pointy problem
Geometry
Level
4
There are 20 points inside a square. They are connected by non intersecting segments with each other and with the vertices of the square ,in such a way that the square is dissected into triangles. How many triangles do we have?
The answer is 42.
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1 solution
The first point to be placed inside a square will have to be connected to all four vertices in order to have only triangles. The next point will be placed inside a triangle and it will take three connections to the vertices of that triangle to make all into squares. This will result in net addition of two triangles, since one will be made into three. The same will be true of all additional points. So we have 4 + 1 9 × 2 = 4 2 triangles.
This is only one way of dividing the square for a given set of points. Any neighboring triangles form a quadrilateral with a diagonal, and the same quadrilateral could have instead the other diagonal and be two different triangles. But there would still be exactly two triangles making up that quadrilateral.
The problem should probably specify that no three points are colinear. While colinear points can be connected using the same number of triangles as non-colinear points, they can also form fewer triangles than the 42 which would result otherwise.