Square a 2 digits number within a second in mnemonic sense

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.


@bjoern.gumboldt , @Daniel_360 , @Rajadodve786 , @Kinma .
Those are respectful and keen math learner/expert, I wish you can lend me a helping hand.

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@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…

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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.

@Antelex

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)

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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

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Similar like

73² = 5329

77² = 5929

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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.

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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

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Thanks all, the replies are useful! I appreciate it.

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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

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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)

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I deleted my previous post because I mistakenly written there some spelling mistakes.

This topic was helpful for me☺️

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