Problem of the Week

A constellation of six numbers 2, 3, 7, 1, 5, 6, has the interesting property that

\begin{align} 2+3+7 &= 1+5+6
2^2+3^2+7^2 &= 1^2 +5^2 + 6^2 \end{align}

Countless constellations have the property

\begin{align} x_1 +x_2 +x_3 &= x_4+x_5+x_6
x_1^2 +x_2^2 +x_3^2 &= x_4^2+x_5^2+x_6^2
\end{align}

Find another such example.

Added *Spring Break Challenge* (for amusement, not to be included in the points ladder, but interesting submissions will be posted on the board). Over 200 years ago, Leonard Euler and Christian Goldbach (separately) developed formulas that would generate such constellations, including ones that would contain eight or ten numbers and extending the property to include cubes as well.

1. Can you find a six-numered constellation that includes the cude property?
2. Can you find an eight-numbered constellation?
3. (Hard) Can you make any progress on finding formulas to generate a six-numbered constellation?

Reminder: I am looking for a well-reasoned solution that walks me through the mathematical reasoning, not just an answer with a box around it.