I know this is not a new topic, but I need your help in a distinct approach.
My question is there any pattern in 6x & 7x is there any trick to tell the first 2 digits correctly.
@Antelex
I didn’t understand your 50 number series.
Why not use like this
25 + x | x^2
Example - 53^2
25 + 3 | 3^2
2809 (last digit always 2 no that’s why 0 add before 9)
56^2
25 + 6 | 6^2
3136
59^2
25 + 9 | 9^2
3481
And so on…
I mean it. the same method as you are doing. Sorry for my poor presentation. The last surrounding blocks are basically my mnemonics.
In that I got 2 extra columns means I have 2 approaches, one is on the answers right, it’s the method you mentioned, and the left side is a faster method I think like 59^2=5+9=14>10 so 3481, or 53^2=5+3=8<10 so 2809.
Just like 5 no. Series , we can do 6 no series easily.
61 = 3721
36 + x | 21
Note : 1) first digit is 6 so 6^2 = 36)
2) add next no. In them.
Last 2 digit you can easily write . (I guess you already know 11 to 19 square, so write last 2 digit of squares)
Example - 61²
36 + 1 | 21 (11² = last 2 digit)
3721
62² = 3844
36 + 2 | 44 (12² last 2 digit)
63² = 3969
36 + 3 | 69 (13² last 2 digit)
64² = 4096
36 + 4 | 96 (14² last 2 digit)
•these are the case when in 6 no series next no is bigger than 6 .
Just remember - 66 , 67 = +1 (gap 1 )
68 , 69 = +2 (gap 2 or more)
66² = 4356
36 + (6 + 1) | 56 (16² last 2 digit)
67² = 4489
36 + (7 + 1) | 89 (17² last 2 digit)
68² = 4624 (18² last 2 digit)
36 + (8 + 2) | 24
69² = 4761
36 + (9 + 2) | 61 (19² last digit)
I learned these using flashcards on Mnemosyne, although ultimately I made a testing tool to drill them faster, and now I’m basically instant with all of them. I spend more time typing the answers than trying to recall them.
E.g. 7921 is easy as 79 + 21 = 100
E.g. 7744 is easy as it’s two double digits
E.g. 6561 is easy as 65 and 61 both are in the 60s
There’s also a nice line of reflection around 25 and around 75, so you can learn these in pairs:
74² = 5476; 76² = 5776
Similar like
73² = 5329
77² = 5929
Number ends with 1 can be do by this.
Square - double - 1
Example - 21² = 441
2² = 4
Double of 2 = 4
1 = 1
31² = 961
3² = 9
Double of 3 = 6
1 = 1
41² = 1681
4² = 16
Double of 4 = 8
1 = 1
51² = 2601
5² = 25
Double of 5 = 10 (carry 1 )
1 = 1
71² = 5041
7² = 49
Double of 7 = 14 (carry 1 )
1 = 1
Well , I always do like this (not take a single sec to do)
51² = 26 01
26 = 5² + 1
01 = reverse digit of 5’s double
61² = 37 21
37 = 6² + 1
21 = reverse digit of 6’s double
71² = 5041
50 = 7² + 1
41 = reverse digit of 7’s double
81² = 6561
65 = 8² + 1
61 = reverse digit of 8’s double
91² = 8281
82 = 9² + 1
81 = reverse digit of 9’s double
Note : I am using this only when numbers double is 2 digit no
Like 5’s double = 10
If you not wanna use this you can use first one, applicable to all numbers.
There’s always the formula x^2 = (x+d)(x-d) + d^2
Set d so that either (x+d) or (x-d) have a zero at the end.
91^2 = 90*92 + 1^2
Thanks all, the replies are useful! I appreciate it.
The other one that I didn’t see mentioned (might have overlooked!) is for squares ending in 5:
(10x + 5)^2 = 100x² + 100x + 25 = 100x(x + 1) + 25
Therefore multiply the tens digit(s) by its successor and write 25 at the end:
85² = [8×9]|25 = 7225
55² = [5×6]|25 = 3025
Also for squares of this form greater than 99²:
255² = [25×26]|25 = [50×13]|25 = 65025
Yep,
I guess antelex already mentioned this, I read that yesterday.
Well I am doing the same for bigger numbers that ending with 5 .
Little change
255² = 25 × 26 | 25
25² + 25 | 25
625 + 25 | 25 = 65025 (rather than 25 × 26 = 50 × 13 = 650 like you mentioned.)
Other example -
3945² = 15563025
394² + 394 | 25
155236 + 394 | 25 (it’s easier to just square the no. and add the same digit and put 25 in the last)
I deleted my previous post because I mistakenly written there some spelling mistakes.
This topic was helpful for me
i didnt understand how are you able to come up with the answer with this can you explain ?please
sorry for my late reply.
50s series square must be greater than 2500, right?
57^2 = 3249 ~ 5+7=12 ~ 20+12=32
54^2 = 2904 ~ 5+4=9 ~ 20+9=29
these two examples have a common base of 2000,
That’s why we can simply add 20 to get their first two digits.
Do you mean this part?
51² => 5² + 1 | 1²
=> 25 + 1 | 01
=> 2601
52² => 5² + 2 | 2²
=> 25 + 2 | 04
=> 2704
53² => 25 + 3 | 3²
=> 28 09
54² => 25 + 4 | 4²
=> 2916
55² => 25 + 5 | 5²
=> 3025
56² => 25 + 6 | 6²
=> 3136
57² => 25 + 7 | 7²
=> 3249
58² => 25 + 8 | 8²
=> 3364
59² => 25 + 9 | 9²
=> 3481
If you want to find squares of numbers between 40 - 49 (just do subtraction in between instead of adding)
for example => 48² => 5² - 2 | (-2)²
=> 25 - 2 | 04
=> 2304
46² => 5² - 4 | (-4)²
=> 25 - 4 | 16
=> 2116
Thanks Raja. @only4u I hopeyou can get it. 
