# Arrangement of Numbers to Get a Common Sum

I'm having trouble with a math problem. I need to arrange 6 numbers on a certain diagram:

At every intersection of two circles, I have to put one of these six numbers: 4, 5, 5, 6, 6, or 7. The sum of all of the numbers on each circle must be the same. Is it possible to arrange these numbers this way?

-
You mean no number that belongs to all three? – Jason Chen May 9 '14 at 4:06
@JoelReyesNoche: Oh, you misunderstood. Each section does not get its own number. The intersections get numbers, and each number is on one of the circles. – Jason Chen May 9 '14 at 4:08

The sum of the numbers is $33$. Each occurs twice, total $66$. Three circles, four numbers adding up to $22$ on each. The $7$ must be matched with $6,5,4$. The $7$ is on the intersection of two circles; each of these circles must include a $4$; but there is only one $4$, so it must be on the other intersection of the same two circles. The rest is easy, and there are various solutions.