# The Perfect Star

As you see in the graph below, there are $10$ empty circles in our star and you are asked to fill these circles with the numbers between $1$ to $12$ (you can choose which two numbers you will not use among these) in them:

But while putting these values inside the circles, you need to consider the special property of this star:

The five sums along the dash lines (such as $C+G+F+E$) and the sum around the pentagon inside the star ($H+I+J+F+G$) are all equal to each other.

This star is called the perfect star.

FYI: Forgot to put dash line through B to D and very tired to draw this again. so assume there is dash line over there too.

$A=2, B=3, C=9, D=12, E=10, F=4, G=1, H=8, I=5, J=6$
The sum of all the lines, and the inner pentagon, are each $24$
This has a left/right mirror equivalent (reverse B and E, I and J, ...). And then both this solution and its mirror can be rotated around the 5 star points, for $2\times 5=10$ solutions in all.
$A+J+F+D=2+6+4+12=24$
$B+I+J+E=3+5+6+10=24$
$C+H+I+A=9+8+5+2=24$
$D+G+H+B=12+1+8+3=24$
$E+F+G+C=10+4+1+9=24$
$H+I+J+F+G=8+5+6+4+1=24$