Every challenging effort achieved seems to be impossible in the beginning.

Yes,there is a definite way to make a phase change from the state of impossible to a state of possible.

Impossible is of course, a zero energy state,where as possible is at higher energy state. The required phase change to take place, one must necessarily spend a quantum of energy.

2.Recreational mathematics

odd squares

A number multiplied by itself gives its square. If n is a number then its square is n x n. It is found that no square numbers ends with 2,3,7 and 8. The square numbers have digital roots of either 1,4,7 or 9 only.A signle digit number obtained after repeatedly adding the digits of a given number is called its digital root.Another important property of square numbers is that they have an odd number of divisors.

We know that 1,9,25,49,81,121,169,225,..... are the first few odd square numbers. They are all ending with 1 or 5 or 9. A kind of regularity is inherent among them. They are all one excess over 4 times the product of any two successive numbers and it is equal to the square of sum of such two successive numbers.

4(0x1) + 1 = 1 = (0+1)(0+1) = 1 x 1

4(1x2) + 1 = 9 = (1+2)(1+2) = 3 x 3

4(2x3) +1 = 25 = (2x3)(2x3) = 5 x 5

There is no strange at all when we seek the help of Algebra. The general form of an odd number is (2n-l), where n takes any value from 1 to n. The square of an odd number (2n-l) is given by 4(nxn) - 4n + 1=

4n(n-1) + 1 and it explains the expression given above.`

The square of an odd number N can be expressed as,

NxN = (N/2)[(N-1) +(N+1)]

where (N-1) and (N+1) are the even numbers, the lower and higher neighbours to the given odd number.

Thus,

3x3 = (3/2) (2+4) = 3(1+2) = 9

5x5 = (5/2)(4+6) = 5(2+3) = 25

7x7 = (7/2)(6+8) = 7(3+4) = 49

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