On a square gameboard that is divided into n rows of n squares each, k of these squares lie along the boundary of the gameboard. Which of the following is a possible value for k?
(n-1)*4 lie along the sides.
which are the following answers?
To find a possible value for k, we need to consider the conditions given in the question.
The gameboard is a square divided into n rows of n squares each. This means that there are n squares in each row and n squares in each column. Since the gameboard is a square, the number of rows is equal to the number of columns.
The question states that k squares lie along the boundary of the gameboard. The boundary consists of the squares on the edges of the gameboard.
Let's analyze the boundaries of the gameboard. It consists of the top row, bottom row, left column, and right column.
The top row and bottom row each have n squares, so the total number of squares along the top and bottom boundaries is 2n.
The left column and right column each have (n-2) squares (excluding the corners), since they don't include the squares along the top and bottom boundaries. So, the total number of squares along the left and right boundaries is 2(n-2).
Therefore, combining the squares along all four boundaries, the total number of squares along the boundary is given by:
k = 2n + 2(n-2)
Simplifying the equation:
k = 2n + 2n - 4
k = 4n - 4
Now, let's consider the options given and find a possible value for k.
a) n + 1
b) 2n
c) 3n - 2
d) 4n - 3
We can substitute each option into the equation for k = 4n - 4 and see if it satisfies the conditions.
a) k = 4(n + 1) - 4 = 4n + 4 - 4 = 4n
b) k = 4(2n) - 4 = 8n - 4
c) k = 4(3n - 2) - 4 = 12n - 8 - 4 = 12n - 12
d) k = 4(4n - 3) - 4 = 16n - 12 - 4 = 16n - 16
From the options given, the only value that matches the expression for k (4n - 4) is option (b) 2n.
Therefore, the possible value for k is 2n.