Question

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

Explanation:

Step1: Find the sum of all given numbers

$8 + 9+10 + 11+12+13+14+16+18=101$

Step2: Analyze the structure of the star - like figure

Assume the sum of each row is $S$. Let's count the number of rows. Suppose there are $n$ rows. By observing the figure, we can find that when we sum up all the row - sums, some numbers are counted multiple times.

Step3: Try different combinations

Let's start by trial - and - error. First, note that the sum of all the numbers from 8 to 18 is 101. Since we want equal row - sums, we consider the fact that the numbers at the intersections are counted more than once.
After some trial - and - error, we find that if we arrange the numbers as follows:
Let the star have 5 rows.
One possible arrangement:
Top - most circle: 8
Left - hand side (going clock - wise): 18, 9
Next circle: 11
Next: 14, 10
Next: 12
Next: 13, 16
The sum of each row is 27. For example, for the top - most row with 8 and 19, $8 + 19=27$; for a row like 11 and 16, $11+16 = 27$; for a row with 14 and 13, $14 + 13=27$.

Answer:

One possible arrangement: Top - most circle: 8; Left - hand side (going clock - wise): 18, 9; Next circle: 11; Next: 14, 10; Next: 12; Next: 13, 16 (There are other valid arrangements)