclusion: we already know the contents of their first two boxes. With the help of these two boxes can get one, two or three dinars.
Now to the third box. How much should it contain? The answer requires immediately: four dinars. Indeed, if the third box close down more than four dinars would not be possible, keeping intact the boxes, four separate dinars of the total. For the first three, we therefore:
1st box: 1 dinar;
2nd box: two dinars;
3rd box: 4 dinars.
With the help of these three cases, we can form all amounts from one to seven dinars. Seven represents the total of the first three boxes, ie one plus two plus four.
Repeating the same reasoning, we are led to say that the next box, ie the fourth, will contain eight dinars. The inclusion of this box of eight dinars will separate the total all amounts from one to fifteen. The fifteen is formed by the contents of the first four boxes.
And the fifth box? Does not provide the calculation of its content is less difficult. Once demonstrated that the first four boxes totaling fifteen, it is clear that the fifth box should close sixteen dinars. The inclusion of the fifth group box allows the first four that we form any number from one to thirty-one, inclusive. The total thirty-one is obtained by adding the top five.
At this point, the calculating made a very quick break, and then continued:
- Let's look at natural chain of our reasoning, it is possible to discover a law or rule by which to calculate the contents of their other remaining boxes. To this should recap:
1st case: a coin;
2nd box: 2 coins;
3rd box: 4 coins;
4 th box: 8 coins;
5 th box: 16 currencies.
Notice that each box, from the second, always contains twice the number of coins from the previous case. Mathematicians say that numbers 1, 2, 4, 8 and 16 form an increasing geometric progression, whose ratio is two - a binary system, therefore. Given the nature of the problem is easy to prove that stays the same progression by setting the contents of the four boxes below. Then we have:
6 th box: 32 coins;
7 th box: 64 coins;
8 th box: 128 currencies;
And the total of coins in all cases, therefore, is 255.
- Uassalã!
(Adapted from Malba Tahan, The Man Who Counted - Conquest, Rio, 1965)
(This is the binary system, basic operation of computers. Bit is the smallest unit of information corresponding to the contents of a box, either of them; byte is the information (number) obtained with one or more selected from these eight cases. The system eight bits (or eight boxes, as described) allows to compute from 0 to 255. The ten digits of the decimal system occupy only ten possibilities, and the remaining 245 are used by programmers to match letters, signals, commands, etc.).
***
thanks to Allah CC for the opportunity, for to have a father ( may Allahcc forgive his sins and make his grave large), who give education , to his daughter, speaking histories like that. elhamdulillah
