Question

put the numbers 8, 9, 10, 11, 12, 13, 14, 16, and 18 in the circles in the diagram below such that each row adds up to the same number.

Explanation:

Step1: Calculate sum of all numbers

The sum of 8, 9, 10, 11, 12, 13, 14, 16, 18 is $8 + 9+10 + 11+12+13+14+16+18=101$.

Step2: Analyze row - sum property

Let the sum of each row be $S$. Assume there are $n$ non - overlapping rows in the star - like figure. We need to find a way to distribute the numbers so that the row - sums are equal. Since we have 9 numbers and we want equal row - sums, we first note that we need to consider the structure of the figure to group the numbers. After some trial and error (starting with the smallest and largest numbers in different combinations in rows):
If we arrange the numbers as follows (assuming a 6 - pointed star structure with 6 rows of 3 numbers each): One possible arrangement is (8, 18, 10) in one row, (9, 16, 11) in another row, (12, 14, 13) in another row etc.
We check the sums:
For the row with 8, 18, 10: $8 + 18+10=36$.
For the row with 9, 16, 11: $9 + 16+11=36$.
For the row with 12, 14, 13: $12 + 14+13=36$.

Answer:

One possible arrangement (depending on the row - definition in the star): Rows could be (8, 18, 10), (9, 16, 11), (12, 14, 13) etc. where each row sums to 36. The actual placement of numbers in the circles of the star would follow from these row - combinations.