Squaring: A New Way

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.

 

  • 112  = 1 2 1
  • 122  = 1 4 4
  • 132  = 1 6 9
  • 142  = 1 8 16 =  100 + 80 + 16 = 196
  • 152  = 1 10 25 = 100 + 100 + 25 = 225
  • 162  = 1 12 36 = 100 + 120 + 36 = 256
  • 172  = 1 14 49 = 100 + 140 + 49 = 289
  • 182  = 1 16 64 = 100 + 160 + 64 = 324
  • 192  = 1 18 81 = 100 + 180 + 81 = 361
  • 202  = 1 102 = 1 20 100 = 100 + 200 + 100 = 400
  • 212  = 1 112 = 1 22 121 = 100 + 220 + 121 = 441
  • 222  = 1 122 = 1 24 144 = 100 + 240 + 144 = 484
  • 232  = 1 132 = 1 26 169 = 100 + 260 + 169 = 529
  • 242  = 1 142 = 1 28 196 = 100 + 280 + 196 = 576
  • 252  = 1 152 = 1 30 225 = 100 + 300 + 225 = 625
  • 262  = 1 162 = 1 32 256 = 100 + 320 + 256 = 676
  • 272  = 1 172 = 1 34 289 = 100 + 34 + 289 = 729
  • 282  = 1 182 = 1 36 324 = 100 + 360 + 324 = 784
  • 292  = 1 192 = 1 38 361 = 100 + 380 + 361 = 841
  • 302  = 1 202 = 1 40 400 = 100 + 400 + 400 = 900

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

  • 312  =  1 212 = 1 42 (1 11)2  =  1 42 (1 22 121)

=  961  = 121 + 220 + 100

= 441 + 420 + 100 = 961

  • 412  = 1 312 =  1 62 (1 21)2 =  162 (1 42) (1 11)2
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=  1 62 (1 42) (1 22 121)

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

  • 512  = 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

  • 612  = 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

  • 712  = 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

  • 812  =  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

  • 912  = 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

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

=   6561 + 1620 + 100  =  8281

  • 1012  = 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

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