A New Way of Squaring.....

If we square 11, it is very simple put 1(21)(12)get 121 same as square 12 put 1(22)(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

=  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

= 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