A square, where s is an odd integer, is divided into unit squares. All the unit squares along the edges and the 2 diagonals of a square are discarded. Find a fully simplified expression, in terms of s, for the number of unit squares remaining answer.
To find the number of unit squares remaining after discarding the squares along the edges and diagonals of a square, we need to determine the total number of unit squares in the original square and subtract the squares that are removed.
Let's start by calculating the total number of unit squares in the original square. Since each side of the square has a length of s units, the total number of unit squares is given by the area of the square, which is s * s = s^2.
Now, let's consider the squares along the edges. There are 4 edges in the square, and each edge has s unit squares. So, the number of squares along the edges is 4s.
Next, we need to count the squares on the diagonals. Since there are 2 diagonals in a square, we can determine the number of squares on each diagonal separately.
For an odd integer, s, the number of squares on the diagonal is (s - 1) / 2. This is because we discard the squares at the ends of the diagonal, and for an odd value of s, there is an odd number of squares remaining. Thus, the number of squares on each diagonal is (s - 1) / 2.
Therefore, the total number of squares on both diagonals is 2 * [(s - 1) / 2] = s - 1.
To find the number of squares remaining, we subtract the squares along the edges and the squares on the diagonals from the total number of unit squares:
Remaining squares = Total squares - (Edges + Diagonals)
= s^2 - (4s + s - 1)
= s^2 - 5s + 1.
Therefore, the fully simplified expression, in terms of s, for the number of unit squares remaining is s^2 - 5s + 1.
I'll assume that s is the length of a side.
After the borders are discarded, an (s-2)x(s-2) square remains.
There are 2(s-2)-1 = 2s-5 squares on the diagonal, so the remaining squares number
(s-2)^2 - (2s-5) = s^2-6s+9 = (s-3)^2
you can easily check this for small s.