Question

In the diagram, there are 6 vertices and 15 different lines. A diagram like this is called a mystic rose.
Another mystic rose has 153 different lines. How many vertices does it have?

Answer 1

Answer:

  18

Step-by-step explanation:

You want the number of vertices such that the number of pairs of vertices is 153.

Combinations

The number of n objects taken 2 at a time is ...

  N = (n)(n -1)/2

Application

You want to find n for N=153.

We notice that multiplying by 2 gives a near square:

  2N = n(n -1)

The square root of this will have a value approximately equal to n-1/2:

  n -1/2 ≈ √(2·153) = √306 ≈ 17.5   ⇒   n = 18

We can check this:

  (18)(17)/2 = 9·17 = 153

The mystic rose with 153 lines has 18 vertices.

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Additional comment

We could solve the quadratic equation 306 = n(n-1) by any of several means. The method used here is about the fastest.

If you want to get formal about solving the equation this way, you can define z = n-1/2, then the equation becomes ...

  306 = (z +1/2)(z -1/2) = z² -1/4

  306.25 = z²   ⇒   z = 17.5   ⇒   n = 18

By leaving off the 0.25 addition to 306, the root we get is slightly low, but still close enough to 17.5 to show us what the answer is.

Answer 2

Step-by-step explanation:

A Mystic Rose consists of twenty points evenly spaced around a circle and every point is joined to every other point.

How many lines are required to construct the Mystic Rose?

Solution

Starting at the first point, it must be joined to nineteen other points. The second point only needs to be joined to eighteen points now. The third point needs to join seventeen points and so on.