Each side of square $$ABCD$$ with side length of $$5\text{ cm}$$ is divided into equal five parts by four points. Choose one of the four points from each side, and connect the points consecutively to obtain a quadrilateral (One example is shown in the adjoining figure). Write all possible areas of the quadrilateral so obtained.

Let the chosen points be at distances $$a, b, c, d \in \{1, 2, 3, 4\}$$ along the four sides, measured in the same direction round the square. The quadrilateral is the square minus four right triangles with leg pairs $$(a, 5 - d)$$, $$(5 - a, b)$$, $$(5 - b, c)$$ and $$(5 - c, d)$$, whose total area is $$\frac{1}{2}[5(a + b + c + d) - (a + c)(b + d)]$$. With $$s_1 = a + c$$ and $$s_2 = b + d$$, the area is $$25 - \frac{1}{2}[5(s_1 + s_2) - s_1 s_2] = \frac{1}{2}[25 + (s_1 - 5)(s_2 - 5)]$$. Here $$s_1, s_2$$ each take every value from 2 to 8, so $$(s_1 - 5)(s_2 - 5)$$ takes the values $$-9, -6, -4, -3, -2, -1, 0, 1, 2, 3, 4, 6, 9$$. Hence the 13 possible areas (in $$\text{cm}^2$$) are 8, 9.5, 10.5, 11, 11.5, 12, 12.5, 13, 13.5, 14, 14.5, 15.5 and 17.