What is the pattern so we can make the next stars?

Question

I found the following pattern question in a group! It took me a lot of time but unfortunately I don't have any ideas to find any logical thing here.

Here's the picture of the question:

The question provides three sets of star patterns, each with five vertices (vertices have value) and a number written inside the star. The task is to find the pattern in these stars to determine the numbers inside the next set of stars.

The only logical pattern I found is the sum of vertices:

Star 1: Sum = 13

Star 2: Sum = 16

Star 3: Sum = 22

Which is like it's increasing by 3n so sum of staar 4 I think would be $22+(3 \times 3) = 31$.

Is there a way we can find the pattern?

Answer 1

I think if you have a star with vertices labeled $\{V1,V2,V3,V4,V5\}$ and middle number $N$ like this

then $N=|V5\cdot V2 - V1\cdot V4| -V3 -1$ where $||$ is the absolute value.

It seems to work for each one

(1) $|5\cdot 3 - 1\cdot 2| - 2 -1 = |15-2|-3 = 13-3=10$

(2) $|4\cdot 6 - 3\cdot 1| - 2 -1 = |24-3|-3 = 21-3 =18$

(3) $|4\cdot 4 -6\cdot 5|—3-1 = |16-30|-4 = |-14|-4 = 14-4 =10$

Hope this helps!