## Wednesday, March 25, 2015

### Patterns-Finding squares in the hard way

The methods 1-5 are available in most aptitude books.

Basic Assumptions:

## 1.You know how to caluclate squares ending with number 5

Many of us know how to caluclate square of numbers ending in 5

Ex: x5*x5=(x*x+1) 25

1. 45*45 = 4*(4+1)  25 = 2025
2. 75*75=7*8  25=5625
3. 85*85 = 8*9 25=7225

## 2.You know how to caluclate squares of numbers near to 100.

Ex1:
102 * 102 = 10404

Number           By how much it is away from 100

102                          +2                            Line 1
102                          +2                            Line 2

-------                          -----
04(2*2) ==>last two digits

Take 102 from first line and add 2 from 2nd line(cross)
102+2=104 ==> First first 3 digits

Ex:2

112*112

Number           By how much it is away from 100

112                          +12                            Line 1
112                          +12                            Line 2

-------                          -----
144(12*12) ==>last two digits

Take 112 from first line and add 12 from 2nd line(cross)
112+12=124 ==> First first 3 digits

## 3.You know how to caluclate squares of numbers from 31-50

47*47=??

• 47 = 50-3
• (-3)*(-3) = 09 ==> Last two digits
•  for first two digits  ==>[25 is used as standard]
•  So use 25-3 = 22

## 4.You know how to caluclate squares of numbers from 51-80

64*64 =??

• 64 = 50+14 (for 66 write as 50+16)
• 14*14=196 take 96 for last two digits
• First two digits ==> 25+14+1(this 1 is from 196)  = 40
[25 is used as standard]

## 5. You know how to caluclate squares of numbers from 81-100

88*88=??

• 88=100-12
• Last two digits: (-12)*(-12)=144 ==>44(1 carried)
• First two digits:88-12+1=77

The above methods 4 and 5 involves two steps.If you can remember some squares by heart then Instead of two steps we can acheive in one step.The above methods are easy,recommended.The methods below are for mnemonic guys.
So for academic purposes use the above methods (4,5)
1,2 methods are common and the only easy methods.

I've seen some aptitude books,finding squares,it's fine but i want to find out a hard way.
I was in my class way back in  2012 in mining lecture.I wanted to play with numbers .So after many caluclations i thought i found a pattern.But i didn't go beyond some huge numbers.Just confined to 3 digit caluclations.

This is a good method to find squares upto 100.You can extend upto 200 though.

``` for i in range(10,127):

print (str(i)+"*"+str(i)+"=",i*i)

print(((i*i)/100)%10)

print "difference=",(((i+1)*(i+1))/100 - (i*i)/100)

print("=======================================")

```
This piece of got bought to me to the above conclusions

"""
Values to Memorise

base             Number             Square                 Squares you can find

4                 17                289                        [17 to 23] (base for these numbers is 4)

6                 27                 729                        [27-33] (base for these numbers is 6)

8                 37                 1369                    [37-43] (base for these numbers is 8)

10                 47                 2209                    [47-53] (base for these numbers is 10)

12                 57                 3249                    [57-63] (base for these numbers is 12)

14                 67                 4489                    [67-73] (base for these numbers is 14)

16                 77                 5929                    [77-83] (base for these numbers is 16)

18                 87                 7569                    [87-93] (base for these numbers is 18)

"""

"""
45*45 =(4*5)(5*5) = 2025 i guess everyone know for numbers ending

Type 3 Example-1

Type 2 Example-1

31*31=?

base of 31 is 6

first two digits

Now I know 30*30=900;i will take first two digits 90
31 is one number away from 30;So i need to add 6 to first two digits;
90+6=96

last digit
31*31 = 1

961

Type 1 Example-1

28*28 =

first two digits

I know 27*27 = 729,it's base is 6;Now to get the answer i will take 72 from 729 and add 6 to it
72+6=78 ==> first two digits

last digit
8*8=4(last digit)