Arithmetic Techniques Part 3: Techniques for calculating n2

Created by Sho Sengoku, 2002

The calculation n2 that usualy takes long can be quickly and easily done using following formulas:

 (a+b)2 = a2 + b2 + 2ab (a-b)2 = a2 + b2 - 2ab

 n2 (a+b)2 or (a-b)2 a2+2ab+b2 or a2-2ab+b2 Results 11 2 (10 + 1) 2 100 + 20 + 1 121 12 2 (10 + 2) 2 100 + 40 + 4 144 13 2 (10 + 3) 2 100 + 60 + 9 169 14 2 (10 + 4) 2 100 + 80 + 16 196 15 2 (10 + 5) 2 100 + 100 + 25 225 16 2 (10 + 6)2 or   (24) 2 100 + 120 + 36 or   28 256 17 2 (10 + 7) or (16 + 1) 2 100 + 140 + 49 or 256 + 32 + 1 289 18 2 (20 - 2) 2 400 - 80 + 4 324 19 2 (20 - 1) 2 400 - 40 + 1 361 20 2 400 21 2 (20 + 1) 2 400 + 40 + 1 441 22 2 (20 + 2) 2 400 + 80 + 4 484 23 2 (20 + 3) 2 400 + 120 + 9 529 24 2 (20 + 4) 2 400 + 160 + 16 576 25 2 (20 + 5) 2 400 + 200 + 25 625 26 2 (20 + 6) 2 400 + 240 + 36 676 27 2 (30 - 3) 2 900 - 180 + 9 729 28 2 (30 - 2) 2 900 - 120 + 4 784 29 2 (30 - 1) 2 900 - 60 + 1 841 30 2 900 31 2 (30 + 1) 2 900 + 60 + 1 961 32 2 (30 + 2)2 or   (25) 2 900 + 120 + 4 or   210 1024 33 2 (30 + 3) 2 900 + 180 + 9 1089 34 2 (30 + 4) 2 900 + 240 + 16 1156 35 2 (30 + 5) 2 900 + 300 + 25 1225 36 2 (30 + 6) 2 900 + 360 + 36 1296 37 2 (40 - 3) 2 1600 - 240 + 9 1369 38 2 (40 - 2) 2 1600 - 160 + 4 1444 39 2 (40 - 1) 2 1600 - 80 + 1 1521 40 2 1600 41 2 (40 + 1) 2 1600 + 80 + 1 1681 42 2 (40 + 2) 2 1600 + 160 + 4 1764

In most case, you can omit the squre calculation in a Kleinman count by memorising the numbers colored in blue in the table (from 112 to 172).

Thanks to Michihito Kageyama who suggested this method to me as a part of the "real time" Kleinman count techniques.