Hello, James!

Prove that . . . the number of 9s is the same.

is the sum of the digits of the number .

. . Then: .

Since , then: .

. . That is: .

This is the number "moved" places to the left, plus a 1.

. . It contains 9's and an 8.

Therefore: . n-1)9 + 8 + 1\quad\Rightarrow\quad\boxed{ S(N^2) \;=\;9n}" alt="S(N^2) \:=\n-1)9 + 8 + 1\quad\Rightarrow\quad\boxed{ S(N^2) \;=\;9n}" />