Arithmetic Techniques Part 3:
Techniques for calculating
n
2
Created by Sho Sengoku, 2002
The calculation
n
2
that usualy takes long can be quickly and easily done using following formulas:
(
a
+
b
)
2
=
a
2
+
b
2
+ 2
ab
(
a
-
b
)
2
=
a
2
+
b
2
- 2
ab
n
2
(
a
+
b
)
2
or
(
a
-
b
)
2
a
2
+2
ab
+
b
2
or
a
2
-2
ab
+
b
2
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 (2
4
)
2
100 + 120 + 36
or 2
8
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 (2
5
)
2
900 + 120 + 4
or 2
10
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 11
2
to 17
2
).
Thanks to Michihito Kageyama who suggested this method to me as a part of the "real time" Kleinman count techniques.