Answer:
18
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.