On a square game-board that is divided into n rows of n squares each, k of these squares lie along the boundary of the game-board. Which of the following is a possible value of k?
I have 25 (B). I don't have much reasoning behind my choice. Not sure if it requires much reasoning anyway. It mentions squares and the only number that is a perfect square is 25.
I think I follow the rest: a square has 4 sides, so you replaced n with each answer choice in the formula and checked which numbers would give a multiple of 4. Am I following this correctly?
We add all four side lengths there.
The problem here is that if you add up all the squares on the 2 horizontal sides, that's 2n.
In adding the squares left to be counted on the vertical sides,
notice that we've already accounted for the 4 squares at the corners!
So the amount of squares remaining to be counted is (n-2)+(n-2).
Another way to count the perimeter . . .
. .Code:: - - n-1 - - : ♥ ♥ ♥ ♥ . . . ♥ ♥ ♠ - - ♣ ♠ : : ♣ ♠ : : . ♠ : : . .n-1 n-1. . : : ♣ . : : ♣ ♠ : : ♣ ♠ - - ♣ ◊ ◊ . . . ◊ ◊ ◊ ◊ : - - n-1 - - :
Each of the four sides has squares.
Hence, there are: . boundary squares, a multiple of 4.
The only multiple of 4 is: .
Here is the way I would do this problem on the SAT:
First I draw some pictures.
Above are pictures for n=2,3,4
The corresponding values for k are as follows:
Draw as many values of n as you need to until you realize that the possible values of k are all positive multiples of 4.
Choice (E) is the only multiple of 4.