Hello, Hellbent!

Another way to count the perimeter . . .

$\displaystyle \text{On an }n\times n\text{ chessboard, }k\text{ squares lie on the boundary of the board.}$

$\displaystyle \text}{Which of the following is a possible value of }k?$

. . $\displaystyle (A)\; 10 \qquad (B)\;25 \qquad (C)\;34 \qquad (D)\;42 \qquad (E)\;52$

Code:

: - - n-1 - - :
♥ ♥ ♥ ♥ . . . ♥ ♥ ♠ -
- ♣ ♠ :
: ♣ ♠ :
: . ♠ :
: . .n-1
n-1. . :
: ♣ . :
: ♣ ♠ :
: ♣ ♠ -
- ♣ ◊ ◊ . . . ◊ ◊ ◊ ◊
: - - n-1 - - :

Each of the four sides has $\displaystyle n-1$ squares.

Hence, there are: .$\displaystyle 4(n-1)$ boundary squares, a multiple of 4.

The only multiple of 4 is: .$\displaystyle (E)\;52$