How Many?

1.

S = S1 + S2 + S3 + S4

S1 = # of squares of the largest size (4 by 4) = 1

S2 = # of squares of the next smaller size (3 by 3) = 4

S3 = # of squares of the next smaller size (2 by 2) = 9

S4 = # of squares of the next smaller size (4 by 4) = 16

S = 1 + 4 + 9 + 16 = 30

2.

S = 1**1 + 2**2 + 3**2 + 4**2 +5**2 + .......... + N**2 =

N(N+1)(2N+1)/6