This topic is motivated by the trolley813's answer on my question.
Question. Is it possible to put the numbers $1,2,3,...,23$ in circles so that the sum of the four numbers on $9$ sides of $3$ large triangles give the same sum?
Edit. All numbers should be used one time.
My attempt is:
The total sum is $1+2+...+23=276$, but $13$ numbers (yellow and blue circles) are totalled for each of the sides twice.
The possible sum of $4$ numbers in each side is $276 : 4 = 69$, if one takes the minimal number $1$ and add three maximal numbers $21, 22, 23$, then $1 + 21 + 22 + 23 = 67 < 69$.
Also it is known that $276 \mod 9 = 6$. Now I do not know is it possible to decrease the sum of $4$ numbers on each side from $69$?