Square Of Number (Fastest Approach)

Number containing 5 at the end:

Getting square of a number ending in 5 is simple.If the last digit contain 5, the last 2 digit of square will be 25 ,whatever the earlier part of number multiply it with one more than itself.

For example: 352=(3*4)|25
452=(4*5)|25


Number containing 1 and 6 at the end:
First observe the following:


12=1=1
22=4=1+3
32=9=1+3+5
42=16=1+3+5+7
52=25=1+3+5+7+9

Above shows that to get n2 ,we add upto first n odd number. for example :

162=sum of first 16 odd number

There from above we conclude that:

262=252+26th odd number i,e 625+51=676

312=900+31st odd number i,e 900+61=961


Number containing 4 and 9 at the end:

Let's take example:

292=302- 30th odd number

342=352- 35th odd number


Number containing 7 at the end:

272=262+27 odd nomber= 252+26th odd number+27th odd number

The sum of the 26th odd number and 27th odd number is same as 4 times 26. Hence,

272=252+4*26 =625+104 =729

772=5625+304(4 times 76) = 5929


Number containing 3 at the end:

532= 552-4 times 76 = 3025-216=2809

932=9025- 376=8649


Source