MULTIPLY NUMBERS FROM 101 THROUGH 109
      In this case the examples would be:-
         101x101
         104x107
         1 05 x105
        106x108
        109x109
And so on----
The total examples will be 45.
Procedure:-
•  The answer will be a five digit number beginning with 1.
•  The next two digits of the answer will equal the sum of ones digits.
•  The last two digits of the answer will equal the product of the ones digits.
Let’s use the trick.
Consider the problem: 102x107.
Step1:- Begin the answer with 1 ---.>   1
Step 2:-Add the ones digits    ------>2+7= 09.
Step 3:- Multiply the ones digits --->2x7=14.
Step 4:-Combine the amount from steps 1,2.and3 writing from left ------>10,914.
Answer 10,914.
Let’s try one more example.
109x105
Step 1:-Begin the answer with 1------------ .>1
Step 2:- Add the ones digits----------- ----->9+5=14.
Step 3:- Multiply the ones digits----------->9x5=45.
Step 4:-Combine the amounts for the steps 1, 2, and 3 .Writing from left to right----->11,445.
Answer is 11,445.
Note :-the sum or the product of ones digits is written in two  ,digits as01,02,03,04,05,06,07,08, and 09.
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SQUARING ANY TWO DIGIT NUMBER ENDING IN ONE OR NINE

Squaring any two digit number ending in 1 or 9 is very interesting and easy method. Two digit numbers ending in 1 are from 11 to 91.Two digit numbers ending in 9 are from 19 to 99.

   The trick is to multiply together the two whole numbers on either side of the number you are squaring, and then add one.
Let’s see how it works.
Problem:-
 31x31 (square of 31)
Step 1:-find the two whole numbers on either of 31 that are 30 and 32
Step 2:-multiply the two whole numbers.30x32=960.
Step 3:- add one to 960 The answer is 961.

Square of 31 is 961.
One more example:
61x61
The two whole numbers on either side of 61 are 60 and 62.
Product of 60 and 62 is 3720.
Adding 1 to 3720 we get 3721.
The answer is 3721.

SQUARE OF ANY TWO DIGIT NUMBER ENDING IN 9..
The method is same as above.
Consider an example of any two digit number ending in 9.

Suppose the number is 19.
Step 1:- find the two whole numbers on either side of 19, that are 18 and 20.
Step 2:- multiply the two whole numbers.
 18x20=360.
Step 3:- adding 1 to 360 ,we get 361.
Square of 19 is 361.

This trick makes the calculation easier by changing one of the numbers into a multiple of 10..

WONDERFUL FOUR

Think of any number.
The number is written in figures and in words.
Suppose we are thinking of 345.
 345 is in figures.
 In words this can be written as ‘three hundred forty five.’
How many letters are in these words?
Count the words.
The letters are 20.
Write 20 in words ‘TWENTY’
Count the letters in twenty.
Number of letters in twenty are 6.
6 is written in words ‘SIX’
Again count the letters in .six.
There are 3 letters in six.
3 is written in words as three.
Three has 5 letters.
5 is written in words as   five.
Five has 4 letters.
4 is written in words a four.
Four has 4 letters.
Try another example.
At last we will come to FOUR.
Does this always happen? What about other languages?

ALWAYS FIVE AND HALF

Think of any digit 1 through 9 in your mind..

Multiply  the number by 11.

Divide the product by its sum of digits.

Answer is always 5.5(five and half)

Example:
Suppose we have chosen the number 8.
8 multiplied by 11.The product is 88.
The sum of digits of 88 is8+8=16.
The product 88 is divided by 16 is 5.5
Answer is 5.5.

Let’s try one more example.
We have chosen the number 7.
7*11=77
Sum of digits =7+7=14.
77/14=5.5.
The answer is five and half.

Triangular Numbers

Pythagoras, the Greek mathematician is known as ‘The father of numbers.’ He explored the mathematical relationship with figurate numbers.
Pythagoras discovered triangular numbers. Triangular numbers are also called Triangle numbers. Pythagoras studied the triangular numbers under figurate numbers in geometry.
First natural number is 1.
Sum of the first two consecutive natural numbers is 1+2 =3
Sum of the first three natural numbers is 1+2+3=6
Sum of the first four natural numbers is 1+2+3+4=10
Sum of the first five natural numbers is 1+2+3+4+5=15
1, 3, 6, 10, 15…are triangular numbers. A    triangle number is the sum of first n natural numbers.
Fredric Gauss discovered the formula for nth triangular number.

Nth triangular number=n (n+1)/2.
For example: 6 th triangular number= 6(6+1)/2=  6(7)/2=  21

Let’s try one more example.
36 th triangular number =36 (36+1) /2
                                         =36 (37) /2= 666.
             666 is called the beast number.

• Every triangular number is a natural number.

• Every triangular number except 1 is divisible by 3 or when divided by 9 the remainder is 1.

• Triangular numbers can never end in 2, 4, 7, or 9.

• Eight times of any triangular number plus 1 is always a perfect square number.

• Nine times of any triangular number plus 1 is always a triangular number.

• Sum of any two consecutive triangular numbers is always a perfect square number.

• Difference between two consecutive squares of any triangular numbers is always a perfect cube of a small triangular number.

• Some triangular numbers are there which reversals are also triangular numbers. The examples are 1,3,6,10,55,66,120,153,171,190 300,595,630,666,820.3003,etc.