The problem presented in the image is about arranging the numbers from 1 to 6 in circles placed at the vertices and midpoints of the sides of a triangle, such that the sum of the numbers on each side is the same, and no number is repeated.

This type of problem can be solved using logical deduction and trial-and-error, but here's a general approach using a mathematical method:

Steps to Solve:

1. Understanding the Setup:

   • You have a triangle with six positions (three vertices and three midpoints).
   • The goal is to place the numbers 1 through 6 in these six positions.
   • The sum of the numbers on each side of the triangle must be the same.

2. Label the Positions:

   • Let's label the positions as follows: $P_1, P_2, P_3$ for the vertices and $P_4, P_5, P_6$ for the midpoints of the sides of the triangle.

3. Define the Side Sums:

   • The sums of the numbers on each side can be represented as:
     • $P_1 + P_4 + P_2$ (side 1),
     • $P_2 + P_5 + P_3$ (side 2),
     • $P_3 + P_6 + P_1$ (side 3).
   • Each of these sums must be the same.

4. Calculate the Target Sum:

   • The total sum of the numbers from 1 to 6 is $1 + 2 + 3 + 4 + 5 + 6 = 21$.
   • Since there are three sides, each side sum should be $\frac{21}{3} = 7$.

5. Find a Valid Arrangement:

   • Start by placing numbers logically and adjusting to ensure that the sums on all three sides equal 7.

By trial and error or using system of equations, you can adjust the positions of the numbers to meet the condition. Would you like me to go into a detailed solution for the arrangement of these numbers?

Here are some follow-up questions to consider:

1. Can you think of another puzzle involving a triangle and sums?
2. What happens if the total numbers are changed to a different range?
3. How would the puzzle change with a quadrilateral instead of a triangle?
4. Can this be generalized to an $n$-gon, such as a pentagon?
5. What patterns do you observe in the possible solutions?

Tip: When dealing with combinatorial problems like this, breaking the problem down into manageable steps and using logical deductions can simplify the solution process.