Symmetry of Digit "2" In Squaring--- by Piyush Goel

[mathspassion]

Symmetry of Digit "2" In Squaring--- by Piyush Goel

If we square 11, it is very simple put 1(2*1) (12) get 121 same as square 12 put 1(2*2)(22) get 144 again for 13 we get 169 and for 14 we get 1 8 16=196 and so on.

When we go deep, we find that there is symmetry of two types

(2, 4, 6, 8, 10, 12, 14, 16, 18, 20 …. Diff is always 2) &

(1, 4 , 9 , 16, 25, 36, 49, 64, 81, 100 ) diff. is 3 5 7 9 11 13 15 17 19 and diff. of 3 5 7 9 11 always 2, so there is true symmetry .

Up to 19 it is right but at 20 how we can put 1 20 100 just because of symmetry.


11^2 = 1 2 1
12^2 = 1 4 4
13^2 = 1 6 9
14^2 = 1 8 16 = 100 + 80 + 16 = 196
15^2 = 1 10 25 = 100 + 100 + 25 = 225
16^2 = 1 12 36 = 100 + 120 + 36 = 256
17^2 = 1 14 49 = 100 + 140 + 49 = 289
18^2 = 1 16 64 = 100 + 160 + 64 = 324
19^2 = 1 18 81 = 100 + 180 + 81 = 361
20^2 = 1 102 = 1 20 100 = 100 + 200 + 100 = 400
21^2 = 1 112 = 1 22 121 = 100 + 220 + 121 = 441
22^2 = 1 122 = 1 24 144 = 100 + 240 + 144 = 484
23^2 = 1 132 = 1 26 169 = 100 + 260 + 169 = 529
24^2 = 1 142 = 1 28 196 = 100 + 280 + 196 = 576
25^2 = 1 152 = 1 30 225 = 100 + 300 + 225 = 625
26^2 = 1 162 = 1 32 256 = 100 + 320 + 256 = 676
27^2 = 1 172 = 1 34 289 = 100 + 34 + 289 = 729
28^2 = 1 182 = 1 36 324 = 100 + 360 + 324 = 784
29^2 = 1 192 = 1 38 361 = 100 + 380 + 361 = 841
30^2 = 1 202 = 1 40 400 = 100 + 400 + 400 = 900


There is a symmetry,he got a method, which is shown as below for 31, 41, 51 and so on.

31^2 = 1 212 = 1 42 (1 11)2 = 1 42 (1 22 121)
= 961 = 121 + 220 + 100

= 441 + 420 + 100 = 961

41^2 = 1 312 = 1 62 (1 21)2 = 162 (1 42) (1 11)2
= 1 62 (1 42) (1 22 121)

= 1 62 (961) = 961 + 620 + 100 = 1681

51^2 = 1 412 = 1 82 (1 31)2 = 1 82 (1 62) (1 21)2
= 1 82 (1 62) (1 42) (1 11)2

= 1 82 (1 62) (1 42) (1 22 121)

= 1 82(1681) = 1681 + 820 + 100 = 2601

61^2 = 1 512 = 1 102 (41)2 = 1 102 (1 82) (1 31)2
= 1 102 (1 82) (1 62) (1 21)2

= 1 102 (1 82) (1 62) (1 42) (1 11)2

= 1 102 (1 82) (1 62) (1 42) (1 22 121)

= 1 102 (2601) = 2601 + 1020 + 100 = 3721

71^2 = 1 612 = 1 122 (1 51)2 = 1 122 (1 102) (41)2
= 1 122 (1 102) (1 82) (1 31)2 = 1 122 (1 102) (1 82) (1 62) (1 21)2

= 1 122 (1 102) (1 82) (1 62) (1 42)(1 11)2

= 1 122 (1 102) (1 82) (1 62) (1 42) (1 11)2

= 1 122 (1 102) (1 82) (1 62) (1 42) (1 22 121)

= 3721 + 1220 + 100 = 5041

81^2 = 1 712 = 1 142 (1 612) = 1 122 (1 51)2 = 1 122 (1 102) (41)2
= 1 122 (1 102) (1 82) (1 31)2 = 1 122 (1 102) (1 82) (1 62) (1 21)2

= 1 122 (1 102) (1 82) (1 62) (1 42)(1 11)2

= 1 122 (1 102) (1 82) (1 62) (1 42) (1 11)2

= 1 122 (1 102) (1 82) (1 62) (1 42) (1 22 121)

= 5041 + 1420 + 100 = 6561

91^2 = 1 812 = 1 162(1 712 ) = 1 142 (1 612) = 1 122 (1 51)2
= 1 122 (1 102) (41)2 = 1 122 (1 102) (1 82) (1 31)2

= 1 122 (1 102) (1 82) (1 62) (1 21)2

= 1 122 (1 102) (1 82) (1 62) (1 42)(1 11)2

= 1 122 (1 102) (1 82) (1 62) (1 42) (1 11)2

= 1 122 (1 102) (1 82) (1 62) (1 42) (1 22 121)

= 6561 + 1620 + 100 = 8281

101^2 = 1 912 = 1 182(1 812) = 1 162(1 712 ) = 1 142 (1 612) = 1 122 (1 51)2
= 1 122 (1 102) (41)2 = 1 122 (1 102) (1 82) (1 31)2

= 1 122 (1 102) (1 82) (1 62) (1 21)2

= 1 122 (1 102) (1 82) (1 62) (1 42)(1 11)2

= 1 122 (1 102) (1 82) (1 62) (1 42) (1 11)2

= 1 122 (1 102) (1 82) (1 62) (1 42) (1 22 121)

= 8281 + 1820 + 100 = 10201