Mathematics can be recreation. Apparent paradoxes of pitiless logic, torsions and the turns d' arithmetic simple, are the delicious spines in each person& #039 intelligent; flesh d' intellectual of S. the truth is this we& #039; about all the intellectual snobs, and plays d' addresses and the chance (where competence is with us, the chance l' other fellow& #039; S) intrigue us all, from nine to ninety, studying with the dilettante. Here a mine of the puzzles and brain-trainers, while far l' hour with vacuum by improving your mathematical qualifications. & amp; #13;
Places in the set of places: The task is d' to write such numbers in the diagram that the sum of the places of two adjacent numbers is identical that the sum of the places of both on the side opposite of the diagram. For example, put 16 in the place has and 2 in the place B. 16 2 = 256; 2 2 = 4; 256 + 4 = 260. We said that F2 4~ G 2 must be identical. The suitable numbers would be 8 and 14, because 8 2 = 64; I4 2 == 196; . & amp 64 + 196 = 260; #13;
Of the same B 2 + C2 must be equal to G 2 + H2; also, 2 + K has 2 = & amp of F2 + d' E 2. ; #13;
Which numbers do we have to write in the empty places? Only integers can be employed. Since 2 + B has 2 = F2 + G 2, then has 2 F2 = G 2 B 2; in d' other terms, the difference between the places of the numbers on the same diagonal must always be identical. In our case, the difference is i6 2 – 8 2 = I4 2 2 & amp 2 = 192; #13;
In the same way, C2 – H2 =. & amp 192; #13;
But the difference between the places in two numbers must be equal to the sum of these numbers multiplied by their difference. Using symbols: (of x/y) (X – F there) = # & amp 2 2. jy; #13;
Consequently, we can write: (C + H) (~ of C H) =. & amp 192; #13;
Result 192 also states to us that (C + H) and (C H) cannot both be odd numbers; otherwise, their product would not be equal. If one (known as, C + H) is equal, then l' other must be as well, because the sum of the difference of the two numbers can be even only if & amp; #13;
the two numbers, C and H, are even or if both are odd. & amp; #13;
Increase 192, using even figures: 2 X 96,4 X 48,6 X 32,8 X 24,12 X 16. Consequently: And these numbers then can be written instead of the & amp of C and H.; #13;
C + H = & AMP 48; #13;
& amp of C-H= 4; #13;
Other: C + H = & AMP 48; #13;
H = & amp 22; #13;
The broken council: When we insert the numbers in their positions, it seems like if it n' do not be essential that we take in so much qu' I and which like H. Cependant, we must pay attention. So in a pair the number greater east in the higher half of the diagram, we must be sure that the number greater east in the pairs beside him in the lower half, since the sum of the places of two greater numbers cannot give the same result as the sum of two smaller those. Continuing this fa4con, we can obtain the other numbers too. & amp; #13;
Susan was very interested by the way in which numbers are reported between them. As of qu' she saw a number, her imagination started to function jusqu' with this qu' she found something interesting about it. & quot; Look at, Claire, & quot; she said to her friend. & quot; Look at what j' noted. Can you see this broken council? & quot; Claire indicated, & quot; Yes, I can see it. And him? It indicates. the & quot 3.025; & amp; #13;
& quot; See how two numbers were left when the council was broken, 30 and 25. If we add them together, we obtain 55. And 55 X 55 (c' be-with-to say, 55*) is 3.025, which is the original number, & quot; Proudly said Susan. & amp; #13;
& quot; Yes, you have reason, & quot; Claire known as. & quot; Let& #039; S find d' other numbers which are similar, and then we can about it say the professor to the next lesson of maths. & quot; & amp; #13;
Thus they took the pencils and paper and tested various numbers. Suddenly howled Claire, & quot; Eureka! . & quot 9.801; Indeed, ==99, and 99 X 99 = 9.801. & amp 98 + 1; #13;
A few days later with l' Susan school noted the numbers in question on the council. & quot; What do you think? & quot; asked the professor. & quot; Are there other numbers of this type? & quot; & amp; #13;
& quot; Please, & quot; George said, & quot; is there a manner of finding of such numbers without employing a method trial-and-error? & quot; & quot; Yes, & quot; the professor said. & quot; George thinks of this just like a mathematician makes when it continues to try to find a general rule to cover all the possible solutions. Let& #039; S will see 2.025: . & quot 20 + 25 = 45 and 45 X 45 = 2.025; & amp; #13;
& quot; But our numbers are better, & quot; Shouted Claire. & amp; #13;
& quot; What do you want to say by better? & quot; & amp; #13;
