Brain Games For Children

The mathematical ones can be funny. The apparent paradoxes of pitiless logic, the torsions and the turns of simple arithmetic, are the thorns charming in every flesh of the intelligent intellectual person. The truth is that we are all snobs intellectuals, and all games of competence and of chance (where competence is the ours, the chance the other friend) intrigues us all, of nine to ninety, the student to the amateur. Here an expression of headache and here the brains trainers, to while far the unoccupied hour in to improve your mathematical competences.

The squares In Ensconces Game: The task is to write the same to such numbers in the diagram that the sum of the squares of two adjacent numbers is as the sum of the squares of the two on the opposed side of the diagram. For example, put 16 in to ensconce ONE and 2 in B. of square 16 2 = 256; 2 2 = 4; 256 + 4 = 260. We said that F 2 4~ G 2 must be the same. The suitable numbers would be 8 and 14, because 8 2 = 64; I4 2 == 196; 64 + 196 = 260.

Of same B 2 + C 2 must be equal to G 2 + O'CLOCK 2; also, A 2 + K 2 = F 2 + E 2.

Which numbers must we write in the squares empty? The only entire numbers could be used. Since A 2 + B 2 = F 2 + G 2, then A 2 francs 2 = G 2 B 2; in of other terms, the difference between the number squares on the same diagonal always must be the same. In our case, the difference is i6 2 - 8 2 = I4 2 2 2 = 192

All the same, C 2 - O'CLOCK 2 = 192.

But the difference between the squares of two numbers must be equals to the sum of these numbers multiplied by their difference. Usage symbols: (X there) (x -f there) = # 2 jy 2.

Therefore, we can write: (C + O'CLOCK) (C~ O'CLOCK) = 192.

The result 192 say us also as (C + O'CLOCK) and (C O'CLOCK) not the two can be curious numbers; otherwise, their product would not be equal. If the one (says, C + O'CLOCK) is equal, then the other must be also, because the sum of the difference of the two numbers can be even alone if
The two numbers, C and O'CLOCK, are equal or if the two are curious.

Increase 192, using even numbers: 2 X'S 96, 4 x's 48, 6 x's 32, 8 x's 24, 12 X'S 16. Therefore: And these numbers then can be written instead of C and O'CLOCK.

C + O'CLOCK = 48
C-h = 4

More: C + O'CLOCK = 48

O'clock = 22

The Counsel Broken: When we insert the numbers in their positions, it seems as if it has not any importance that we take as I and that as O'CLOCK. Nevertheless, we must watch out. If in a pair more big number is in the superior half of the diagram, we must be sure that the biggest number is in the pair next to him in the lower half, since the sum of the squares of two bigger numbers cannot give the same result as the sum of two smaller one a. Continue in this method, we can obtain the other numbers also.

Susan was very interested in how the numbers are related to every other. Immediately that she saw a number, his imagination began the functioning until she found to interest something of him. "The look, Clear," she said to his friend. "The look that I noticed. Can you see that the counsel broken"? Clear said, "Yes, I can see it. And it? It says 3,025".

"To see how two numbers were leave when the counsel was broken, 30 and 25. If we add them together, we obtain 55. And 55 X'S 55 (that is, 55*) is 3,025, that is the original number," said Susan proudly.

"Yes, you have reason," said Clear. "We to allow finding other numbers that are similar, and then we can say the professor of him to the next lesson of maths".

Themselves they took pencils and the paper and tried various numbers. Suddenly Clear shouted, "Euredka! 9,801". In fact, 98 + 1 ==99, and 99 x's 99 = 9,801.

Some days later to school Susan noted the numbers in the question on the counsel. "That do you think"? asked the professor. "Are there no other numbers of this type"?

"To please," said George, "is there a manner to find such numbers without using a d'essai-et-la method of error"? "Yes," said the professor. "George thinks about this just as a mathedmaticien does when it keeps to try to find a general rule to cover all possible solutions. We to allow having one to look at 2,025: 20 + 25 = 45 and 45 X'S 45 = 2,025".

"But our numbers are better," screamed Clear.

"That do you mean by better"?

"Well, in our numbers all the figures are different".

"You have reason," said the professor. "But 2,025 cannot be excluded for this reason; we to allow seeing how much of the numbers of this type there is".

They tried and tried, but outside of 3,025 (55 X'S 55), 9,801 (99 x's 99), and 2,025 (45 X'S 45), they could not find any others. The professor explained while there is no.

Why? The number of four face must be given by the square of a number of two face; we to allow calling this # 2. We to allow calling the two numbers to two figures x and there. We say that the numbers of two face are added, that the result is square, and that we return to the number of four face of original one. C'est-a -dire: (X + 3;) 2 = & r x + there = one a d there = a x.

As can see us example of Clear one, we can think about 01 as a number of two face, and even oooo is a number of four satisfactory face.

On the other hand, in the number of four face of original, x can be considered as the number of hundreds (expressing the thousands as the hundreds) and there unities (expressing the tens as unities). Therefore, 2 (the original number) can be written as: 1OO# + there = one

We know that jy equals one - #;, substituting therefore: + one - x = a 2 = a 2 a

As said us to the beginning, x must be an entire number. This can arrive only if one (an I) can be divided by 99 without a remainder left (99 can be expressed as 9 X'S n) #. can be an entire number in four cases:
1. one = 99 when the fraction if is simplified let us obtain us x = 98 and there = I, giving the number of four face as 9,801.

2. an I = 99. But then one = 100, that we cannot use, as a 2 == 10,000, that is a number of five face.

3. one is divisible by 9, and (an I) by n. How do we find a number as that?

We to allow writing down below the one and the numbers to two figures that are divisible by nine and the numbers that the one are less than these: 8,18,17,27,26,36,35,54,53,63,62,72,71,81,80,90, 99 and 98.

The only pair of numbers that satisfies all our conditions is 45 and 44. In this case, when we simplify our equation, we obtain: x = 20 and there = 25, giving 2,025 as the number of four face.