& quot; Well, in our numbers all the figures are different. & quot; & amp; #13;
& quot; You are right, & quot; the professor said. & quot; But 2.025 cannot be excluded for this reason; let& #039; S see how much numbers of this type there are. & quot; & amp; #13;
They tested and tested, but independently of 3.025 (55 X 55), of 9.801 (99 X 99), and of 2.025 (45 X 45), they could not find any others. The professor then explained qu' it n' there none has. & amp; #13;
Why? The furnace-figure number must be given by the place d' a two-figure number; let& #039; call of S this # 2. Let us ask the two numbers two digits X and Y. We say that the two-figure numbers are added, that the result is adjusted, and that we return to the number furnace-figure original. C' is: (X +3; ) 2 = & amp; amp; R X + there = D there = a & amp of X.; #13;
As we can see of Claire& #039; example of S, we can think of 01 like two-figure number, and l' oooo equal is a satisfying furnace-figure number. & amp; #13;
D' a share, in the number furnace-figure original, X can be regarded as the number of hundreds (expressing the thousands like hundreds) and of there the units (expressing the ten like units). Consequently, 2 (the original number) can be written like: 1oo# + there = a & amp; #13;
We know who equal jy has them – #; therefore, substituent: + – X = of 2 = of the 2 has a & amp; #13;
As we said to the beginning, X must be an integer. This can only occur if has (an I) can be divided by 99 without remainder having left (99 can be expressed as 9 X N). # can be an integer in four cases: & amp; #13;
1. has = 99 when the fraction is simplified thus we obtain X = 98 and there = I, giving the furnace-figure number like. & amp 9.801; #13;
2. I = 99. But d' another share has = 100, that we cannot employ, like 2 == 10.000, which is a five-figure number. & amp; #13;
3. has is divisible by 9, and (an I) by the NR. How let us find us a number like that? & amp; #13;
Let us note that and the numbers with two digits which are divisible by new and the numbers which are a least that the latter: . & amp 8.18.17.27 .26.36.35.54.53.63.62.72.71.81.80.90, 99 and 98; #13;
The only pair of numbers which answers all our requirements is 45 and 44. In this case, when we simplify our equation, we obtain: X = 20 and there = 25, giving 2.025 like furnace-figure number.

If  ™ of  € of youâ about seeking téléchargements of free plays, téléchargements of music, film téléchargements, free play of téléchargement of PC; I will expose you à all last téléchargements. Téléchargement something that you want, when you want. More than 4 million people are constantly reliés, dividing more than 800 million files!. When j' have dÔ m' to register à a good site d' adhésion so that téléchargements of play m' à help  résoudre all my problèmes of téléchargement of free play, j' have dÔ to make a research complète on line and to found thereafter téléchargements illimités. Trouvaille below some of the devices that j' have découverts in the téléchargements illimités which m' have donné l' accès with the téléchargements of free play. 1. The plays illimités télécharge m' allowed to seek thousands on thousands of files of play, intégraux films of qualité of DVD, software your songs préférées and much moreover. It n' there forever no limit on the quantité research which you can make and aucunes limiting on the things which you can seek. Aucuns articles d' spy or articles d' announces. I could obtain à 100% of the faster téléchargements of play and the files more available of 100% to téléchargement! 2. Another était good thing which I could également télécharger something that I m' wanted à any moment wanted. More than 4 million people are constantly reliés, dividing more than 800 million files! they m' provided the software of téléchargement of play, free music, & amp of copy of DVD; ampère; Software of help, MP3 players, software brÔ lant CD, online search of music, texts, radio d' Internet, & amp automatic; ampère; Solvents d' articles d' spy, customer support and much indeed j' étais really excited to join a so marvellous adhésion of play on Internet of téléchargement. 3. An important note très – in the sector d' adhésions they provide you the tools to play n' import which play téléchargé, song téléchargée, film téléchargé or exposure of TV téléchargée on your computer. and you easily obtain également the software for brÔ ler your plays vidéo, music and films with CD, no brÔ their expensive of DVD prié. The joint of l' adhésion était indeed l' the wisest investment that j' have réalisé in terms of source illimitée tools of entertainment, if you new êtes with the plays of téléchargement I give à téléchargement illimité of plays my recommendation plus élevée.